A topological structure involving hesitant fuzzy sets

Some Results on Submodules Using (μ,ν,ω)-Single-Valued Neutrosophic Environment

The use of a single-valued neutrosophic set (svns) makes it much easier to manage situations in which one must deal with incorrect, unexpected, susceptible, faulty, vulnerable, and complicated information. This is a result of the fact that the specific forms of material being discussed here are more likely to include errors. This new theory has directly contributed to the expansion of both the concept of fuzzy sets and intuitionistic fuzzy sets, both of which have experienced additional development as a direct consequence of the creation of this new theory. In svns, indeterminacy is correctly assessed in a way that is both subtle and unambiguous. Furthermore, membership in the truth, indeterminacy, and falsity are all completely independent of one another. In the context of algebraic analysis, certain binary operations may be regarded as interacting with algebraic modules. These modules have pervasive and complicated designs. Modules may be put to use in a wide variety of different applications. Modules have applications in a diverse range of industries and market subsets due to their adaptability and versatility. Under the umbrella of the triplet (μ,ν,ω) structure, we investigate the concept of svns and establish a relationship between it and the single-valued neutrosophic module and the single-valued neutrosophic submodule, respectively. The purpose of this study is to gain an understanding of the algebraic structures of single-valued neutrosophic submodules under the triplet structure of a classical module and to improve the validity of this method by analyzing a variety of important facets. In this article, numerous symmetrical features of modules are also investigated, which demonstrates the usefulness and practicality of these qualities. The results of this research will allow for the successful completion of both of these objectives. The tactics that we have devised for use in this article are more applicable to a wide variety of situations than those that have been used in the past. Fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets are some of the tactics that fall under this category.

An approach to [Formula: see text]-single-valued neutrosophic submodules

Dealing with erroneous, unexpected, susceptible, flawed, vulnerable, and intricate information is simplified with the use of a single-valued neutrosophic set (svns). This is because of the fact that these types of information are more sensitive to error. This is due to the fact that these particular kinds of information are more prone to error. The ideas of fuzzy sets and intuitionistic fuzzy sets have both undergone further development as a direct result of the development of this new theory. In svns, indeterminacy is quantified in a way that is both obvious and unambiguous, and truth membership, indeterminacy membership, and falsity membership are all completely independent of one another. In algebraic analysis, certain binary operations can be thought of as interacting with algebraic modules. These modules are intricate and ubiquitous structures. There are many different applications for modules to be used in. Modules find use in an extremely wide variety of different kinds of businesses and market segments. We investigate the idea of [Formula: see text]-svns and relate it to [Formula: see text]-single-valued neutrosophic module and [Formula: see text]-single-valued neutrosophic submodule, respectively. The goals of this research are to comprehend the algebraic structures of a [Formula: see text]-single-valued neutrosophic submodule of a classical module and enhance the legitimacy of this technique by discussing numerous essential aspects. Both of these goals will be accomplished through the course of this study. The strategies that we have developed in this manuscript are more generalizable than those that have been utilized in the past. These strategies include fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets.

Topological Data Analysis with Cubic Hesitant Fuzzy TOPSIS Approach

A hesitant fuzzy set (HFS) and a cubic set (CS) are two independent approaches to deal with hesitancy and vagueness simultaneously. An HFS assigns an essential hesitant grade to each object in the universe, whereas a CS deals with uncertain information in terms of fuzzy sets as well as interval-valued fuzzy sets. A cubic hesitant fuzzy set (CHFS) is a new computational intelligence approach that combines CS and HFS. The primary objective of this paper is to define topological structure of CHFSs under P(R)-order as well as to develop a new topological data analysis technique. For these objectives, we propose the concept of “cubic hesitant fuzzy topology (CHF topology)”, which is based on CHFSs with both P(R)-order. The idea of CHF points gives rise to the study of several properties of CHF topology, such as CHF closure, CHF exterior, CHF interior, CHF frontier, etc. We also define the notion of CHF subspace and CHF base in CHF topology and related results. We proposed two algorithms for extended cubic hesitant fuzzy TOPSIS and CHF topology method, respectively. The symmetry of optimal decision is analyzed by computations with both algorithms. A numerical analysis is illustrated to discuss similar medical diagnoses. We also discuss a case study of heart failure diagnosis based on CHF information and the modified TOPSIS approach.

