Bonferroni Mean Research Articles

A data-driven group emergency decision-making method based on interval-valued intuitionistic hesitant fuzzy sets and its application in COVID-19 pandemic

The outbreak of Corona Virus Disease 2019 (COVID-19) makes people more concerned about the validity and timeliness of emergency decision making. When an emergency occurs, it is difficult for decision makers (DMs) to give accurate assessment information in the early stage due to the urgency of time, the incompleteness of information, and the limitations of DMs’ cognition and knowledge. Hence, we use interval-valued intuitionistic hesitant fuzzy sets rather than exact numbers to better characterize the fuzziness and uncertainty of emergencies. In addition, the Internet has become a major platform for the public to express their opinions or concerns, so we can collect the user-generated content on social media to help DMs determine appropriate emergency decision-making criteria which are the premise and basis of scientific decisions. However, there is likely to be some correlation between the obtained criteria. To this end, we first extend the Bonferroni mean (BM) operator to the interval-valued intuitionistic hesitant fuzzy environment, and propose three interval-valued intuitionistic hesitant fuzzy BM operators to capture the interrelation of fuzzy input variables, including an interval-valued intuitionistic hesitant fuzzy BM operator, a simplified interval-valued intuitionistic hesitant fuzzy BM operator, and a simplified interval-valued intuitionistic hesitant fuzzy weighted BM (SIVIHFWBM) operator. Then, a new group emergency decision-making method based on the SIVIHFWBM operator and social media data is proposed, and the specific steps of ranking all emergency plans are put forward. Moreover, our method is applied to evaluate emergency plans for the prevention and control of COVID-19. Finally, the effectiveness and feasibility of the method are verified by the sensitivity analysis, validity test, and comparative analysis.

Schur m-power convexity for general geometric Bonferroni mean of multiple parameters and comparison inequalities between several means

Abstract In the paper, the authors present the Schur m-power convexity and concavity for the general geometric Bonferroni mean of multiple parameters and establish comparison inequalities for bounding the general geometric Bonferroni mean in terms of the arithmetic, geometric, and harmonic means. These Schur convexity and concavity provide a unified generalization of the Schur convexity and concavity for the geometric Bonferroni means of two or three parameters.

A novel extended Portuguese of Interactive and Multi‐Criteria Decision Making and Archimedean Bonferroni mean operators based on prospect theory to select green supplier with complex q‐rung orthopair fuzzy information

AbstractThe word sustainable or green supply chain refers to the concept of incorporating sustainable environmental procedures into the traditional supply chain. Green supply chain management gives a chance to revise procedures, materials and operational ideas. Choosing the fuzziness of assessing data and the spiritual situations of experts in the decision‐making procedure are two important issues. The main contribution of this analysis is to derive the theory of Archimedean Bonferroni mean operator for complex q‐rung orthopair fuzzy (CQROF) information, called the CQROF Archimedean Bonferroni mean and CQROF weighted Archimedean Bonferroni mean operators which are very valuable, dominant and classical type of aggregation operators used for examining the interrelationship among the finite number of attributes in modern data fusion theory. Inspirational and well‐used properties of the initiated theories are also diagnosed with some special cases. Additionally, the theory of extended TODIM tool using the prospect theory based on CQROF information was discovered, which play an essential and critical role in the environment of fuzzy set theory. Finally, a real life problem by computing a green supply chain management based on the initiated CQROF operators was evaluated and fully illustrating the feasibility and efficiency of the diagnosed work with the help of a comparison between existing and prevailing theories.

A hybrid filter-wrapper feature selection using Fuzzy KNN based on Bonferroni mean for medical datasets classification: A COVID-19 case study

