Generalized Heronian Mean Operators Based on Archimedean T-Norms of the Complex Picture Fuzzy Information and Their Application to Decision-Making

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Introduction
Te multiattribute decision-making (MADM) technique is the subpart of the decision-making which is a massive signifcant technique for choosing the best optimal from the family of alternatives.Te process of the MADM technique has been widely employed in all sorts of regions.In genuine life dilemmas, it is massive signifcant how we show the attribute value more dominant and efcient.Because of inconsistency, it is enough to state the values of the attributes by exact values.To address it, Atanassov [1] modifed the principle of the fuzzy set (FS) [2] to initiate the intuitionistic FS (IFS).Under the truth grade (TG) M After the appearance of IFS, researchers widely utilized it in the diferent regions and presented diferent methods such as complete ranking technique [3], Dombi aggregation operators [4], generalized entropy measures [5], distance measure [6], failure mode, and their afection [7], TODIM technique [8], hamming distance [9], aggregation of infnite chains [10], parameterized interval-valued intuitionistic soft sets [11], telecom services givers [12], multiattribute border approximation area comparison approach [13], and distance measures [14] to solve the decision-making problems.
In various instances, it is massive challenging to manage three sorts of data in the shape of singleton sets.For illustration, if an individual faces data in the shape of yes, abstinence, and no, then the IFS has been neglected.For this, the principle of picture FS (PFS) was initiated by Cuong [15,16] which covers the TG M, abstinence grade (AG) A and FG N with a well-known principle of 0 ≤ M + A + N ≤ 1. IFS and FS are the specifc cases of the PFS, and several persons have employed them in the region of distinct areas such as geometric aggregation operators [17], aggregation operators [18], Dombi aggregation operators [19], weighted aggregation operators [20], Hamacher aggregation operators [21], evidence theory [22], picture fuzzy soft set [23], analytical hierarchy processes [24], and distance measures [25].
Alkouri and Salleh [26] revised the assumption of complex FS (CFS) [27] is to introduce the complex IFS (CIFS).Te CIFS is massively preferable to employ it in the region of information measures [28], correlation measures [29], complex fuzzy soft set [30], decision making [31], distance measures [32], power aggregation operators [33], complex intuitionistic fuzzy preference relation [34], and aggregation operator [35].In diferent occurrences, the principle of CIFS is neglected.For illustration, if a person gives data in the shape of complex-valued TG, AG, and FG, then the theory of CIFS has been unsuccessful.For this, the principle of complex PFS (CPFS) was initiated by Akram et al. [36] which covers the TG M , abstinence grade (AG) , and FG N with a well-known principle of 0 ≤ M IFS, CIFS, FS, and CFS are the specifc cases of the CPFS.
Hamy mean (HM) operators [37] are massive consistent and dominant to initiate the order among any element of items.Te simple averaging aggregation and geometric aggregation operators are the specifc cases of the investigated HM operators.Keeping the supremacy of the HM operators, several individuals have employed it in the region of separated areas.For illustration, HM operators for IFSs [38], geometric HM operators for IFSs [39], Dombi HM operators for IFSs [40], HM operators for interval-valued IFSs [41], Heronian aggregation operators for IFS by using Archimedean t-norm and t-conorm [42], power geometric HM operators for IFSs [43], power HM operators for interval-valued IFSs [44], frank HM operators for IFSs [45], HM operators for PFSs [46], Dombi HM operators for PFSs [47], and interactional partitioned HM operator for PFSs.Prevailing operators were invented in the availability of the existing theories of IFS, CIFS, and PFS due to their weak and complicated structure.In the CPFS, TG, AG, and FG are in the shape of the polar form and stated in complex numbers.Te amplitude object consisting of the TG (AG and FG) includes the extent of supporting (neutral and not supporting) of a term in a CPFS and the phase/ periodic object consisting of the TG (AG and FG) includes the extra, morally stating the periodicity in the terms of supporting (neutral and not supporting) of a term in a CPFS.Te prevailing theories have a lot of ambiguity because some ideas cannot include the FG, and some cannot include the phase terms.Due to this reason, we faced to lose a lot of data during decision-making strategy, for instance, numerous enterprises are wanted to novel information processing and analysis software.For this, the enterprise refers to an intellectual who provides the data concerning (i) dissimilar features of the software and (ii) diagnosed data of software.Te enterprise wants to suggest the massive benefcial optimal(s) of software with its new and latest version continually.Here, the dilemma is two-dimensional, called to choose the benefcial optimal of the software and their new version.Tis dilemma cannot be addressed efciently in the presence of the prevailing IFS and PFS theories.So, the benefcial path to express all the data given by the intellectual is by using the prevailing IFS and PFS.Te amplitude objects in CPFS may be utilized to provide enterprises decision-making concerning alternatives of software, and the phase object may be stated in the form of the version of the software.Moreover, the CPFS is massive general than the prevailing IFSs, CIFSs, and PFSs to manage inconstant and awkward data in genuine life dilemmas.Keeping the supremacy of the CPFS and HM operators, in this analysis, we combine these two ideas.Te major contribution of this analysis is initiated as follows.
(1) To explore the algebraic, Einstein, Hamacher, and Frank operational laws under the CPFS.(2) To elaborate the principle of complex picture fuzzy Archimedean Heronian aggregation (CPFAHA) operator and complex picture fuzzy weighted Archimedean Heronian aggregation (CPFWAHA) operator is also elaborated by using Archimedean t-norm (TN) and t-conorm (TCN).(3) To explore a MADM technique by using the initiated operators is to elaborate the consistency and reliability of the explored works.(4) To illustrate many numerical examples for discussing the advantages, sensitive analysis, and graphical representation of the investigated works.
Te rest of the paper is organized as follows: in Section 2, we briefy describe the principles of CPFSs.In Section 3, we explored the several norm operational laws under the CPFS.In Section 4, the principle of CPFAHA operator and CPFWAHA operator is also elaborated by using Archimedean TN and TCN.In Section 5, by using the elaborated operators, a MADM technique is presented to elaborate the consistency and reliability of the explored works.Finally, many examples are illustrated for discussing the advantages, sensitive analysis, and graphical representation of the investigated works.Te ending of the evaluation of this analysis is examined in Section 6.

