A Novel Approached Based on T-Spherical Fuzzy Schweizer-Sklar Power Heronian Mean Operator for Evaluating Water Reuse Applications under Uncertainty

: The T-Spherical Fuzzy set (T-SPHFS) is one of the core simplifications of quite a lot of fuzzy concepts such as fuzzy set (FS), intuitionistic fuzzy set (ITFS), picture fuzzy set (PIFS), Q-rung orthopair fuzzy set (Q-RUOFS), etc. T-SPHFS reveals fuzzy judgment by the degree of positive membership, degree of abstinence, degree of negative membership, and degree of refusal with re-laxed conditions, and this is a more powerful mathematical tool to pair with inconsistent, indecisive, and indistinguishable information. In this article, several novel operational laws for T-SPFNs based on the Schweizer – Sklar t-norm (SSTN) and the Schweizer – Sklar t-conorm (SSTCN) are initiated, and some desirable characteristics of these operational laws are investigated. Further, maintaining the dominance of the power aggregation (POA) operators that confiscate the ramifications of the inappropriate data and Heronian mean (HEM) operators that consider the interrelationship among the input information being aggregated, we intend to focus on the T-Spherical fuzzy Schweizer – Sklar power Heronian mean (T-SPHFSSPHEM) operator, the T-Spherical fuzzy Schweizer – Sklar power geometric Heronian mean (T-SPHFSSPGHEM) operator, the T-Spherical fuzzy Schweizer – Sklar power weighted Heronian mean (T-SPHFSSPWHEM) operator, the T-Spherical fuzzy Schweizer – Sklar power weighted geometric Heronian mean (T-SPHFSSPWGHEM) operator, and their core properties and exceptional cases in connection with the parameters. Additionally, deployed on these newly initiated aggregation operators (AOs), a novel multiple attribute decision making (MADM) model is proposed. Then, the initiated model is applied to the City of Penticton (British Columbia, Canada) to select the best choice among the accessible seven water reuse choices to manifest the practicality and potency of the preferred model and a comparison with the proffered models is also particularized.


Introduction
The purification and restoration of non-traditional or polluted water for constructive use are known as water reuse [1]. Water recycling is associated through the use of recovered water, which might help to alleviate water shortage, particularly in anticipation of the recent influence of climate change and increased human activity. Water recycling has become commonplace across the world to address the degradation of water supplies, which has resulted in a reduction in water availability. Water reuse application evaluation application evaluation. Water reuse applications are challenging to assess since they are varied in nature and may have competing requirements. To achieve sustainable judgments, the review will now include a study of many factors involving social, scientific, financial, political, geological, and engineering factors. When evaluating water reuse applications, ambiguity arises when decision-makers are undecided about which choice they prefer based on a given criterion or even how much they favor a certain alternative. To deal with such a situation, the fuzzy set theory is the most appropriate set.
Zadeh [3] was the founder of the Fuzzy set (FS) as a mechanism for illustrating and reassigning unevenness and vagueness. Since its origination, FS has managed notable contemplation from scholars throughout the world, who premeditated its actual and theoretical features. Some of the latest research efforts on the theory and applications of FSs have been launched in economics and business [12][13][14], genetic algorithms [15,16], and supply chain management [17,18], etc. Following the introduction of the concept of FS, a number of extensions of FSs were expected, such as interval-valued FS [19], which elucidated the membership degree is a subset of [0, 1] , and Atanassov's ITFS [20], which elucidated the membership degree (MED) and non-membership degree (NOMD) by a single number in the   0, 1 , with the total of these two degrees having to be less than or equal to one. As a result, ITFS explains uncertainty and unreliability in greater depth than FS. The appealing scenario arises when such an entity's MED and NOMD are in the [0, 1] , but somehow the total of two such functions exceeds one. The standard ITFS fails to manage such types of data under such scenarios. To address the scenario described earlier, Yager [21][22][23] suggested Pythagorean fuzzy sets (PytFSs), which might be considered an augmentation of ITFSs. The main distinction amid both PytFS and ITFS would be that in PytFS, the addition of squares of the MED and NOMD will be less than or equal to one, whereas in IFS, the sum of MED and NOMD would be less than or equal to one. After the commencement of PytFS, a variety of studies have been carried out by numerous investigators, such as distance measure [24][25][26][27] and correlation coefficient [28,29]. Zhang and Xu [30] conveyed the ample mathematical representation for PytFS and anticipated the initiative of PytF number (PytFN), then they also anticipated a MADM algorithm deployed on PytF TOPSIS to pair with PytFNs. Some other studies about PytFS have been conducted by different investigators and give their applications in different areas [31][32][33]. Q-RUOFS [34] is another conception of ITFS and PytFS. Q-RUOFS is a more dominant and reliable set to pact with unclear and conflicting information.
In various areas, ITFS, PytFS and Q-RUOFS have been addressed. Unfortunately, there have been some circumstances for which IFS, PyFS, and Q-ROFS are not compatible. For example, the members of three departments, namely the department of mathematics, department of physics, and department of information technology, vote for the selection of the dean post among one of these departments. From these faculties, 100 teachers (Assistant professors and associate professors) are chosen for voting, and 1 professor from each department is chosen for the selection of the dean post. Thirty teachers support professors from the department of mathematics, forty teachers oppose professors from the department of mathematics, twenty teachers abstain from voting, and ten teachers decline to vote. ITFS, PytFS, and Q-RUOFS fail to cope with these sort of data. Cuong and Kreinovich [35] initiated and termed picture fuzzy set (PIFS), another conception of ITFS and FS to cope with such information. Furthermore, there will be instances whenever an entity's MED, neutral MED, and NMED are permissible from a unit interval, yet the sum of these three degrees is greater than one. Under such cases, the traditional PFS is unable to manage the data. As a result, dealing with such data necessitates the use of a more sophisticated mathematical tool. For this intention, Mahmood et al. created the spherical fuzzy set (SPHFS) and T-SPHFS, which is a further expansion of PytFS, Q-RUOFS, and PIFS. T-SPHFS has the same structure as PIFS, with the exception that the total of q th power of these three degrees must be less than or equal to one. As a result, T-SPHFS is a more dominant tool for handling ambiguous, conflicting, and unclear data.