Hesitant fuzzy C-means algorithm and its application in image segmentation†

Image segmentation plays an important role in many fields such as computer vision, pattern recognition, machine learning and so on. In recent years, many variants of standard fuzzy C-means (FCM) algorithm have been proposed to explore how to remove noise and reduce uncertainty. In fact, there are uncertainty on the boundary between different patches in images. Considering that hesitant fuzzy set is a useful tool to deal with uncertainty, in this paper, we merge hesitant fuzzy set with fuzzy C-means algorithm, introduce a new kind of method of fuzzification and defuzzification of image and the distance measure between hesitant fuzzy elements of pixels, present a method to establish hesitant membership degree of hesitant fuzzy element, and propose hesitant fuzzy C-means (HFCM) algorithm. Finally, we compare our proposed HFCM algorithm with some existing fuzzy C-means (FCM) algorithms, and apply HFCM algorithm in natural image, BSDS dataset image, different size images and multi-attribute decision making. These numerical examples illustrate the validity and applicability of our proposed algorithm including its comprehensive performance, reducing running time and almost without loss of accuracy.

Solar energy plant project selection with AHP decision-making method based on hesitant fuzzy linguistic evaluation

Increased energy demand is expected to be met by reliable and continuous energy sources. Renewable energy which is obtained from nature and can continuously reload itself from natural sources is a new generation energy type. The sun, which is the main source of renewable energies and produces heat and electricity by direct and indirect methods, is an important renewable energy source. The installation of solar energy systems takes place under the basic technical, economic and political factors. Alternative solar energy plant projects are evaluated linguistically under the main criteria based on the knowledge and experience of the experts. Hesitant fuzzy linguistic terms are used to incorporate the uncertain and hesitant expressions into the decision-making process. The decision-making process that takes place with hesitant linguistic expressions in multiple sub-criteria is based on the AHP model. The inclusion of hesitant statements in the decision-making process with the AHP model enables more realistic choices among the alternatives. System technology (0.18), energy policy (0.15) and energy price change (0.13) appear as the most important factors in the pairwise comparison of the factors based on hesitant fuzzy linguistic evaluations. The results coincide with the need for high efficiency in solar energy systems, the importance of governmental supportive policies and the effects of price competition in the energy sector. Also, the closeness of the overall priority values of all projects (0.189, 0.23, 0.287, 0.135, 0.158) indicates that the decision makers take into account the effective factors.

Hesitant Fuzzy Topological Spaces

In this study, we define a hesitant fuzzy topology and base, obtain some of their properties, respectively, and give some examples. Next, we introduce the concepts of a hesitant fuzzy neighborhood, Q-neighborhood, closure, and interior and obtain some of their properties, respectively. Furthermore, we define a hesitant fuzzy continuous mapping and investigate some of its properties. Furthermore, we define a hesitant fuzzy subspace and obtain some of its properties. In particular, we obtain the Pasting lemma. We investigate the concept of hesitant fuzzy product space and study some of its properties.

Evaluation and Selection of HazMat Transportation Alternatives: A PHFLTS- and TOPSIS-Integrated Multi-Perspective Approach

Outsourcing the hazardous materials (HazMat) transportation is an effective way for manufacturing enterprises to avoid risks and accidents as well as to retain sustainable development in economic growth and social inclusion while not bringing negative impacts on the public and the environment. It is imperative to develop viable and effective approaches to selecting the most appropriate HazMat transportation alternatives. This paper aims at proposing an integrated multi-criteria group decision making approach that combines proportional hesitant fuzzy linguistic term set (PHFLTS) and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) to address the problem of HazMat transportation alternative evaluation and selection. PHFLTSs are adopted to represent the congregated individual evaluations in a bid to avoid information loss and increase the reliability of results. Two weight assignment models are then proposed to determine the comprehensive weights of experts and criteria. Furthermore, several novel manipulations of PHFLTS are also defined to enrich its applicability. The TOPSIS method is subsequently extended to the context of PHFLTSs to rank alternatives and choose the best one. Eventually, the feasibility and validity of the proposed approach are verified by a practical case study of a HazMat transportation alternative evaluation and selection decision and further comparison analyses.