Several feature selection methods have been developed to extract the optimal features from a dataset in medical datasets classification. Creating an efficient technique has become a challenge because of the high dimensions, noise, and redundant information. In this paper, we propose a hybrid filter-wrapper approach for feature selection. An ensemble of filter methods, ReliefF and Fuzzy Entropy (RFE) is developed, and the union of top-n features from each method are considered. The Equilibrium Optimizer (EO) technique is combined with Opposition Based Learning (OBL), Cauchy Mutation operator and a novel search strategy to enhance its capabilities. The OBL strategy improves the diversity of the population in the search space. The Cauchy Mutation operator enhances its ability to evade the local optima during the search, and the novel search strategy improves the exploration capability of the algorithm. This enhanced form of EO is integrated with eight time-varying S and V-shaped transfer functions to convert the solutions into binary form, Binary Enhanced Equilibrium Optimizer (BEE). The features from the ensemble are given as input to the Binary Enhanced Equilibrium Optimizer to extract the essential features. Fuzzy KNN based on Bonferroni mean is used as the learning algorithm. Twenty-two benchmark datasets and four microarray datasets are used to test the algorithm’s efficiency. This method is also applied to a COVID-19 case study. The results demonstrate the superiority of the proposed approach, RFE-BEE, among other methods in terms of fitness values, accuracy, precision, sensitivity, and F-measure, among several other state-of-the-art algorithms. RFE-BEE can be applied to various biomedical, computer vision and engineering applications such as electromyography pattern recognition, COVID-19 diagnosis, face recognition and fault diagnosis.

Identifying native endemic plant species in Nilgiris using the interval type 2 q-rung orthopair fuzzy Bonferroni mean operator

Identifying native endemic plant species in Nilgiris using the interval type 2 q-rung orthopair fuzzy Bonferroni mean operator

Generalized q-rung orthopair fuzzy interactive Hamacher power average and Heronian means for MADM.

In this paper, we establish a novel q-rung orthopair fuzzy (q-ROF) multi-attribute decision making (MADM) model on the basis of the proposed q-ROF interactive Hamacher weighted adjustable power average (q-ROFIHWAPA) and q-ROF interactive Hamacher weighted coordinated Heronian means (HMs), which (1) can reflect the correlations among multiple attributes; (2) weakens the impacts of the extreme evaluation values more reasonably; (3) considers the interactions between the membership degree (MD) and non-membership degree (N-MD) of different q-ROF numbers (q-ROFNs); (4) has the characteristic of generality (It can generate different methods by different operations). Firstly, the q-ROF interactive Hamacher operations, improved score function and new q-ROF entropy (q-ROFE) formula, which are the necessary raw materials for the implementation of MADM, are presented. Secondly, we introduce the adjustable power average (APA) and its weight form (WAPA) to remedy the deficiencies of the classical power averages (PAs). Afterwards we extend the WAPA to q-ROF circumstance and propose the q-ROF interactive Hamacher WAPA (q-ROFIHWAPA), and its basic properties are analyzed. Further, the entropy weight fitting method is presented to determine the parameter carried by the q-ROFIHWAPA. Thirdly, inspired by the evolutionary process of Bonferroni means (BMs), we define the weighted coordinated HM (WCHM) and weighted geometric coordinated HM (WGCHM) based on the traditional HMs, respectively, which eliminate the redundancy of the dual generalized weighted BM (DGWBM) and dual generalized weighted Bonferroni geometric mean (DGWBGM), i.e., the case of Then we develop the q-ROF interactive Hamacher WCHM (q-ROFIHWCHM) and q-ROF interactive Hamacher WGCHM (q-ROFIHWGCHM) by combining them with the q-ROF interactive Hamacher operations, and the common properties and special cases are also investigated. Finally, we create a MADM algorithm relied on the q-ROFIHWAPA and q-ROFIHWCHM (resp. q-ROFIHWGCHM), and a practical example is introduced to illustrate the effectiveness and superiority of the proposed method.

Dombi Bonferroni Mean Operators Under Bipolar Complex Fuzzy Environment and Their Applications in Internet World

Internet is almost a necessary facility and tool to solve daily life problems in every field life. Whether at the individual level or national and international level sale purchase of any kind of object has always been of much importance, especially after Corona Pandemic, when online business is at its peak. Because of the enhancement of online sales and purchases, various businessmen are looking for suitable internet websites for their businesses, and the selection of the most suitable internet websites is one of the multi-attribute decision-making (MADM) dilemmas. Thus, in this script, we take benefits of three various concepts that are Bonferroni mean (BM) operator which is a significant technique to catch the interrelatedness among any number of inputs, Dombi operations which are based on Dombi t-norm and t-conorm and the ability to create an aggregation procedure more flexible because of the parameter, bipolar complex fuzzy set (BCFS) which is an outstanding model for tackling two-dimensional information with negative aspect and interpret bipolar complex fuzzy (BCF) Dombi Bonferroni mean (BCFDBM), BCF weighted Dombi Bonferroni mean (BCFWDBM), BCF Dombi geometric Bonferroni mean (BCFDGBM), and BCF weighted Dombi geometric Bonferroni mean (BCFWDGBM) operators. After ward, in this script, for tackling MADM dilemmas in the setting of BCFS, we investigate a MADM procedure based on the investigated operators and solve a MADM dilemma (selection of a suitable internet website for businessmen). Further, to display the superiority and efficiency of our work, we compare our approach and operators with a few current approaches and operators.

