Applications of Cubic Schweizer–Sklar Power Heronian Mean to Multiple Attribute Decision-Making

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Introduction
Zadeh [1] proposed the fuzzy set (FS) as a procedure for expressing and transmitting precariousness and ambiguity. Since its inception, FS has attracted signifcant attention from intellectuals all over the world, who have calculated its factual and theoretical characteristics. Economic and business [2][3][4], genetic algorithms [5,6], and supply chain management [7,8] etc., are some of the most recent academic attempts at the theory and implementations of FSs. Ensuring the insertion of the notion of FS, various modifcations of FSs were predicted, namely interval-valued FS [9], which explained the membership degree (MD) as a subclass of [0, 1] and Atanassov's intuitionistic fuzzy set (AIFS) [10], which clarifed the MD and nonmembership degree (NMD) as a single number in the [0, 1], with the constraint that sum of the two degrees must be less or equal to 1. As a consequence, IFS goes farther into explaining uncertainty and unreliability than FS. Te attractive scenario occurs when the MDs of such an object is expressed in the form of IVFS and FS. Under such settings, the conformist IFS is unable to manage such data. To handle the aforementioned situation, Jun et al. [11] initiated the perception of the cubic set (CS). Te aforementioned sets are special cases of CS. Mahmood et al. [12] proposed the concept of CNs and initiate some weighted aggregation operators (AOs) and apply these AOs to resolve multiple attribute decision-making (MADM) problems under a cubic environment.
One of the key elements in the MADM process is the AOs. Te AOs can blend many real numbers into a single one. Various AOs have diferent properties, namely, the PA operator ofered by Yager [13], have the ability to remove the negative efects of uncomfortable information from last ranking results, and have been enlarged by numerous researchers from all over the world to fgure out how to deal with diferent situations. Xu [14] enlarged the ordinary PA operator and delivered the IF power aggregation operator and implemented it in multiple attribute group decisionmaking (MAGDM), which can minimize the efects of inaccurate information. Some AOs, such as the Bonferroni mean (BM) operators [15], Heronian Mean (HM) operators [16], and Muirhead mean (MM) operators [17], as well as the Maclaurin symmetric mean (MSM) operator [18], ended up taking the connection between input arguments into consideration. BM and HM can take into account the connection between two input arguments, but MSM and MM operators can take into account the connection between any number of input opinions. Tese AOs were later extended to deal with a wide range of ambiguous circumstances [19][20][21][22][23].
Te majority of AOs use algebraic T-norm (TN) and Tconorm (TCN) to aggregate CNs. Currently, Ayub et al. [24] have presented a set of cubic fuzzy Dombi AOs that have been implemented on Dombi [25] TN and TCN and utilized to resolve MADM issues in a cubic fuzzy context. Dombi TN and Dombi TCN, as well as other TN and TCN, such as algebraic, Einstein, Hamacher, and Frank, are simplifed in Archimedean TN (ATN) and Archimedean TCN (ATCN). On a generic parameter, Dombi TN and TCN outperform generic TN and TCN, providing more fexibility in the input dataset. Fahmi et al. [26], anticipated Einstein AOs for CNs and apply these AOs to solve MADM problems unde cubic information. Wan [27] and Wan and Dong [28] developed some power average/geometric operators for trapezoidal intuitionistic fuzzy (trIF) numbers and apply them to solve MAGDM problems under a trIF environment. Wan and Yi [29] initiated PA operators for trIFNs using strict t-norm and strict t-conorm. CS was further extended by Ali et al. [30] who introduced the concept of neutrosophic cubic set and give its applications in pattern recognition.
Schweizer-Sklar (SS), TN (SSTN), and Schweizer-Sklar TCN (SSTCN) [31] are thorough ATN and ATCN instances, similar to the TN and TCN mentioned above. Because they contain a parameter that may be changed, SSTN and SSTCN are more fexible and superior to the prior techniques. Despite this, the majority of SS research has been on identifying the underlying theory and forms of SSTN and SSTCN [32,33]. Recently, SS operational laws (OLs) were anticipated for interval-valued IFS (IVIFS) and IFS by Liu et al. [34] and Zhang [35], respectively, and predicted various power aggregation operators for these fuzzy structures. On the basis of SS OLs, Wang and Liu [36] projected MSM operators for IFS and apply them to resolve MADM problems. Liu et al. [37] also proposed SS OLs for singlevalued neutrosophic (SVN) elements, as well as a variety of SS prioritised AOs for dealing with MADM issues in an SVN context. Zhang et al. [38] predicted and used certain MM operators for SVNS identifed on SS OLs to solve MADM issues. By capturing the variable parameter from [0, ∞], Nagarajan et al. [39] developed a couple of SS OLs for interval neutrosophic set (INS). For IN numbers, they anticipated various WA/WG AOs implemented on these SS OLs. Te COPRAS was enhanced by Rong et al. [40], who predicted a new MAGDM technique based on SS OLs.
From the above literature, it has been observed that the existing aggregation operators for CNs have only the capacity of removing the efect of awkward data or have the capacity of taking interrelationships among input arguments and a generic parameter. Yet, there are no such aggregation operators for CNs, which have the capacity of removing the efect of awkward data, can consider the interrelationship among input CNs, and also consist of the generic parameter. It has been observed that studies on various implementations of fuzzy MADM AOs depending on SS OLs have been published rapidly. Yet, no one has attempted to defne cubic SS OLs and merge them with a power HM operator to deal with cubic information. As a consequence, we propose the following: (1) Te SS operations are considerably more adaptable and superior than the prior methods in terms of a variable parameter.
(2) Fortunately, there are many MADM difculties in which the characteristics are linked, and many existing AOs can only alleviate such scenarios when the attributes are in the shape of real integers or other fuzzy formations.
(3) In the current situation, no such AOs exist which are drawn on SS OLs. In response to this limitation, we combined PA and HM operators with SS OLs to address MADM problems utilising cubic information.
Te subsequent are the urgencies and contributions of this efort as a result of important impacts from earlier studies as follows: (1) Developing innovative SS ALs for CNs, describing their basic features, and using them in SS ALs that anticipate CSS power HM operators, CSS power geometric HM operators, and their weighted form (2) Examining the commencing AOs' basic features and exceptional cases (3) Expecting the deployment of a MADM model on these commencing AOs (4) Assessing enterprise resource planning (ERP) applications using a MADM model (5) Confrming the feasibility and appropriateness of the launched MADM model Tis paper is structured in the following way to achieve these goals. Section 2 introduces a variety of key concepts such as CSS, score and accuracy functions, PA, and HM operators. In Section 3, we look at a few SS OLs for CNs with 2 Complexity general parameters that take values from [− ∞, 0]. Section 4 introduces the CSSPHM and CSSPGHM operators, as well as their weighted variants, and examines limited properties and detailed instances of the proposed AOs. In Section 5, a novel MADM model is established on these new AOs. A numerical example of enterprise resource planning is provided in order to verify the unassailability and compensations of the initiated approach. Finally, in Section 6, a brief conclusion is provided.

