Geometric Heronian Mean Research Articles

Advancements in MADM with college English teaching quality evaluation: Integrating 2-tuple linguistic neutrosophic models with prioritized heronian mean techniques

With the continuous development of Internet era, teaching modes are also being innovated and developed, and online and offline blended teaching (OOBT) is one of them, which is mainly achieved by combining OOBT. By combining online and offline teaching modes with college English courses, teachers not only help students cultivate independent learning habits, but also help them think alone and improve their practical skills. This is also true for vocational and technical education, and it can be said that this approach is more relevant. The OOBT evaluation in college English teaching courses is deemed to multiple-attribute decision-making (MADM). In this study, we connected the geometric Heronian mean (GHM) technique and prioritized aggregation (PA) in line with 2-tuple linguistic neutrosophic numbers (2TLNNs) to produce the generalized 2TLNN power GHM (G2TLNPGHM) technique. Then, the G2TLNPGHM technique is implemented to process the MADM under 2TLNNs. Finally, the G2TLNPGHM technique is implemented as the numerical example of evaluating the OOBT in college English courses. Some comparative analysis of the impact of parameters are fully addressed.

Decision algorithm for educational institute selection with spherical fuzzy heronian mean operators and Aczel-Alsina triangular norm

This article presents a novel study of spherical fuzzy sets (SFSs), a more comprehensive framework of intuitionistic fuzzy sets and picture fuzzy sets. The SFS allows the decision-makers (DMs) to cope with complicated and insufficient information during the aggregation process. The Heronian mean (HrM) model theory is also utilized to express correlation among different input arguments or characteristics. Recently, the theory of Aczel Alsina triangular norms gained a lot of attention from various research scholars and has many capabilities to provide smooth approximations during decision analysis. In this article, we developed some appropriate operations of Aczel Alsina t-norms and t-conorms in light of spherical fuzzy (SF) information. We develop new mathematical ways to look at SF data to keep clarity and sufficient information. These are the SF Aczel Alsina Heronian mean (SFAAHrM) and SF Aczel Alsina weighted Heronian mean (SFAAWHrM) operators. Furthermore, we also present a list of new strategies based on Aczel Alsina operations, such as SF Aczel Alsina geometric Heronian mean (SFAAGHrM) and SF Aczel Alsina weighted geometric Heronian mean (SFAAWGHrM) operators. Some notable properties are also characterized to show the validity and effectiveness of our derived mathematical approaches. Considering our derived strategies, an algorithm for the multiple attribute decision-making (MADM) problem is established to resolve complicated real-life applications. A numerical example presents the compatibility of derived approaches and provides a solid mechanism to improve the performance of educational institutes. A comparison technique is also demonstrated to show the applicability and consistency of diagnosed approaches by contrasting the findings of pioneered approaches with existing methodologies.

An integrated group decision-making framework for evaluating the urban fire emergency management capability based on PDHLWPHM and PDHLWPGHM operator

Urban fire accident is a common dangerous accident in urban sudden accidents, which threatens the safety of people’s lives and property. For this reason, in recent years, all cities have incorporated the prevention and emergency management of urban fire accidents into their urban development planning, and actively improved their fire accident emergency management capabilities. However, how to evaluate the urban fire accident emergency management capacity of each city to ensure that people’s lives and property are protected to the greatest extent is an urgent problem to be considered and solved. Therefore, this paper defines a class of probabilistic double hierarchy linguistic Heronian mean (PDHLHM) operator, probabilistic double hierarchy linguistic Power Heronian mean (PDHLPHM) operators, and their dual operators that can reflect the relationship between two attributes during aggregation. Taking urban fire accident risk monitoring and early warning capability, fire infrastructure and communication system, fire-fighting and rescue capability, recovery and reconstruction capability as evaluation attributes, the probabilistic double hierarchy linguistic weight Power Heronian mean (PDHLWPHM) operator model and the probabilistic double hierarchy linguistic weight Power geometric Heronian mean (PDHLWPGHM) operator model are constructed for group decision-making. In addition, the idempotence, boundedness and monotonicity of these operators are studied, and the sensitivity of the parameters involved in the operator model is analyzed. Finally, the new model proposed in this paper is compared with the existing model to verify its scientificity.

