Cosine and cotangent similarity measurements are critical in applications for determining degrees of difference and similarity between objects. In the literature, numerous similarity measures for various extensions of fuzzy set, soft set, Intuitionistic Fuzzy Sets (IFSs), Pythagorean Fuzzy Sets (PFSs) and HyperSoft Sets (HSSs) have been explored. Neutrosophic HyperSoft Sets (NHSSs), on the other hand, has fewer cosine and cotangent similarity measures. In this paper, we propose the trigonometric similarity measures of NHSSs. We further investigate the basic operators, theorems, and propositions for the proposed similarity measures. We know that global warming causes environmental problems. One of applications for solving global warming is the concept of renewable energy. To show the effectiveness of the proposed similarity measures, we apply them to renewable energy source selection problems. The study reveals the best geographical area to install the energy production units, under some technical attributive factors. To check the validity and superiority of the proposed work, it is compared with some existing techniques which reveal that, decision-making problems with further bifurcated attributes, have more accurate and precise results and can only be solved with this technique. In the future, the proposed techniques can be applied to case studies, in which attributes are more than one and further bifurcated along with more than one decision-maker. Also, this proposed work can be extended for several existing hybrids of hypersoft sets, intuitionistic hypersoft, neutrosophic hypersoft set, bi-polar hypersoft, m-polar hypersoft sets, and Pythagorean hypersoft set to solve Multi-Criteria Decision Making (MCDM) problems.

The idea of a neutrosophic hypersoft set (NHSS) was coined by Smarandache in 2018 as a generalization of the soft set. This structure is a hybrid of a neutrosophic set with a hypersoft set. It can be a valuable structure for dealing with multi-attributes, multi-objective problems with disjoint attributive values. Similarity measures (SM) play a vital role in measuring the similarity index that how much the things are similar. Different types of similarity measures were developed in literature with different fuzzy, intuitionistic, and neutrosophic theories. It is intended to merge the neutrosophic theory with the hypersoft set theory and propose different similarity measures with the help of new proposed distances with max-min operators. Also, we proved different theorems and properties of distance and similarity measures. Then as solid waste management is a global issue, and there are some Solid Waste Management Systems (SWMS) for environment protection, so an example will be given for the site selection for SWMS to check the validity of proposed techniques. To verify the validity and superiority of the suggested work, it is contrasted to several existing methodologies, which show that decision-making issues with more bifurcation attributes provide more accurate and precise outcomes and can only be solved using this technique. In the future, the presented methodologies could be used in case studies with several qualities that are further bifurcated and multiple decision-makers. This proposed work can also be extended to many existing hypersoft set hybrids, such as Fuzzy hypersoft sets (FHSs), Intuitionistic hypersoft sets (IHSs), Bipolar hypersoft sets (Bi-HSs), m-polar HSs, and Pythagorean hypersoft sets (PHSs).

Intuitionistic fuzzy hypersoft sets (IFHSSs) are a novel model that is projected to address the limitations of Intuitionistic fuzzy soft sets (IFSSs) regarding the entitlement of a multi-argument domain for the approximation of parameters under consideration. It is more flexible and reliable as it considers the further classification of parameters into their relevant parametric valued sets. In this paper, we proposed some trigonometric (cosine and cotangent) similarity measures and their weighted trigonometric similarity measures (SMs). Trigonometric Similarity measures (SMs) for intuitionistic fuzzy hypersoft sets (IFHSSs) are significantly implied to check the similarity measures and help to determine the similarity between different factors. Also, in order to evaluate the validity of the significant study and apply the results to a daily life problem. We use them to solve problems involving the selection of renewable energy sources. According to several technical contributing factors, the analysis identifies the ideal location for the implementation of the energy production units. Future case studies with many features and additional bifurcation along with multiple decision-makers can use the suggested methodologies. Also, several existing structures, such as fuzzy, Pythagorean fuzzy, Neutrosophic theories, etc., can be utilized with the suggested method.

Decision‐making is a complex issue due to the vague, imprecise, and indeterminate environment especially when attributes are more than one and further bifurcated. To solve such types of problems, the concept of neutrosophic hypersoft set is proposed by Smaranndache. In this paper, the primary focus is to extend the concept of neutrosophic hypersoft sets (NHSs) to the neutrosophic hypersoft matrices (NHSMs) with the essential study of matrices with suitable examples. Then, the analytical study of some common operations for NHSM has been created. Lastly, decision‐making issues have been presented by establishing a new algorithm based on a score function, and it has been interpreted with the help of numerical example for the selection of teachers at the college level. In this study, NHSM algorithm is elaborated efficiently and conveniently for optimal choice selection to solve decision‐making problems.

The diagnosis of Type-1 diabetes is a challenging and sophisticated procedure for medical experts. The complexity of this condition necessitates the use of sophisticated decision-making tools, and in this setting, intuitionistic soft set theory and its accompanying matrices prove to be invaluable resources. Our suggested technique uses intuitionistic soft matrices to solve challenging multi-criteria decision-making issues, opening a promising new direction for improving the precision of diabetes diagnosis. Diabetes requires careful evaluation and assessment since it is characterized by several illnesses that interfere with the body's capacity to control blood plasma glucose levels. Our main goal is to use intuitionistic fuzzy soft matrices to thoroughly examine diabetic patients in the decision-making domain. By giving medical professionals more accurate tools to treat this pervasive and difficult health issue, this novel strategy has the potential to revolutionize the diagnosis and management of diabetes.

Pythagorean fuzzy hypersoft sets (PFHSSs) are a novel model that is projected to address the limitations of Pythagorean fuzzy soft sets (PFSSs) regarding the entitlement of a multi-argument domain for the approximation of parameters under consideration. It is more flexible and reliable as it considers the further classification of parameters into their relevant parametric valued sets. This article aims to be multi-faced. Firstly, several axiomatic properties, operational results, and aggregation operations on PFHSSs will be developed. Secondly, matrices are developed for PFHSSs, called Pythagorean fuzzy hypersoft matrices (PFHSMs). The essential basic properties and aggregation operations of PFHSMs are then characterized with the support of numerical examples. Thirdly, the matrix theory of PFHSSs is implemented in real-world decision-making scenarios for Mobile selection using the proposed choice matrix theory. At the end of the article, we go on a real-life problem for wastewater treatment. Wastewater treatment is crucial for preserving the environment and public health. It comprises purifying wastewater of contaminants and pollutants so that it may be utilized for other things or discharged safely into the environment. It is essential to protect the environment and the public health by removing toxins from domestic, industrial, and commercial sewages. We finally apply our proposed algorithm in the selection of wastewater treatment plants by employing the proposed algorithm based on PFHSMs. In fact, PFHSMs are flexible enough to be used in a wide range of fields, including image processing, expert systems, pattern recognition, and medical diagnosis. The future directions are discussed with these PFHSMs to develop MCDM techniques such as TOPSIS, VIKOR, and SAW so that they can be applied in a wider range of fields.