Preliminaries
In this portion, various essential conceptions namely, cubic set (CS), the Heronian mean (HM) operator, and their basic characteristics are reviewed.

Te Cubic Set and Its Operational Laws
Defnition 1 (see [11]). Let U be a universe of discourse set. A CS is classifed and mathematically indicated as follows: where Te OLs for CS were classifed by Jun [11] and are established below as follows: Defnition 2 (see [11]). Let CS 1 and CS 2 be any two CSS. Ten, (2) For the comparison of two CNs cn 1 and cn 2 the score, accuracy functions, and comparison rules are designated as follows: ArY cn 1 � For comparison of two CNs, the comparison rules are listed below.

Heronian Mean (HM)
Operator. HM [16] operator is one of the substantial tools for aggregation, which can exemplify the interrelations of the input elements, and is demarcated as go after.
Defnition 6 (see [16] Ten, the mapping HM p,q is suspected to be an HM operator with constraints. Te HM operator should certify the qualities of idempotency, boundedness, and monotonicity.

Schweizer-Sklar Operational Laws for Cubic Numbers
In this portion, the SS OLs are commenced for CNs based on SSTN and SSTCN, and numerous underlying characteristics of SS OLs for CNs are explored. Te SSTN and SSTCN [28,29] are recognized as go after: where Moreover, some worthy properties of the operational laws can be easily achieved.
Proof. Te proof of 1 and 2 are easy, so we can only prove the remaining formulas.
Te proofs of the other two parts are the same as the above two parts. Terefore, the proofs are omitted here. □
, then the defnition of CSSPHM operator is correspondent to the following type: (1) When a � 2 by Equation (17) and Equation (25), we have By using Equation (17), we get 2 i,j�1 j�i Tat is (22) is true a � 2.
(2) Let us assume that Equation (24) is true a � z.