Preliminaries
In this analysis, we revised the prevailing principles of CPFSs and their algebraic laws and their operational laws.
In this study, we denoted M , and as TD, AD, and FD and the universal is invented by Here, the terms R and I in Ξ ═ R , Ξ ═ I stated the real and imaginary terms of TD, AD, and FD.

Operational Laws for CPFSs under the Different TN and TCNs
Keeping the benefts of the prevailing laws like algebraic, Einstein, Hamacher, and Frank, in this analysis, we initiated these laws for CPFS is to determine the fexibility of the presented works.
) and By using the TN and TCN in the shape of equations ( 9) and (10), then equations (19) to (22) are changed to equations (2) to (5), which invented the algebraic operational laws.Additionally, if we chose the TN and TCN in the shape of equations ( 11) and (12), then equations (19) to (22) are changed to equations 6 Mathematical Problems in Engineering (23) to (26), which invented the Einstein operational laws.
Mathematical Problems in Engineering When we choose the TN and TCN in the shape of equations ( 11) and ( 12), then equations (19) to equations (22) are changed to equations ( 27) to (30), which invented the Hamacher operational laws.
For c SC ═ > 0 , we have 8 Mathematical Problems in Engineering For the value of  δ SC � 1, equations ( 27) to ( 30) are changed to equations (2) to ( 5) and if we choose the value of  δ SC � 2, equations ( 27) to (30) are changed to equations (23) to (26).When we chose the TN and TCN in the shape of equations ( 13) and ( 14), then equations (19) to (22) are changed to equations ( 31) to (34), which invented the Frank operational laws.
Mathematical Problems in Engineering

□ Property 1. Choose any group of CPFNs
Proof.By hypothesis Proof.By hypothesis, we know that if then, Mathematical Problems in Engineering 13 therefore, Similarly, we investigate for an unreal term, such that Moreover, for real and unreal terms of AD, we have Moreover, for real and unreal terms of FD, we have 14 Mathematical Problems in Engineering By using equation ( 6), we easily get the terms ), then by using Property 2, we initiate where Moreover, by using the value of the parameters f ═ and g ═ , we employed distinct cases of the elaborated works.