A Normalized Weighted Bonferroni Mean Aggregation Operator in Neutrosophic Vague Multi-Criteria Decision- Making

Decision-making problems involve uncertain and incomplete information, which can be well represented by the Neutrosophic set (NS). Various extensions of NS are available in the literature for solving such problems. However, the published extensions of NS have some restrictions such as single based membership degree. Neutrosophic vague set (NVS) is a newly developed theory to address the shortcomings of previous set theory. NVS is structured based on interval membership in the context of dependent membership functions. Beside uncertainty information, aggregation operators (AOs) are critical components in real-world decision-making issues. As a generalization to the conventional aggregation functions defined on the set of real numbers, numerous AOs have been presented in the literature. Each AO provides a distinct purpose in effectively resolving decision-making problems. Recently, Bonferroni meant (BM) operator has received great attention among scholars because of its ability to consider interrelationship among criteria available in decision-making problems. Based on the advantages of the NV and BM operator, we would like to fill in the gaps by developing a Neutrosophic vague normalized weighted Bonferroni mean (NV-NWBM). In addition, five mathematical properties related to proposed AO are also examined. Besides that, a three-phase decision-making framework is presented to clarify that the proposed AO can be applied to real world decision-making issues. The NV-NWBM operator along with decision-making framework is applied to the example of investment selection under NV environment. The finding shows a computer company is the best alternative for investment. Finally, influence of parameter is performed to validate the effect of parameter towards ranking order.

Generalized triangular Pythagorean fuzzy weighted Bonferroni operators and their application in multi-attribute decision-making

<abstract><p>The consolidation of evaluations from various decision-makers within a group, concerning multiple attributes of limited schemes, seeks to unify or compromise collective preferences according to specific rules. The superior characteristics of Possibility Fuzzy Sets (PFS) in membership endow it with enhanced capabilities in depicting ambiguous information. The Bonferroni operator proficiently mitigates the influences of interrelations between attributes in decision-making dilemmas. To address the Multi-Attribute Decision Making (MADM) conundrum wherein attribute values are associative Triangular Pythagorean Fuzzy Numbers (TPFNs), a novel methodology leveraging the Generalized Triangular Pythagorean Fuzzy Weighted Bonferroni Mean (GTPFWBM) operator and the Generalized Triangular Pythagorean Fuzzy Weighted Bonferroni Geometric Mean (GTPFWBGM) operator is advanced. Initiating with the foundational Triangular Pythagorean Fuzzy Set and the Generalized Bonferroni Mean (GBM) operator, both the GTPFWBM and GTPFWBGM operators are delineated. Subsequent exploration dives into the intrinsic properties of these pioneering operators, encompassing facets like reducibility, permutation invariance, idempotency, monotonicity and boundedness. Building upon this foundation, a MADM methodology predicated on the GTPFWBM and GTPFWBGM operators is conceptualized. The culmination of this research underscores the method's rationality and practicality, illustrated through a venture capital investment exemplar.</p></abstract>

Intelligent System for Optimal Geometric Design Using Fuzzy Soft PDE

In this research, a methodology is discussed to develop an intelligent system for optimal geometric design for the culinary product. An optimum geometric design meets certain specified criteria in the most efficient or cost‐effective way possible. The criterion for an optimum design will depend on the specific goals and constraints of the specific problem. In this methodology, partial differential equation and weighted Bonferroni mean are consolidated to develop the intelligent system for optimal geometric designs for the required product with the blend of fuzzy soft sets. Fuzzy soft sets allow capturing the incorporation of subjective or personal opinions into decision‐making processes, as well as the consideration of multiple conflicting criteria. A parameter known as the smoothness parameter is used to control the shape of the optimal geometric model. The smoothness parameter, used as a fuzzy number, is important in this developed system as it fulfills the requirements for the desired intelligent system for product design according to the industries’ demands. To verify the credibility of this system, an illustrated example is presented to design a culinary product, which is profitable for the hotel industry.