Theoretical Fundamentals
Since 2018, it has been noted that research on different structures of fuzzy MADM AOs based on SS ALs have been released at a quick pace.
It is clear from Table 1 that no authors have sought to define SS ALs for T-SPHFS and combine them with power HEM operator to cope with T-SPHF data. As a result, we put forward: Table 1. Offers the latest literature of different structures of fuzzy MADM AOs based on SS ALs.
As a result of major influences of the previous studies, the following are the priorities and offerings of this work: (1) Initiating novel SS ALs for T-SPHFNs, discussing its basic properties, and deploying it on the SS ALs anticipating T-SPHFSS power Heronian mean operators, T-SPHFSS power geometric Heronian mean operators, and its weighted form.
(2) Inspecting its basic properties and special cases of these initiating AOs.
(3) Anticipating a MADM model deployed on these initiating AOs.
(4) Applying a MADM model to assess water reuse applications. (5) Verifying the initiated approach' effectiveness and practicality.
To accomplish these intentions, this paper is planned as follows. In part 3, we instigate a number of vital ideas of T-SPHFS and score and accuracy functions, POA, and HEM operators. In part 4, we scrutinize a number of SS ALs for TSPHFNs where the general parameter acquires values from   ,0 − . In part 5, we put forward T-SPHFSSPOHEM and T-SPHFSSPOGHEM operators, their weighted forms, and scrutinize a few properties and meticulous cases of the projected AOs. In part 6, we initiate a MADM model recognized on these AOs and applied it to select a water reuse option in the available options to validate the unassailability and compensations of the initiated approach by weighing against other accessible approaches. Lastly, a petite conclusion is prepared in part 7.

Methodology
In this part, some core concepts such as T-SPHFS, the HEM operator, POA operator, and their core properties are discussed.  The ALs for T-SPHFS were classified by Mahmood et al. [28] and are prearranged below:   21 .

Definition 2. ([4]) Let
For comparison of two T-SPFNs 1 2 and tsf tsf , the score, accuracy functions, and judgment rules are illustrated as follows:

The Power Average (POA) Operator
Yagar [36] initiated the perception of the POA operator that is the key AO. The POA operator reduced various unconstructive effects of pointlessly high or unreasonably low opinions specified by experts. The predictable POA operator may combine a collection of crisp integers where the weighting vector is based only on the input data and is characterized as:

Heronian Mean (HEM) Operator
For aggregation, the HEM [9] operator is also a significant AO, which can embody the interrelationships of the input attributes, and is delineated as: HEM is alleged to be HEM operator with parameters , AB.
The HEM operator must ensure the characteristics of idempotency, boundedness, and monotonicity.

