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.
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) in arbitrary set has the formwhere is represented by the positive membership grade.
Definition 2. (see [20]). An interval-valued fuzzy set (IVFS) in a universe set is an object having the formwhere is represented by the positive membership grade.
Definition 3. (see [21]). A cubic set in a universe set is an object having the formwhere and .
Definition 4. (see [3]). An intuitionistic FS in a universe set is an object having the formwhere are positive and are negative membership grades, respectively. In addition, , .
Definition 5. (see [5]). A Pythagorean FS in a universe set is an object having the formwhere are positive and are negative membership grades, respectively. In addition, , .
Definition 6. (see [22]). An interval-valued Pythagorean FS (IVPFS) in a universe set is an object having the formwhere are positive and are negative membership grades, respectively. In addition, .
Definition 7. (see [17]). A cubic Pythagorean fuzzy set in a universe set is an object having the formwhere and .
Maclaurin symmetric mean (MSM) is established by Maclaurin [23] and defined as follows.
Definition 8. (see [23]). Take any collection of nonnegative elements and . Ifthen is said to be MSM operator, where is the binomial coefficient and traversal all the k-tuple combination of .
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 and . Ifthen is said to be DMSM operator, where is the binomial coefficient and traversal of all the k-tuple combination of .
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 in arbitrary set has the formwhere and represent the positive and negative grades of memberships.
For whole study, the list of cubic Pythagorean fuzzy sets are represented by . For simplicity, we write 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 and with . Then, the operational rules are described as(1)(2)(3)(4)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 in arbitrary set has the formwhere and 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. Let and with . Then, the operational rules are described as(1)(2)(3)(4)
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 in arbitrary set , with , is the linguistic term set, where is the odd cardinality, having the formwhere and and represent the positive and negative grades of memberships.
For the whole study, the list of cubic Pythagorean linguistic fuzzy sets are represented by . For simplicity, we write 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 and with . Then, the operational rules are described as(1)(2)(3)(4)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 CLPFS in arbitrary set , with , is the linguistic term set, where is the odd cardinality, having the formwhere and and 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. Let and with . Then, the operational rules are described as(1)(2)(3)(4)
Definition 18. Letbe the score values of . Also,are the accuracy values of . If(a)(b)(c)
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 . Then, the CPLFWA operator is described as follows:where represents any permutation and represents the total numbers of elements.
Theorem 1 (see [19]). Let . Then, using defined operational laws, operator is obtained as
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]). Let . Then, the CGPLFWA operator is described as follows:where represents any permutation and represents the total numbers of elements.
Theorem 2 (see [19]). Let . Then, using defined operational laws, the operator is obtained as
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 . Then, the CPLFWG operator is described as follows:where represents any permutation and represents the total numbers of elements.
Theorem 3 (see [19]). Let . Then, using defined operational laws, the operator is obtained as
6. 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.
6.1. Updated Weighted Averaging AO
Definition 22. Let . Then, the CPLFWA operator is described as follows:where the weights of have and .
Theorem 4. Let . Then, using defined operational laws 17, operator is obtained aswhere the weights of have and .
6.2. Updated Weighted Geometric AO
Definition 23. Let . Then, the CPLFWG operator is described as follows:where the weights of have and .
Theorem 5. Let . Then, using defined operational laws 17, the CPLFWG operator is obtained aswhere the weights of having and .
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 and is weight vector having and . Then, the CPLFMSM operator is described as follows:where represents any permutation and represents the total numbers of elements.
Theorem 6 (see [19]). Let . Then, using defined operational laws, the operator is obtained as
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 and is weight vector having and . Then, the CPLFWMSM operator is described as follows:where represents any permutation and represents the total numbers of elements.
Theorem 7 (see [19]). Let . Then, using defined operational laws, operator is obtained as
8. 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.
Definition 26. Let . Then, the CPLFMSM operator is described as follows:where is the binomial coefficient and traversal of all the k-tuple combination of .
Theorem 8. Let . Then, using defined operational laws 17, the CPLFMSM operator is obtained asUpdated weighted MSM AO is defined as follows.
Definition 27. Let and is weight vector having and . Then, the CPLFWMSM operator is described as follows:where is the binomial coefficient and traversal of all the k-tuple combination of .
Theorem 9. Let . Then, using defined operational laws 17, the CPLFWMSM operator is obtained as
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 and is weight vector having and . Then, the CPLFDWMSMA operator is described as follows:where represents any permutation and represents the total numbers of elements.
Theorem 10 (see [19]). Let . Then, using defined operational laws, the operator is obtained as