Prioritization of types of wireless sensor networks by applying decision-making technique based on bipolar complex fuzzy linguistic heronian mean operators

Wireless sensor networks are flexible monitoring systems that save track of, data, and communicate multipoint digital information interpretations to other devices. Wireless sensor networks meaningly enhance the accuracy, breadth, and extent of local data collection, commonly doing away with the requirement for expensive data wiring and recurring manual checks at risky, remote, or inaccessible locations. As a result, it is utilized to keep an eye on systems and environmental or physical parameters. In this manuscript, we expand the Heronian mean operators in the model of bipolar complex fuzzy linguistic set to concoct bipolar complex fuzzy linguistic arithmetic Heronian mean, bipolar complex fuzzy linguistic weighted arithmetic Heronian mean, bipolar complex fuzzy linguistic geometric Heronian mean and bipolar complex fuzzy linguistic weighted geometric Heronian mean operators. We also inspect the special cases of the invented bipolar complex fuzzy linguistic arithmetic Heronian mean and bipolar complex fuzzy linguistic geometric Heronian mean operators. Moreover, in this manuscript, we concoct a technique of decision-making in the model of a bipolar complex fuzzy linguistic set with the assistance of the invented operators. As the selection and prioritization of the various types of Wireless sensor networks is the decision-making dilemma, we prioritize various types of Wireless sensor networks by employing the concocted technique of decision-making and by taking artificial data in the model of the bipolar complex fuzzy linguistic set. To reveal the influence and excellence of the concocted work, a comparative study is given in this manuscript.

Fermatean fuzzy Archimedean Heronian Mean-Based Model for estimating sustainable urban transport solutions

Public transportation frameworks assume a critical role in the metropolitan region, especially in huge urban communities, where they offer a feasible answer for easing gridlock, moderating commotion contamination, and diminishing CO2 discharges. This paper presents some novel Fermatean fuzzy Heronian mean operators based on Archimedean t-norms, specifically the generalized Fermatean fuzzy Archimedean Heronian mean (GFFAHM) and the Fermatean fuzzy Archimedean geometric Heronian mean (FFAGHM). The study investigates different unique instances of these operators while exploring their essential properties. Besides, a powerful multiattribute decision-making (MADM) method is developed using the proposed operators. This approach offers a key asset for handling complex decision-making problems. Overcoming the capabilities of conventional BM operators, the GFFAHM and FFAGHM operators effectively minimize the potential redundancy in interrelationships during the decision-making process. The inclusion of the flexible parameters ρ and ϕ, which have a big impact on the decision-making process’ outcomes, increases the adaptability and resilience of the Archimedean t-based operators. To depict the feasibility of the proposed MADM method, a thorough quantitative model is introduced, outlining its useful execution. Through an exhaustive case study, the predominance of the proposed approach over existing strategies is experimentally established. Remarkably, the study uncovers that reducing fares is the most compelling factor in expanding the public transport framework, subsequently advancing sustainable urban transport. This study’s findings provide useful insights into decision-making processes and practical implications for policymakers, allowing them to make informed and impactful changes to the public transportation system. The proposed MADM method can play a vital role in upgrading the public transport framework, making way for remarkable strategy interventions in urban transport sustainability.

Methodology for evaluating the ideological and political education quality of chinese universities based on fuzzy number intuitionistic fuzzy multiple-attribute decision-making

The quality evaluation of Chinese universities ideological and political (IAP) education has gone through the stages of defining tasks, proposing standards and exploring and carrying out, and has completed the stage tasks and accumulated practical experience. To construct the quality evaluation system of IAP education of Chinese universities in the new era, it is necessary to find the quality positioning in the fundamental task of establishing moral education and pay attention to the synergy between the internal and external parts of the quality of IAP education of Chinese universities. The IAP education quality evaluation of Chinese universities are the multiple-attribute decision-making (MADM) issue. In this paper, we extend the geometric Heronian mean (GHM) operator to fuzzy number intuitionistic fuzzy numbers (FNIFNs) to propose the fuzzy number intuitionistic fuzzy weighted geometric HM (FNIFWGHM) operator. Then, the MADM method are built on FNIFWGHM operator. Finally, a numerical example for IAP education quality evaluation of Chinese universities and some comparative studies are used to prove the built methods’ credibility and reliability.

