Applications of Cubic Schweizer–Sklar Power Heronian Mean to Multiple Attribute Decision-Making
The cubic set (CS) is a basic simplification of several fuzzy notions, including fuzzy set (FS), interval-valued FS (IVFS), and intuitionistic FS (IFS). By the degrees of IVFS and FS, CS exposes fuzzy judgement, and this is a much more potent mathematical approach for dealing with information that is unclear, ambiguous, or indistinguishable. The article provides many innovative operational laws for cubic numbers (CNs) drawn on the Schweizer–Sklar (SS) t-norm (SSTN), and the SS t-conorm (SSTCN), as well as several desired properties of these operational laws. We also plan to emphasise on the cubic Schweizer–Sklar power Heronian mean (CSSPHM) operator, as well as the cubic Schweizer–Sklar power geometric Heronian mean (CSSPGHM) operator, in order to maintain the supremacy of the power aggregation (PA) operators that seize the complications of the unsuitable information and Heronian mean (HM) operators that contemplate the interrelationship between the input data being aggregated. A novel multiple attribute decision-making (MADM) model is anticipated for these freshly launched aggregation operators (AOs). Finally, a numerical example of enterprise resource planning is used to validate the approach’s relevance and usefulness. There is also a comparison with existing decision-making models.
Zadeh  proposed the fuzzy set (FS) as a procedure for expressing and transmitting precariousness and ambiguity. Since its inception, FS has attracted significant attention from intellectuals all over the world, who have calculated its factual and theoretical characteristics. Economic and business [2–4], 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 modifications of FSs were predicted, namely interval-valued FS , which explained the membership degree (MD) as a subclass of [0, 1] and Atanassov’s intuitionistic fuzzy set (AIFS) , which clarified 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. The 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.  initiated the perception of the cubic set (CS). The aforementioned sets are special cases of CS. Mahmood et al.  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. The AOs can blend many real numbers into a single one. Various AOs have different properties, namely, the PA operator offered by Yager , have the ability to remove the negative effects of uncomfortable information from last ranking results, and have been enlarged by numerous researchers from all over the world to figure out how to deal with different situations. Xu  enlarged the ordinary PA operator and delivered the IF power aggregation operator and implemented it in multiple attribute group decision-making (MAGDM), which can minimize the effects of inaccurate information. Some AOs, such as the Bonferroni mean (BM) operators , Heronian Mean (HM) operators , and Muirhead mean (MM) operators , as well as the Maclaurin symmetric mean (MSM) operator , 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. These AOs were later extended to deal with a wide range of ambiguous circumstances [19–23].
The majority of AOs use algebraic T-norm (TN) and T-conorm (TCN) to aggregate CNs. Currently, Ayub et al.  have presented a set of cubic fuzzy Dombi AOs that have been implemented on Dombi  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 simplified in Archimedean TN (ATN) and Archimedean TCN (ATCN). On a generic parameter, Dombi TN and TCN outperform generic TN and TCN, providing more flexibility in the input dataset. Fahmi et al. , anticipated Einstein AOs for CNs and apply these AOs to solve MADM problems unde cubic information. Wan  and Wan and Dong  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  initiated PA operators for trIFNs using strict t-norm and strict t-conorm. CS was further extended by Ali et al.  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)  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 flexible 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.  and Zhang , respectively, and predicted various power aggregation operators for these fuzzy structures. On the basis of SS OLs, Wang and Liu  projected MSM operators for IFS and apply them to resolve MADM problems. Liu et al.  also proposed SS OLs for single-valued neutrosophic (SVN) elements, as well as a variety of SS prioritised AOs for dealing with MADM issues in an SVN context. Zhang et al.  predicted and used certain MM operators for SVNS identified on SS OLs to solve MADM issues. By capturing the variable parameter from , Nagarajan et al.  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. The COPRAS was enhanced by Rong et al. , 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 effect 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 effect 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 define cubic SS OLs and merge them with a power HM operator to deal with cubic information. As a consequence, we propose the following:(1)The SS operations are considerably more adaptable and superior than the prior methods in terms of a variable parameter.(2)Fortunately, there are many MADM difficulties 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.