Mathematical Problems in Engineering
It is stated as the CPFHM operator.
(6) By using equations ( 11) and ( 12), equation ( 36) is changed to and , y 18 Mathematical Problems in Engineering It is stated as the CPF Einstein HM operator.

Mathematical Problems in Engineering
It is stated the CPF Hamacher HM operator.
(8) By using equations ( 15) and ( 16), equation ( 36) is changed to • and then the CPFWAHA operator is invented by where

Theorem 2. Choose any group of CPFNs
then by using equation ( 57), we initiate Proof.Omitted.

MADM Technique under the Initiated Operators
A bundle of intellectuals has discussed the MADM technique in the region of PFSs and CPFSs, but under the HM operators based on Archimedean TN and TCN for CPFSs is not developed up to date.Te major contribution of this analysis is to initiate the HM operators are employed in the region of the CPF environment.
To elaborate on the above data, we initiate the decision-making procedure, whose stages are illustrated as follows.
Stage 1: Standardize the matrix.In genuine decisionmaking, the characteristic qualities are partitioned into two kinds, i.e., the expense characteristic and the advantage property.To kill the distinction in the characteristic qualities, we need to convert them to the same sort.As a rule, because most properties are the advantage type, we need to change over the expense type into the advantage type.Te matrix is standardized by using the following formula: Mathematical Problems in Engineering If the matrix covers the beneft sorts of data, then it is not needed to be standardized.Stage 4: To initiate the best optimal, we rank to all alternatives.
To manage inconsistent and ambiguous data, we illustrate a numerical example based on presented operators to determine the supremacy and consistency of the initiated works.

Illustrated Example.
A media communications network is a gathering of hubs interconnected by broadcast communications connections that are utilized to trade messages between the hubs.Te connections might utilize an assortment of advancements dependent on the techniques of circuit exchanging, message exchanging, or parcel exchanging to pass messages and signals.
Various hubs might participate to pass the message from a beginning hub to the objective hub, through numerous organization bounces.For this steering capacity, every hub in the organization is appointed an organization address for ID and fnding it on the organization.Te assortment of addresses in the organization is known as the location space of the organization.A telecommunication network wants to invest some money into an enterprise, where there are fve possible enterprises in the shape of alternatives others.An expert is hired to provide the rating corresponding to each alternative with respect to the given attributes.Te rating provided by them is summarized in Table 1.To elaborate on the above data, we initiate the decision-making procedure, whose stages are illustrated as follows.
Stage 5: Standardize the matrix.In genuine decisionmaking, the characteristic qualities are partitioned into two kinds, i.e., the expense characteristic and the advantage property.To kill the distinction in the characteristic qualities, we need to convert them to the same sort.As a rule, because most properties are the advantage type, we need to change over the expense type into the advantage type.Te matrix is standardized by using the following formula: Since the matrix has covered all the beneft sorts of data, so it is not needed to be standardized.
Stage 8: To initiate the best optimal, we rank to all alternatives, which is illustrated as follows: Te best optimal is Ξ ═ AL− 3 .Moreover, to investigate the supremacy and consistency of the initiated operators, we choose the intuitionistic fuzzy sorts of data and elaborate it by using the presented operators.Te graphical representation of ranking of the alternatives is shown in Figure 1.
If the decision-maker considers only the real component during the initial rating values of Table 1, then this representation is summarized as Table 2.
Mathematical Problems in Engineering 27 Te best optimal is Ξ ═ AL− 3 .Te data are in Table 1; if we choose the unreal part or not, the obtained results are the same.To improve the quality of the presented works, we discussed the sensitive analysis of the elaborated works with some prevailing operators to investigate the supremacy of the proposed works.Te graphical representation of such information is given in Figure 2.