Complexity 9
Furthermore, when a � z + 1 Firstly, we will show that We shall prove (31), on mathematical induction on z.

be a faction of CNs, and cn
By specifying distinctive values of the parameters H,Ü andÕ, numerous specifc AOs are obtained from the CSSPGHM operator, and are stated as given below: Case 1. If H � 0, then the CSSPGHM operator reverts to the CPGHM operator and is stated as follows: 20 Complexity Case 2. IfÕ ⟶ 0, then the CSSPGHM operator reverts to the cubic descending PG operator and is stated as follows: 22 Complexity Case 6. IfÜ �Õ � 1/2 and sup(cn i , cn j ) � h(h ∈ [0, 1])(∀i ≠ j), then the CSSPGHM operator reverts to the cubic GHM operator and is stated as follows: Case 7. IfÜ �Õ � 1 and sup(cn i , cn j ) � h(h ∈ [0, 1])(∀i ≠ j), then the CSSPGHM operator reverts to the cubic basic GHM operator and is stated as follows: Complexity

MADM with Known Weight Vectors of Attributes.
In this section, to pact with real decision circumstances in which the importance degrees of attributes are known in advance, we apply the CSSWPHM operator and CSSWPGHM operator to launch the following approach to solve MADM problems under cubic environments. To do so, instantly follow the steps given: Step 1. Locate support Sup(cn de , cn dx ) by utilizing the following formula: where DNE(cn de , cn ex ) is the distance measure and is intended by utilizing (6).
Step 2. Discover the weighted support degree T(cn de  Step 5. Locate the scores SoF(cn d ) for the overall CNs of the alternatives tr d (d � 1, 2, . . . , g) by manipulating Defnition 2.

Illustrative Example.
In this subpart, a numerical example adapted from [26] about enterprise resource planning in order to verify the unassailability and compensations of the initiated approach. Let us say a corporation decides to use an ERP system (enterprise resource planning). Te specialist's panel chose tr d (g � 1, 2, 3, 4, 5) fve prospective investors after gathering all relevant information on ERP dealers and systems. Some external decision-making specialists are among the organization's members. Te group decides on fve attributes Cei e (e � 1, 2, 3, 4, 5). To assess the alternatives, (1) function and technology Cei 1 , (2) strategic ftness Cei 2 , (3) the ability of the vendor Cei 3 , (4) reputation of the vendor Cei 4 , and (5) growth analysis of the vendor Cei 5 , with weight vectors of the attributes are (0.2, 0.15, 0.15, 0.25, 0.25) T . CFN's will be used by the expert's committee to create the initial decision matrix given in Table 1. To solve this decision making, the following steps to be followed: Step Step 4Manipulating Equation (64)  (83) Step 5. Operating Equation (6) Step 6. According to the score values, ranking order of the alternatives tr d (d � 1, 2, . . . , 5) is as follows: Terefore, according to the ranking order, the best alternative is tr 5 , while the worst one is tr 2 .

Impact of the Parameter H on Final Ranking Orders
Applying CSrSrWPHM and CSrSrWPGHM Operators. In this subportion, the impact of the parameter H on last ranking orders employing CSSWPHM and CSSWPGHM operators is explored, and the value of parametersÜ �Õ � 2 are permanent. For distinct values of the parameter H, the score values and ranking orders while operating CSSWPHM and CSSWPGHM operators are given in Table 2. One can observe from Table 2 that for diferent values of the parameter H the ranking orders are diferent. Tat is, 28 Complexity   30 Complexity Complexity employing the CSSWPHM operator and CSSWPGHM operator the best alternative is either tr 5 or tr 4 for diferent values of H while the worst alternative is either tr 2 or tr 1 . One can also see from Table 2 that when the values of the parameter H decrease, the score values of the alternative tr g (g � 1, . . . , 5) increases while exploiting the CSSWPHM operator. Similarly, from Table 2, when the values of the parameter H decrease, the score values of the alternatives tr g (g � 1, . . . , 5) decreases.

Efect of the ParametersÜ,Õ on Final Ranking Order.
In this subsegment, the efect of the parametersÜ,Õ on last ranking orders operating CSSWPHM and CSSWPGHM operators are inspected, and the value of parameter H � − 2 are permanent. For distinct values of the parameters A, B, the score values and ranking orders while operating CSSWPHM and CSSWPGHM operators are quantifed in Table 3. From Table 3, one can observe that when the values of the parametersÜ,Õ are diferent, the ranking orders obtained are diferent. Tat is, to employ the CSSWPHM operator and CSSWPGHM operator, the best alternative is either tr 5 or tr 4 or tr 1 for diferent values of the parameters U,Õ, while the worst alternative is either tr 2 or tr 1 or tr 3 . One can also see from Table 3 that when the values of the parametersÜ,Õ increase, the score values of the alternatives tr g (g � 1, . . . , 5) decrease, while exploiting the CSSWPHM operator. Similarly, from Table 3, when the values of the parameterÜ,Õ decrease, the score values of the alternatives tr g (g � 1, . . . , 5) increase, while exploiting the CSSWPHM operator. Te basic reason for this is because the above AOs are more adjustable since they are constituted of generic parameters, limit the infuence of inconvenient information, and take into account the relationship between input information. As a result, the MADM model developed on these aggregation operators is more adaptable. As a consequence, the decision-maker may modify the values of these parameters to the specifc requirements of the scenario.