An Approach to MADM Based on Aczel-Alsina Power Bonferroni Aggregation Operators for q-Rung Orthopair Fuzzy Sets

The power Bonferroni mean (PBM) is a combined networking system that can benefit both the power average (PA) operator, which might attenuate the effects of irrational data provided by the prejudiced decision-makers (DMs), and the BM operator, which may incorporate the interconnections between any two attributes. In this paper, we defined the aggregation operators (AOs), such as the power average (PA) operator and the power Bonferroni mean (PBM) operator, based on the Aczel-Alsina (AA) operational rules under the environment of the q-Rung Orthopair fuzzy set (q-ROFS), and provided their specific characteristics for the solution of numerical problems. We extend the power Bonferroni mean (PBM) operator under the framework of q-ROFVs to establish the q-Rung Orthopair Fuzzy Aczel-Alsina power Bonferroni mean (q-ROFAAPBM) operator and the q-Rung Orthopair Fuzzy Aczel-Alsina weighted power Bonferroni mean (q-ROFAAWPBM) operator and discuss a few of the most useful properties of these AOs. After that, the q-ROFAAWPBM operator is used to solve the multi-attribute decision-making steps for q-ROFVs. The disease is diagnosed with the MADM method, which shows its usefulness and effectiveness. Finally, the comparison of our developed approach has been demonstrated to be superior and more effective than the existing literature, and we can conclude the result.

Integrated Single-Valued Neutrosophic Normalized Weighted Bonferroni Mean (SVNNWBM)-DEMATEL for Analyzing the Key Barriers to Halal Certification Adoption in Malaysia

The implementation of halal certification in Muslim-owned dining establishments in Malaysia faces various obstacles. This research aimed to analyze and investigate these challenges. The study used the DEMATEL approach with the integrated single-valued neutrosophic normalized weighted Bonferroni mean (SVNNWBM) to identify and understand the interrelationships among the obstacles. Three groups comprising academicians, industry professionals, and Muslim owner-restaurant representatives provided their perspectives. The findings highlighted three key obstacles hindering the acceptance and implementation of halal certification: customer perception, halal perception, and inadequate awareness. Fiscal constraints, inadequate awareness, and halal perception were significant barriers in Malaysia's restaurant sector. These obstacles were categorized as causal factors, generating additional barriers in the industry. By identifying these challenges, effective strategies can be developed to overcome them and improve the adoption of halal certification. This research provides valuable insights for policymakers, restaurant owners, and other stakeholders, enabling them to understand the obstacles better and develop targeted interventions. Employing the integrated SVNNWBM-DEMATEL approach and incorporating perspectives from experts in academia, industry, and Muslim owner-restaurant representatives ensures a comprehensive analysis of the obstacles faced in halal certification implementation.

Novel Fermatean Fuzzy Bonferroni Mean aggregation operators for selecting optimal health care waste treatment technology

Fermatean fuzzy set (FFS) is an expedient tool in today’s world for coping with uncertainty, ambiguity, and incompleteness that emerge in many decision-making processes. In light of the relevance of FFS, this work offers a Multi-Criteria Group Decision Making (MCGDM) method together with Bonferroni mean and weighted Bonferroni mean operators in FF environment. These operators consider the interdependencies between the parameters into account during aggregation process in group decision making. On the other hand, health care waste (HCW) is an unavoidable byproduct of healthcare institutions, and so HCW management is vital owing to its negative influence on public health and environment. In this study, the proposed FFMCGDM has been applied to determine the optimal HCW treatment technologies (TT) for assuring sustainable environmental development, considering six TTs as alternatives assessing against nine criteria. Moreover, the proposed approach has been illustrated through an empirical case study of many district hospitals in Tripura, India. The obtained results are compared to other two well-known MCDM techniques as well as existing Fermatean Fuzzy Aggregation Operators (FFAOs). A sensitivity analysis has also been carried out to examine the stability of the model. From the analysis, it has been found that the proposed technique is competitive to obtain the optimal HCW TT. For further validation of the proposed method, a well-known MCDM problem, namely Green Supplier Selection (GSS) has been modeled and solved.