Intelligent decision methodology for business English teaching quality evaluation based on GHM and PG operators with 2-tuple linguistic neutrosophic numbers

Business English teaching quality evaluation Business English is a new type of composite specialty, which is a discipline innovation made by China’s higher education to adapt to the new market demand and international standards since the reform and opening up. Over the past 20 years, it has cultivated a number of compound talents for the cause of China’s reform and opening up. However, the backwardness of business English theoretical research has greatly restricted the development of business English. At present, Business English has been officially approved as a new major for undergraduate enrollment by the Ministry of Education of the People’s Republic of China. Its subject nature, specialty structure, training objectives, and specialty compound characteristics need to be qualitatively studied theoretically. The business English teaching quality evaluation is viewed as the multiple attribute decision making (MADM) issue. In this paper, we connect the geometric Heronian mean (GHM) operator and power geometric (PG) with 2-tuple linguistic neutrosophic numbers (2TLNNs) to propose the generalized 2-tuple linguistic neutrosophic power geometric HM (G2TLNPGHM) operator. Then, the G2TLNGHM operator is applied to deal with the MADM problems under 2TLNNs. Finally, an example for business English teaching quality evaluation is used to show the proposed methods. Some comparative analysis and parameter influence analysis are fully given. The results show that the built algorithms method is useful for business English teaching quality evaluation.

Assessment of Solar Panel Using Multiattribute Decision-Making Approach Based on Intuitionistic Fuzzy Aczel Alsina Heronian Mean Operator

Multiattribute decision-making (MADM) approach is an effective method for handling ambiguous information in a practical situation. The process of the MADM technique has drawn a lot of interest from various academic and selection processes of extensive analysis. The aggregation operators (AOs) are the best mathematical tools and received a lot of attention from researchers. This article explored the theory of intuitionistic fuzzy IF sets (IFSs) and their certain fundamental operations. The theory of triangular norms also explores Aczel Alsina operations (AAOs) in advanced mathematical tools. The concepts of Heronian mean (HM) and geometric HM (GHM) operators are presented to define interrelationships among different opinions. We developed a list of certain AOs by utilizing AAOs under the system IF information, namely, IF Aczel Alsina HM (IFAAHM), IF Aczel Alsina weighted HM (IFAAWHM), IF Aczel Alsina GHM (IFAAGHM), and IF Aczel Alsina weighted GHM (IFAAWGHM) operators. Some particular characteristics of our invented methodologies are also presented. Solar energy is an effective, efficient resource to enhance electricity production and the country’s economic growth. Therefore, we studied an application of solar panel systems to solve real-life problems under a robust technique of the MADM approach by utilizing our invented approaches of IFAAWHM and IFAAWGHM operators. A numerical example was also given to select more suitable solar panels under our proposed methodologies. To find the competitiveness and feasibility of discussed methodologies, we make an inclusive comparative study in which we contrast the results of existing AOs with the consequences of current approaches.

A study and performance evaluation of computer network under the environment of bipolar complex fuzzy partition Heronian mean operators

The goal of this paper is to study and assessment of the performance of a computer network with the assistance of partition Heronian mean (PHM) operators in the environment of bipolar complex fuzzy (BCF) information. In today's world computers play a significant role in both professional and personal lives. With the advancement of technology, networking arrived in the space and we continuously advanced from wired network innovation to wireless network innovation. A computer network is a gathering of two or more computer systems that are associated together to share information and resources. It can be considered the base of communication in the area of information technology. Thus, for studying and performance evaluation of a computer network, firstly, this paper contains BCF partition Heronian mean (BCFPHM), BCF weighted partition Heronian mean (BCFWPHM), BCF partition geometric Heronian mean (BCFPGHM), and BCF weighted partition geometric Heronian mean (BCFWPGHM) and their corresponding properties. After that, this paper contains a decision-making (DM) mechanism constructed on the explored operators and a genuine-life example to evaluate the performance of a computer network based on the DM mechanism in the environment of BCF information. The last part of the paper contains the comparison of the explored mechanism of DM and operators with certain current work for exhibiting the capacity and profits of the explored work.