The subsequent are the urgencies and contributions of this effort 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)Confirming the feasibility and appropriateness of the launched MADM model
This 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 general parameters that take values from . 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.
In this portion, various essential conceptions namely, cubic set (CS), the Heronian mean (HM) operator, and their basic characteristics are reviewed.
2.1. The Cubic Set and Its Operational Laws
Definition 1. (see ). Let be a universe of discourse set. A is classified and mathematically indicated as follows:where are IVFS and FS, respectively. For computational affluence, we shall label a cubic number (CN) by the ordered pair where are IVFN and FN, respectively. If , then it is said to be an internal cubic number and if , then it is said to be an external cubic number.
The OLs for CS were classified by Jun  and are established below as follows:
Definition 2. (see ). Let be any two CSS. Then,For the comparison of two CNs the score, accuracy functions, and comparison rules are designated as follows:For comparison of two CNs, the comparison rules are listed below.(i)If then is superior to and is labelled by (ii)If then is superior to and is labelled by (iii)If then is same to and is labelled by
Definition 3. (see ). Let the two CNs be and . Then, the OLs for CNs are identified as go after:
Definition 4. Assume that the two CNs be and . Then, the normalized Hamming distance among and is labelled as go after:
2.2. The PA Operator
Yagar  originated the acuity of the PA operator which is the vital AOs. The PA operator concentrated a variety of ineffectual consequences of inanely high or awkwardly low sentiments stated by professionals. The anticipated PA operator can merge a set of crisp numbers where the weighting vector is simply on the input information and is classified as go after.
Definition 5. (see ). Let be a faction of nonnegative real numbers. The PA operator is a function delineated bywhereand is the support degree for from which meets the following axioms. (1) , (2) , and (3) if
2.3. Heronian Mean (HM) Operator
HM  operator is one of the substantial tools for aggregation, which can exemplify the interrelations of the input elements, and is demarcated as go after.
Definition 6. (see ). Let if satisfiesThen, the mapping is suspected to be an HM operator with constraints. The HM operator should certify the qualities of idempotency, boundedness, and monotonicity.
3. 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.
Additionally, when we have and . That is, SS TN and TCN reduce to algebraic TN and TCN.
Now, based on SSTN and TCN , we can permit the following definition of SRSR ALS of CNs.
Definition 7. Let the three CNs be , and . Then, the SS ALS for CNs are classified as follows:Moreover, some worthy properties of the operational laws can be easily achieved.
Theorem 1. Let and be any three CNs. Then,
Proof. The proof of 1 and 2 are easy, so we can only prove the remaining formulas.In the meantime, we can acquire thatTherefore, holds.The proofs of the other two parts are the same as the above two parts. Therefore, the proofs are omitted here.
4. The Cubic Schweizer–Sklar Power HM Operators
4.1. The CSSPHM and CSSPGHM Operators
In this segment, numerous new AOs namely, CSSPHM and CSSPGHM operators are anticipated by combining HM and GHM operators with PA operators to anticipate.
Definition 8. Let be a faction of CNs, and then the cubic Schweizer–Sklar power HM (CSSPHM) operator is clarified as follows:where and can be figured by (6).
Let , then the definition of CSSPHM operator is correspondent to the following type:
Theorem 2. Let and grab no more than one value of 0 at a time, be a set of CNs. Then, exploiting the CSSPHM operator, their merged values are CNs, and
Proof. Firstly, we need to prove the following equation:By SS OLs defined in Definition 7, we getandSimilarly,So,(1)When by Equation (17) and Equation (25), we have By using Equation (17), we get That is (22) is true .(2)Let us assume that Equation (24) is true .Furthermore, when Firstly, we will show thatWe shall prove (31), on mathematical induction on (i)For we have By using (11), we get(ii)Suppose that Equation (28) is true .Then, when we have