Schweizer-Sklar ALs for T-SPHFNs
In this subpart, the SS ALs are initiated for T-SPHFNs deployed on SSTNO and SSTCNO, and various core properties of SS ALs for T-SPHFNs are investigated.
The SSTNO and SSTCNO [48] are identified as: Likewise, a number of attractive properties of the T-SPHFNs ALs can be easily obtained. 11

The T-Spherical Fuzzy Schweizer-Sklar Power Heronian Mean Operators
The 1 , , ,..., Now, we confer various core properties of the anticipated T-SPHFSSPOHEM operator.
pm as nm tsf ==    , various special cases of the T-SPHFSSPOHEM operator can be obtained, which can be expressed as given below: Case 1. If 0  = , then the T-SPHFSSPOHEM operator degenerates to the T-SPHFPOHEM operator, which can be expressed as follows: ,1 2 , ,..., 1 pm as Case 2. If 1 q = , then the T-SPHFSSPOHEM operator degenerates to the picture fuzzy Schweizer-Sklar power Heronian mean (PIFSSPOHEM) operator, which can be expressed as follows: Case 5. If 0 A → , then the T-SPHFSSPOHEM operator degenerates to the T-SPF ascending POA operator, which can be expressed as follows: , then the T-SPHFSSPOHEM operator degenerates to the T-SPHF linear descending WA operator, which can be expressed as follows: ,..., lim 1 tor degenerates to the T-SPHF linear ascending WA operator, which can be expressed as follows: ,..., 1 ,..., 1 be a group of T-SPHFNs, and then the T-SPHF Schweizer-Sklar power geometric Heronian mean (T-SPHFSSPOGHEM) operator is explained as follows: , ,..., Proof. From the Schweizer-Sklar operational laws for T-SPFS, we have   , ,..., Case 2. If 1 q = , then the T-SPHFSSPOGHEM operator degenerates to the picture fuzzy Schweizer-Sklar power Heronian mean (PIFSSPOGHEM) operator, which can be expressed as follows: , ,..., , ,...,

T-SPHFSSWPOHEM and T-SPHFSSWPOGHEM Operators
In this subpart, we initiate the T-SPHFSSWPOHEM operator and the T-SPHFSW-POGHEM operator by taking the importance of the attributes.  , ,..., , ,..., The proofs of Theorems 10 and 11 are the same as Theorems 2 and 6. Therefore, here we omit their proofs.

The Approach to Solve MADM Problems
In this part, we initiate an innovative process for MADM problems, which is based on the initiated T-SPHFSSWPOHEM and T-SPHFSSWPOGHEM. The main goal of the MADM problem is to choose the best option between several available options. The procedure of the initiated process can be articulated as follows: Alternatives The initiated process is in accord with the following steps: Step Step 2. Determine the Support Degree by utilizing the formula:

Numerical Example
In this subpart, we utilize the initiated algorithm based on these newly initiated aggregation operators for assessing water reuse applications. These data are taken from reference [4].
The suggested algorithm is applied for the City of Penticton (CoP) in British Columbia (BC), Canada. Residents, businesses, institutions, and industries all use public water. The wastewater is processed and treated using biological nutrient removal equipment at an automated wastewater treatment facility. Half of the filtered water is reused, while the rest is released into a river. There are seven alternatives (Water reuses) such as  Table 3.  Now, we will utilize the suggested algorithm to solve the assessment of the water reuse application. The following steps should be followed.    2  2  2  2  2  2  14  41  15  51  23  32   2  2  2  2  2  2  2  2  2  2  24  42  25  52  34  43  35  53  45 3  3  3  3  3  3  3  3  3  3  12  21  13  31  14  41  15  51  23  32   3  3  3  3  3  3  3  3  3  3  24  42  25  52  34  43  35  53 T  T  T  T  T   T  T  T  T  T   T  T  T  T  T   T  T  T Tables 6 and 7. From Table 6, one can notice that the ranking order is slightly different for different values of the parameter , AB, although the best and the worst alternative remains the same. From Table 6, we also noticed that when the values of the parameters enlarge, the score values of the alternatives decline. Similarly, From Table 7, one can notice that the ranking order is different for different values of the parameter , AB. When ,1 AB= , the best one is 3 VS , while the worst alternative remains the same. From Table 7 we also noticed that when the values of the parameters increase, the score values of the alternatives increase.   VS while the worst alternative remains the same, which is 7 VS . We can also notice that when the values of the parameter decline, the score values of the alternative decline.   Table 10. One can notice from Table 10, that is, for distinct values of the parameter, q , the ranking order is totally different. That is, utilizing T-SPFSSPWHM and T-SPFPWGHM operators for distinct values of q , the best alternative is either 15 ,