Method for education value evaluation with regional culture integration into international students’ ideological education under fuzzy number intuitionistic fuzzy environment

For thousands of years, the Chinese people have accumulated and inherited profound cultural traditions. The uniqueness of this cultural tradition lies in its amazing creative wisdom and power. “The ideological and political education of the integration of Chinese regional culture into international students refers to the educative influence of excellent regional culture that can run through the entire international education management system, curriculum system and extracurricular practice system to achieve “all-round, full-process, full-staff” Education goals. The sustainable education value evaluation based on the integration of regional culture into international students’ ideological education is a classical multiple-attribute decision-making (MADM) issue. In this paper, we extend the geometric Heronian mean (GHM) operator to fuzzy number intuitionistic fuzzy numbers (FNIFNs) to propose the fuzzy number intuitionistic fuzzy GHM (FNIFGHM) operator. Then, the multiple-attribute decision-making (MADM) methods are built on FNIFGHM operator. Finally, a numerical example for sustainable education value evaluation based on the integration of regional culture into international students’ ideological education and some comparative analysis are used to prove the built methods’ credibility and reliability.

Extended MABAC method based on 2-tuple linguistic $ T $-spherical fuzzy sets and Heronian mean operators: An application to alternative fuel selection

<abstract><p>In recent years, fossil fuel resources have become increasingly rare and caused a variety of problems, with a global impact on economy, society and environment. To tackle this challenge, we must promote the development and diffusion of alternative fuel technologies. The use of cleaner fuels can reduce not only economic cost but also the emission of gaseous pollutants that deplete the ozone layer and accelerate global warming. To select an optimal alternative fuel, different fuzzy decision analysis methodologies can be utilized. In comparison to other extensions of fuzzy sets, the $ T $-spherical fuzzy set is an emerging tool to cope with uncertainty by quantifying acceptance, abstention and rejection jointly. It provides a general framework to unify various fuzzy models including fuzzy sets, picture fuzzy sets, spherical fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets and generalized orthopair fuzzy sets. Meanwhile, decision makers prefer to employ linguistic terms when expressing qualitative evaluation in real-life applications. In view of these facts, we develop an extended multi-attributive border approximation area comparison (MABAC) method for solving multiple attribute group decision-making problems in this study. Firstly, the combination of $ T $-spherical fuzzy sets with 2-tuple linguistic representation is presented, which provides a general framework for expressing and computing qualitative evaluation. Secondly, we put forward four kinds of 2-tuple linguistic $ T $-spherical fuzzy aggregation operators by considering the Heronian mean operator. We investigate some fundamental properties of the proposed 2-tuple linguistic $ T $-spherical fuzzy aggregation operators. Lastly, an extended MABAC method based on the 2-tuple linguistic $ T $-spherical fuzzy generalized weighted Heronian mean and the 2-tuple linguistic $ T $-spherical fuzzy weighted geometric Heronian mean operators is developed. For illustration, a case study on fuel technology selection with 2-tuple linguistic $ T $-spherical fuzzy information is also conducted. Moreover, we show the validity and feasibility of our approach by comparing it with several existing approaches.</p></abstract>

A study on the Heronian mean operators for managing complex picture fuzzy uncertain linguistic settings and their application in decision making

The major influence of this manuscript is to diagnose a valuable and considerable technique of Complex Picture Fuzzy Uncertain Linguistic setting and described its useful and valuable operational laws. The theory of Complex Picture Fuzzy Uncertain Linguistic information is massive modified and generalized than the Complex intuitionistic Fuzzy Uncertain Linguistic, Complex Fuzzy Uncertain Linguistic, Fuzzy Uncertain Linguistic and Uncertain Linguistic information. Keeping the supremacy and dominancy of the Complex Picture Fuzzy Uncertain Linguistic information, we investigated the Complex Picture Fuzzy Uncertain Linguistic Arithmetic Heronian Mean, Complex Picture Fuzzy Uncertain Linguistic Weighted Arithmetic Heronian Mean, Complex Picture Fuzzy Uncertain Linguistic Geometric Heronian Mean and Complex Picture Fuzzy Uncertain Linguistic Weighted Geometric Heronian Mean operators. The property of idempotency, boundedness, monotonicity, and various well-known results with certain specific cases of the invented work are also deliberated. Furthermore, in the availability of the above-proposed analysis, we constructed a multi-attribute decision-making technique by considering the diagnosed operators for complex picture fuzzy uncertain linguistic information to enhance the worth and rationality of the invented theory. Finally, we illustrated the merits and restrictions of the novel operators by comparing them with certain prevailing operators based on fuzzy generalization. Finally, in the presence of evaluated examples, we compared the pioneered operators with various existing operators to enhance the feasibility and worth of the invented operators.