Evaluating alternative low carbon fuel technologies using a stakeholder participation-based q-rung orthopair linguistic multi-criteria framework

It is widely believed that alternative low carbon fuels (ALCF) can be instrumental in achieving the transportation sector’s decarbonization goal. Unlike conventional fossil-based fuels, ALCF can be produced through a combination of different chemical processes and feedstocks. The inherent complexity of the problem justifies the multi-criteria decision-making (MCDM) approach to support decision-making in the presence of multiple criteria and data uncertainty. In this paper, we propose a novel stakeholder participation-based MCDM framework integrating experts' perspectives on ALCF production pathways using the analytics hierarchy process (AHP) and the q-rung orthopair linguistic partition Bonferroni mean (q-ROLPBM) operator. The key merit of our approach lies in treating criteria of different dimensions as heterogeneous indicators while considering the mutual influence between criteria within the same dimension. The proposed framework is applied to evaluate four ALCF production pathways against 13 criteria categorised under economic, environmental, technical, and social dimensions for the case of the United Kingdom (UK). Our analysis revealed the environmental and the economic dimensions to be the most important, followed by the social and technical evaluation dimensions. The e-fuel followed by the e-biofuel are found to be the two top-ranked production pathways that utilise the electrochemical reduction process and its combination with anaerobic digestion. These findings, along with our recommendations, provide decision-makers with guidelines on ALCF production pathway selection and formulate effective policies for investment.

The aggregation of Z-numbers based on overlap functions and grouping functions and its application on group decision-making

In this paper, we propose a series of novel aggregation techniques based on overlap and grouping functions for group decision-making (GDM) issues in the Z-number environment. Firstly, we introduce an optimization model to determine the underlying distribution for Z-numbers, meanwhile, the mean function of Z-numbers is revised to better reflect the three-dimensional structure of the Z-numbers. And then, a series of operations for Z-numbers using overlap and grouping function are defined, and their properties are also investigated. Subsequently, we construct the overlap function-based Z-number weighted Bonferroni mean (OZWBM) operator to integrate the information in the Z-number environment. Furthermore, to verify the efficiency of novel aggregation techniques, a new energy investment selection issue is addressed. Besides giving the rankings of alternatives, by analyzing the calculation results, we can also derive the degree of preference from investment experts in scoring each investment object. Finally, by comparing the existing methodologies, we confirm that the new approach is effective and reasonable for the multi-criteria group decision-making (MCGDM).

An MCDM approach based on some new Pythagorean cubic fuzzy Frank Muirhead mean operators

Pythagorean cubic fuzzy sets (PCFSs) are the most convenient aid to depict ambiguous data in practical decision-making situations. The Muirhead mean (MM) possesses the ability to capture the interrelationship between the criteria. Further, the MM generalizes several significant operators, for instance, the Bonferroni mean (BM) and the Maclaurin symmetric mean (MSM), etc. The Frank triangular norms can offer significant adaptability and robustness due to the presence of an additional parameter as compared to other families of triangular norms. Therefore, Muirhead mean and Frank operations can be combined for obtaining suitable results during decision-making under Pythagorean cubic fuzzy information with interrelated criteria. The primary goal of this study is to introduce some new MM operators based on Frank operations, namely, PCFFMM and PCFFGMM. Then some of the properties of PCFFMM and PCFFGMM are described. Moreover, an MCDM approach is formulated using PCFFMM and PCFFGMM operators. Eventually, the projected MCDM approach is demonstrated with a numerical example. The phenomenon of interrelationships among criteria of real-life problems can be suitably handled by the PCFFMM and PCFFGMM. The overall ranking result obtained by utilizing the proposed operators PCFFMM or PCFFGMM is more appropriate as compared to the ones obtained by using the BM, MSM, Heronian mean (HM), Choquet integral (CI), etc. Moreover, PCFFMM or PCFFGMM can be preferred ahead of BM, MSM, HM, and CI due to its computation simplicity, accuracy, flexibility, and robustness during the aggregation process of multiple correlated input data.