VS VS
or 6 VS while the worst alternative is 1 4 7 , VS VS and VS . The reason behind these different ranking orders is that these AOs are more flexible due to consisting of general parameters. Therefore, the MADM model based on these aggregation operators is more flexible. Hence, the decision-maker may choose the values of these parameters according to the actual needs of the situations.

. Comparison with Existing Approaches
In this subpart, we compare our produced MADM model, which is based on these developed novel AGOs, to some current techniques, such as the approach initiated by Garg et al. [38] and the MADM model initiated by Tahir et al. [4]. The score values and ranking orders are given in Table 11. From Table 11, one can notice that the ranking order obtained from these approaches and the proposed approaches are the same, except the approach developed on the T-SPFWG operator. This shows that the developed MADM decision making model is valid. The initiated MADM model has some advantages over the existing approaches.
(1) The anticipated MADM model is based on the newly initiated aggregation operators.
That is, these aggregation operators are proposed utilizing SS ALs for T-SPFNs, which consist of general parameters that make the decision-making process more flexible. Meanwhile, the existing MADM models are based on the aggregation operators, which are initiated utilizing algebraic ALs. (2) The existing aggregation operators have the characteristic that they can only remove the effect of awkward data by utilizing power weight vector, while the anticipated aggregation operators have the ability to remove the effect of awkward data as well as consider the interrelationship among the input data at the same time. (3) The other advantage of the anticipated AOs is that it consists of general parameters, which make the decision-making process more flexible. Therefore, the initiated AOs are more practical and comparative in their utilization while solving MADM models under T-SPF information. Table 11. Comparison with existing approaches.

Conclusions
One of several implementations of multi-criteria decision-making (MCDM) problems is indeed the assessment of water reuse strategies. Water reuse is a potential method for boosting the urban supply of water, particularly in light of the changing standards such as climate change and increased human activity. A cost-effective, long-term water reuse application should pose an admissible health risk to customers. Data collection is frequently coupled with difficulties of ambiguity, hesitation, and parameterization, making water reuse application evaluation difficult. In this article, a T-Spherical fuzzy set-based decision support is initiated to offer an efficient approach to explain ambiguity, hesitation, and uncertainty. The contribution of this article is fourfold. Firstly, the novel SS ALs for T-SPFN are initiated and some the vital characteristic of these ALs are investigated. Secondly, based on these novel SS ALs, some T-SPFSSPHM operators such as the T-Spherical fuzzy Schweizer-Sklar power Heronian mean operator, the T-Spherical fuzzy Schweizer-Sklar power geometric Heronian mean operator, the T-Spherical fuzzy Schweizer-Sklar power weighted Heronian mean operator, the T-Spherical fuzzy Schweizer-Sklar power weighted geometric Heronian mean operator, and their vital properties and special cases with respect to the parameters are discussed. By giving specific values to the general parameters, we can observe that some of the existing AOs are special cases of these newly initiated AOs. These AOs have advantages over the existing AOs. These AOs have the capacity to remove the influence of awkward data and consider the interrelationship among the input data at the same time, while the existing aggregation operators for T-SPFS can remove the effect of awkward data or create interrelationships among input data. Thirdly, based on these AOs, a MADM model is anticipated. Lastly, the anticipated model is applied to select the best option in water reuse from the available options.
In future, we will apply the anticipated approach to some new applications, such as supply chain management [5,9], public transportation [24], traffic control [54], digital twin model [56], and so on, or extend the anticipated model to some more extended form of T-SPFSs.
Author Contributions: Idea and conceptualization, Q.K. and M.S.; methodology, Q.K. and J.G.; collection of data and result calculation, Q.K and M.K.A.; validation, J.G.; writing-original draft preparation, Q.K. and J.G.; writing-review and editing, Q.K. and J.G. All authors have read and agreed to the published version of the manuscript.