An intelligent decision methodology for physical health evaluation of college students based on generalized 2-tuple linguistic neutrosophic geometric HM operators

Physical Health is an important part of health education and health promotion in my country, and the health literacy level of students majoring in physical education in colleges and universities is an important factor in the development of health education in primary and secondary schools, and also directly affects the implementation of school health education in the future. The physical health evaluation of College students is frequently viewed as the multiple attribute decision making (MADM) issue. In our article, we combine the geometric Heronian mean (GHM) operator, generalized weighted Heronian mean (GWHM) operator with 2-tuple linguistic neutrosophic numbers (2TLNNs) to propose the generalized 2-tuple linguistic neutrosophic geometric HM (G2TLNGHM) operator and generalized 2-tuple linguistic neutrosophic weighted geometric HM (G2TLNWGHM) operator. Meanwhile, some ideal properties of built operator are studied. Then, the G2TLNWGHM operator is applied to deal with the MADM problems under 2TLNNs. Finally, an example for Physical health evaluation of College students is used to show the proposed methods.

MULTIMOORA Method for Addressing Security Algorithms Evaluation Problem under q-Rung Orthopair Fuzzy Environment

How to determine a suitable security algorithm for a special application scenario is a complex problem. In this paper, this complex problem is formulated as a multicriteria decision-making (MCDM) problem, and we propose a novel MULTIMOORA (multiobjective optimization on the basis of a ratio analysis plus the full MULTIpevaluation information in the security algorithms evaluation problem. The MULTIMOORA method is an excellent decision method, which owns strong robustness. However, it has not been used to process the complex information structure of q-rung orthopair fuzzy sets. Moreover, it cannot solve the problem that the extreme values negatively influence the ranking results, and it also cannot capture the interrelationship hiding behind the criteria. To overcome the above challenges, we propose novel q-rung orthopair fuzzy Dombi power Heronian mean (DPHM) operator and q-rung orthopair fuzzy Dombi power geometric Heronian mean (DPGHM) operator. Based on these two operators, the MULTIMOORA method is improved for solving the security algorithms’ evaluation problem. Finally, a practical example for evaluating five security algorithms is used to illustrate the decision process of the proposed q-rung orthopair fuzzy MULTIMOORA method.

A new three-way group decision-making model based on geometric heronian mean operators with q-rung orthopair uncertain linguistic information

As an effective tool for three-way decisions (3WD) problems, decision-theoretic rough sets (DTRSs) have raised increasing attention recently. In view of the advantages of q-rung orthopair uncertain linguistic variables (q-ROULVs) in depicting uncertain information, a new DTRSs model based on q-ROULVs is proposed to solve three-way group decision-making (3WGDM) problems. Firstly, the loss function of DTRSs is depicted by q-ROULVs and a q-rung orthopair uncertain linguistic DTRSs model is constructed subsequently. Secondly, to aggregate different experts’ evaluation results on loss function in group decision-making (GDM) scenario, the q-rung orthopair uncertain linguistic geometric Heronian mean (q-ROULGHM) operator and the q-rung orthopair uncertain linguistic weighted geometric Heronian mean (q-ROULWGHM) operator are presented. Related properties of the proposed operators are investigated. Thirdly, to compare the expected loss of each alternative, a new score function of q-ROULVs is defined and the corresponding decision rules for 3WGDM are deduced. Finally, an illustrative example of venture capital in high-tech projects is provided to verify the rationality and effectiveness of our method. The influence of different conditional probabilities and parameter values on decision results is comprehensively discussed.