Sensitivity Analysis.
Te major contribution of this analysis is to compare the initiated works with numerous prevailing operators based on PFSs and CPFSs to investigate the fexibility and dominancy of the elaborated works.Te prevailing works are followed as the theory of HM operators for IFS which was developed by Liu and Chen [42]; Akram et al. [36] initiated the Hamacher aggregation operators for CPFSs and proposed works based on CPFSs with weight vectors 0.3, 0.3, 0.3, and 0.1.Te sensitive analysis of the explored and prevailing operators is discussed in the shape of Table 3.
By using the data in Table 3, we get the two diferent sorts of results that show the consistency of the initiated works.Moreover, by using ten values of parameters f ═ and g ═ , we discussed the fuency of the elaborated work.For f ═ � 1, the fuency of the g ═ is discussed in the shape of Table 4 by using the data in Table 1.Te graphical illustration on the ranking of the alternatives between the proposed and existing approaches is shown in Figure 3.By changing the value of the parameter, we have gotten the same ranking results; further, for f ═ � 1, the fuency of the g ═ is discussed in the shape of Table 5 by using the data in Table 2. Te graphical representation of the variation of the score values for diferent values of parameter g ═ for the information mentioned in Tables 1 and 2 is shown in Figures 4  and 5, respectively.
However, to show the infuence of the parameter f ═ on to the ranking of the alternatives, we perform an analysis with g ═ � 1, and hence the results are listed in Tables 6 and 7, respectively, for the input information in Tables 1 and 2. Furthermore, the graphical representation of the score values of the given alternatives is shown in Figures 6 and 7.

Conclusion
To handle problematic and convoluted data in genuine life dilemmas, the principle of complex picture fuzzy set (CPFS) is a capable and skillful technique to resolve real-life problematic dilemmas.CPFS handle such sort of dilemmas, which covers the three sorts of data such as yes, abstinence, and no in the shape of fuzzy numbers.Te major infuence obtained from this study is summarized here: (1) We explored the algebraic, Einstein, Hamacher, and Frank operational laws under the CPFS (2) Te principle of CPFAHA operator and CPFWAHA operator is also elaborated by using Archimedean TN and TCN (3) By using the elaborated operators, a MADM technique is presented to elaborate the consistency and reliability of the explored works (4) Many examples are illustrated for discussing the advantages, sensitive analysis, and graphical representation of the investigated works.
From the conducted study, we conclude it is efcient to solve the decision-making problems in an efcient manner.Apart from that, we also observe that the several of the existing studies are considered as a special case of the proposed work.For instance,

Stage 2 :Stage 3 :
By choosing the elaborated operators, we accumulate the matrix to a single set.Elaborate the SV of the accumulated values.

Figure 1 :
Figure 1: Graphical representation of ranking of alternatives of the input.

Figure 3 :
Figure 3: Comparison of the rating of the given alternatives.

Figure 6 :Figure 7 :
Figure 6: Variation of the score values with f ═ by taking g═ � 1 for data in Table1.

Table 1 :
Input decision-matrix provided by the expert.

Table 2 :
Te original matrix is given by the decision-maker.

Table 3 :
Comparative analysis of the proposed and existing works.

Table 4 :
Impact of the parameter g ═ by fxing the value f on the input given in Table1.

Table 5 :
Impact of the parameter g ═ by fxing the value f for the input data in Table2.

Table 6 :
Impact of the parameter f � 1 on ranking of alternatives for Table1.
═with g ═

Table 7 :
Impact of the parameter f � 1 on ranking of alternatives for Table2.