Comparison with Existing Approaches.
In this section, the developed MADM model, which is predicated on this newly established novel AGOs, to various current approaches, namely, Mahmood et al. [12], Ayub et al. [24], and Fahmi et al. [26] MADM models. Te comparison between these approaches and the proposed approach is given in Table 4. From Table 4, we can observe that the best alternative obtained from the existing approaches and the proposed approach is the same except for utilizing the CWHMDA operator, while the worst alternative obtained from the existing approaches and the proposed approach is diferent. Te MADM model that will be implemented is based on the recently launched aggregation operators. To put it another way, these aggregation operators are suggested for CNs using Schweizer-Sklar operational, which are generic parameters that make the decision-making procedure more adaptable. In the meanwhile, existing MADM models are drawn on aggregation operators that are launched using algebraic operational laws or Dombi operational laws. Te aggregation operator proposed by Mahmood et al. [12] is a simple weighted aggregation operator, which does not have the capacity of taking interrelationships or removing the efect of awkward data from the fnal ranking results. While, the aggregation operators developed by Ayub et al. [24], are drawn on Dombi operational laws, which have the capacity of tinterrelationships among input arguments and also consist of a generic parameter. Te aggregation operators developed by Fahmi et al. [26] are simple weighted averaging operators based on Einstein operational laws for CNs. Tese aggregation operators do not have the characteristic of taking interrelationship among input arguments or removing the efect of awkward data from the fnal ranking results. Up to now, the existing aggregation operators for CNs have only the capacity of considering interrelationships among input arguments and cannot remove the efect of awkward data from the fnal ranking results. Whereas, anticipated aggregation operators will be able to remove the efect of awkward data while also taking into account the interrelationship between the input data at the same time. Te predicted AOs also have the beneft of including generic parameters, which sorts the decision-making procedure more supple. As a result, while solving MADM models using cubic information, the initiated AOs are more practical and comparable in their application.

Conclusion
Te evaluation of the ERP (enterprise resource planning) system is one of the numerous applications of MADM. Te goal of this article is to introduce a cubic set-based decision support as a practical way to explain ambiguity, reluctance, and uncertainty. Tis article makes a four-fold contribution. Firstly, inimitable Schweizer-Sklar operational rules for CNs are developed, and some of their key characteristics are examined. Secondly, using these inimitable Schweizer-Sklar operational laws, some CSSPHM operators are discussed, including the cubic Schweizer-Sklar power Heronian mean operator, the cubic Schweizer-Sklar power geometric Heronian mean operator, the cubic Schweizer-Sklar power weighted Heronian mean operator, and the cubic Schweizer-Sklar power weighted geometric Heronian mean operator, as well as their vital properties. We can see that some of the existing AOs are special instances of these freshly launched AOs by supplying particular values to the generic parameters. Tese AOs ofer benefts over current AOs. While the current aggregate operators for CSS can only consider interrelationships among input data, the initiated AOs can remove the efect of awkward data, examine the interrelationship among the input data, and also have a general parameter at the same time. Finally, a MADM model is expected based on these AOs. Te suggested technique is supported by numerical examples from enterprise resource planning. We also investigate the impact of the decision's outcome using the recently released cubic fuzzy Schweizer-Sklar power Heronian mean aggregation operators. Ten, we compare our work to that of others and also discussed the advantages of the proposed aggregation operators. 32 Complexity In future, we will defne Schweizer-Sklar operational laws for trapezoidal intuitionistic fuzzy numbers, dual hesitant fuzzy soft sets [41], Neutrosophic cubic sets [30], complex intuitionistic fuzzy sets [42], and extend several aggregation operators such as power average [27][28][29], robust aggregation operators [43], Choquet Integral for Spherical Fuzzy Sets [44], initiated for these structures and initiate some MADM models and apply these models to solve MADM problem under the said structure. We will also apply the anticipated approach to some new applications, such as detecting hate speech in social media [42], public transportation, technologies selection [45], trafc control, the digital twin model, and so on, or extend the anticipated model to some more extended form of CSS.

Data Availability
Data sharing does not apply to this article as no datasets were generated or analyzed during the current study.