Analyzing the effect of different types of pollution with bipolar complex fuzzy power Bonferroni mean operators

When any amount of harmful materials (any substance or any type of energy) is introduced into the climate at a rate quicker than it very well may be scattered or securely put away, then pollution occurs. These harmful materials are known as pollutants which can be natural and can also be manmade such as trash generated by factories. These harmful materials harm the quality of land, air, and water and cause various types of pollution, which affects the environment. In this article, we analyze the effect of various types of pollution on the environment and evaluate the most harmful type of pollution through an illustrative example by employing power Bonferroni mean (BM) operators in the setting of the bipolar complex fuzzy set (BCFS), like bipolar complex fuzzy (BCF) power BM (BCFPBM), BCF weighted power BM (BCFWPBM), BCF power geometric BM (BCFPGBM), and BCF weighted power geometric BM (BCFWPGBM) operators and a decision-making (DM) procedure created on these operators in the environment of the BCFS which are introduced in this article. Furthermore, we illustrate that the introduced operators and a DM procedure in the environment of the BCFS are more effective and have a wide model and advantages than certain prevailing works.

Pythagorean Fuzzy Bonferroni Mean with Weighted Interaction Operator and Its Application in Fusion of Online Multidimensional Ratings

Aiming at the aggregation problem of online multi-attribute interactive ratings, a decision-making method based on Pythagorean Fuzzy Bonferroni mean with weighted interaction (PFWIBM) operator is developed by mining the advantages of ratings driven decision-making and the quantitative advantages of Pythagorean fuzzy sets on linguistic ratings. First, the shortcomings of existing operators in criterion interactive representation are analyzed, and the improved Bonferroni mean with weighted interaction and its dual operator are defined, which are used to aggregate the membership information and non-membership information under the Pythagorean fuzzy environment, respectively, and then the PFWIBM operator is constructed. Second, a generation method of interaction coefficients embedded in the PFWIBM operator is designed by combining expert knowledge and user ratings to overcome the limitations of the method of relying on subjective setting parameter values. Third, based on the proposed operator and the transformation relationship between Pythagorean fuzzy sets and linguistic ratings, an online multi-dimensional rating aggregation decision-making approach for solving product raking problems is developed. Finally, the proposed method is applied to a passenger car ranking example to show the feasibility of the method, and through some comparative analysis, the advantages of the proposed operator are clarified, and the influence of operator parameters on the decision-making results is analyzed.

Multiple Attribute Decision-Making Based on Bonferroni Mean Operators under Square Root Fuzzy Set Environment

The intuitionistic fuzzy set (IFS), which has a membership and non-membership degree, is a controlling and effective device for dealing with fuzziness and uncertainty. Recently,the square root fuzzy set which is one of the efficient generalizations of an IFS for dealing with uncertainty and haziness in information has beenintroduced. In this study, a novel method for multiple attribute decision-making (MADM) based on SR-fuzzy information isinvestigated. Since aggregation operators are significant in the decision-making (DM) process, to achievethis goal, the current paper suggests a variety of novel Bonferroni mean and weighted Bonferroni mean operators to aggregate the SR-fuzzy values for the various decision-maker preferences. To achieve this goal, the current paper suggests a variety of novel Bonferroni mean and weighted Bonferroni mean operators to aggregate the SR-fuzzy values for the various decision-maker preferences. SR-fuzzyBonferroni mean operator and weighted SR-fuzzy Bonferroni mean operator are established and their properties are described. Then, we constructed a MADM approach using the proposed operators for the SR-fuzzyinformation and proved the approach with a mathematical example. Inthe end, a comparative study ofthe developed and existing approaches has been discussed to evaluate the pertinency and practicality of the proposed DM technique.

An Extension of Bonferroni Mean under Cubic Pythagorean Fuzzy Environment and Its Applications in Selection-Based Problems

Cubic Pythagorean fuzzy (CPF) set (CPFS) is a hybrid set that can describe both interval-valued Pythagorean fuzzy (IVPF) sets (IVPFSs) and Pythagorean fuzzy (PF) sets (PFSs) simultaneously for addressing information ambiguities. Since an aggregation operator (AO) is a significant mathematical approach in decision-making (DM) problems, this article presents some novel Bonferroni mean (BM) and weighted Bonferroni mean averaging operators between CPF-numbers (CPFNs) for aggregating the different preferences of the decision-makers. Then, using the proposed AOs, we develop a DM approach under the CPF environment and demonstrated it with a numerical example. Furthermore, a comparative study between the proposed and existing methods has been performed to demonstrate the practicality and efficiency of the proposed DM approach.