A method to multi-attribute decision making problems by using heronian mean operators based on linear diophantine uncertain linguistic settings

In the real decision process, an important problem is how to express the attribute value more efficiently and accurately. In the real world, because of the complexity of decision-making problems and the fuzziness of decision-making environments, it is not enough to express attribute values of alternatives by exact values. For this managing with such sorts of issues, the principle of Linear Diophantine uncertain linguistic set is a valuable and capable technique to manage awkward and inconsistent information in everyday life problems. In this manuscript, we propose the original idea of Linear Diophantine uncertain linguistic set and elaborated their essential laws. Additionally, to determine the association among any numbers of attributes, we elaborated the Linear Diophantine uncertain linguistic arithmetic Heronian mean operator, Linear Diophantine uncertain linguistic weighted arithmetic Heronian mean operator, Linear Diophantine uncertain linguistic geometric Heronian mean operator, Linear Diophantine uncertain linguistic weighted geometric Heronian mean operator, and their properties are also discovered. By using these operators, we utilize the multi-attribute decision-making procedure by using elaborated operators. To determine the consistency and validity of the elaborated operators, we illustrate some examples by using explored operators. Finally, the superiority and comparative analysis of the elaborated operators with some existing operators are also determined and justified with the help of a graphical point of view.

A new MAGDM method with 2-tuple linguistic bipolar fuzzy Heronian mean operators.

In this article, we introduce the 2-tuple linguistic bipolar fuzzy set (2TLBFS), a new strategy for dealing with uncertainty that incorporates a 2-tuple linguistic term into bipolar fuzzy set. The 2TLBFS is a better way to deal with uncertain and imprecise information in the decision-making environment. We elaborate the operational rules, based on which, the 2-tuple linguistic bipolar fuzzy weighted averaging (2TLBFWA) operator and the 2-tuple linguistic bipolar fuzzy weighted geometric (2TLBFWG) operator are presented to fuse the 2TLBF numbers (2TLBFNs). The Heronian mean (HM) operator, which can reflect the internal correlation between attributes and their influence on decision results, is integrated into the 2TLBF environment to analyze the effect of the correlation between decision factors on decision results. Initially, the generalized 2-tuple linguistic bipolar fuzzy Heronian mean (G2TLBFHM) operator and generalized 2-tuple linguistic bipolar fuzzy weighted Heronian mean (G2TLBFWHM) operator are proposed and properties are explained. Further, 2-tuple linguistic bipolar fuzzy geometric Heronian mean (2TLBFGHM) operator and 2-tuple linguistic bipolar weighted geometric Heronian mean (2TLBFWGHM) operator are proposed along with some of their desirable properties. Then, an approach to multi-attribute group decision-making (MAGDM) based on the proposed aggregation operators under the 2TLBF framework is developed. At last, a numerical illustration is provided for the selection of the best photovoltaic cell to demonstrate the use of the generated technique and exhibit its adequacy.

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.

Interval-Valued Hesitant Fuzzy Linguistic Multiattribute Decision-Making Method Based on Three-Parameter Heronian Mean Operators

Numerous variants have been proposed for sets of linguistic terms and the interval-valued hesitant fuzzy set (IVHFS). In particular, the interval-valued hesitant fuzzy linguistic set (IVHFLS) is more suitable for defining the hesitancy and inconsistency inherent in the human cognitive processes of decision making. A key aggregation operator is Heronian mean (HM), based on which the correlation among aggregated arguments can be captured. However, the existing HM operators partially overlook the correlation among more than two arguments and lack the properties of idempotency and reducibility. In this work, the limitations of HM operators are first analyzed. Then, two new HM variants are introduced: three-parameter weighted Heronian mean (TPWHM) and three-parameter weighted geometric Heronian mean (TPWGHM). Thus, the reducibility, idempotency, monotonicity, and boundedness properties are proven for the two computational procedures, and unique situations are mentioned. Furthermore, two more elaborate operators are also introduced which are called the interval-valued hesitant fuzzy linguistic TPWHM (IVHFLTPWHM) and the interval-valued hesitant fuzzy linguistic TPWGHM (IVHFLTPWGHM). The main properties, as well as unique situations of these two computational procedures, are discussed. Finally, the introduced methods are clarified by illustrative examples. In addition, the parameter effects on the decision-making outcomes are discussed and comparisons with other reference methods are made.

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 relaxed 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.