Set Of Fuzzy Numbers Research Articles

Dini’s Theorem for Fuzzy Number-Valued Continuous Functions

This work aims to provide several versions of Dini’s theorem for fuzzy number-valued continuous functions defined on a compact set K. In this context, there is a wide variety of possibilities since, unlike the real line, we can consider different topologies and orders on the set of fuzzy numbers. For example, we will show that the fuzzy Dini’s theorem holds for the usual partial orders and the most commonly used topologies but does not hold for all orders in general.

On fuzzy inner products constructed by fuzzy numbers in linear spaces

In this paper a new definition for fuzzy inner product spaces is presented. Just as the set of real numbers can be embedded in a set of fuzzy numbers, a crisp inner product space can be considered as a special case of fuzzy inner product spaces. An example is given to demonstrate that, the new definition is a nontrivial generalization for the crisp inner product spaces. Under certain restrictions, a fuzzy inner product space can become a fuzzy normed space. Based on some elementary properties for the families of semi-inner products of endpoints, the linearly topological structure of new spaces is discussed. Moreover, the orthogonality between two vectors is considered and a fuzzy version of Pythagorean theorem is given.

Decision-making techniques based on aggregation operators and similarity measures under q-rung orthopair hesitant fuzzy connection numbers and their application

The concept of q-rung orthopair hesitant fuzzy set represents an advancement and extension of hesitant fuzzy sets, encompassing both fuzzy sets and q-rung orthopair fuzzy sets. q-rung orthopair hesitant fuzzy set characterizes a set of membership and non-membership grades within the interval [0, 1], which enhances its adaptability compared to existing methods. This flexibility proves invaluable in providing more insightful data about various objects. The primary objective of this research is to introduce a decision-making technique in the context of q-RHF using the theory of set pair analysis (SPA). q-RHFS effectively handles ambiguous data by incorporating membership and non-membership grades, while the connection number (CN) based on SPA theory manages the intricacies of uncertainty and certainty structures by relying on "identity", "discrepancy" and "contrary" grades. Building on the relationship between q-RHFS and the connection number of set pair analysis, a comprehensive framework known as q-rung hesitant fuzzy connection number set (qHCNs) is developed. This model not only addresses uncertainty, but also offers valuable insights. Furthermore, this research introduces similarity measures derived from qHCN and examines their advantages through illustrative examples. Additionally, a novel approach to decision modeling utilizing these measures applied to medical diagnosis is also introduced. The application of this established model contributes to an effective approach and demonstrated its soundness and efficiency. In addition, a detailed comparative study is conducted with the existing models and advantages of proposed model. The research concludes with a summary of the authors' findings, highlighting the consistency and effectiveness of their work.

First-order rule partitions-based decomposition technique of type-1 and interval type-2 rule-based fuzzy systems for computational and memory efficiency

Rule-based fuzzy systems are widely adopted in various fields for their effectiveness in reducing uncertainties. However, large rule bases present significant computational and memory challenges, especially for real-time applications. To overcome this limitation, this paper introduces a novel first-order rule partitions-based decomposition technique (FORPs-DT) for type-1 (T1) and interval type-2 (IT2) fuzzy logic systems (FLSs). This method involves decomposing the universe of discourse, membership functions (MFs), and fuzzy sets (FSs) by defining conditions for non-zero firing levels (intervals). Significantly, this approach allows the construction of subfuzzy systems with separate knowledge bases while maintaining the input–output relationship. The paper provides detailed explanations and visual representations of FORPs-DT and proposes a case of identical FORPs to illustrate its simplicity and resource optimization benefits. Extensive experiments demonstrate the superiority of this technique, showcasing not only reduced execution time and memory usage but also enhanced performance through identical partitions, which allows an increase in the number of FSs without additional computational or memory overhead.

On a total order on the set of Z-numbers based on discrete fuzzy numbers

Z-numbers were introduced by Zadeh in 2011 as a pair of fuzzy numbers (A, B), where A is interpreted as a fuzzy restriction on the values of a variable, while B is interpreted as a measure of certainty or sureness of A. From the initial proposal, several other approaches have been introduced in order to reduce the computational cost of the involved operations. One of such approaches is called discrete Z-numbers where A and B are modelled as discrete fuzzy numbers. In this paper, the construction of total orders on the set of discrete Z-numbers is investigated for the first time. Specifically, the total order is designed for discrete Z-numbers where the second component has membership values belonging to a finite and prefixed set of values. The method relies on solid and coherent linguistic criteria and several linguistic properties are analyzed. The order involves the transformation of the first components of the discrete Z-numbers by using the credibility of the second components in the sense that a lower credibility enlarges in a greater extent the uncertainty of the first component. Then a total order on the set of discrete fuzzy numbers is applied. Finally, a practical example on how to order discrete Z-numbers is presented and a comparison with other ranking methods is performed from which the strengths of our method are stressed.

Strategic location analysis for offshore wind farms to sustainably fulfill railway energy demand in Turkey

Providing the energy demands of various sectors through the utilization of renewable energy sources offers a more sustainable and long-term approach compared to fossil fuel consumption. In this study, the proposition is made to address Turkey's rail industry energy demand through Offshore Wind Farms (OWF). Accordingly, the spatial analysis for determining the most suitable OWF location is explored. The research methodology is conducted based on a Geographic Information System-Multi-Criteria Decision Making (GIS-MCDM) hybrid approach. Alternative OWF locations are identified using GIS. Location selection criteria are established based on literature review, and a fuzzy-based decision matrix is constructed, incorporating expert opinions. To leverage two distinct fuzzy number sets, Intuitionistic Fuzzy Weighted Averaging (IFWA) and Spherical Fuzzy Stepwise Weight Assessment Ratio Analysis (SF-SWARA) methods are employed for criteria weighting, followed by the aggregation of criterion weights using the Einstein Operations t-norm technique. To ensure comparable OWF alternative rankings, five different alternative ranking methods are employed to obtain alternative rankings. Ultimately, the most suitable OWF location for meeting the energy demand of the rail industry is determined to be the Izmir region. The research findings contribute to managerial perspectives by proposing recommendations concerning strategic location selection, optimized resource allocation, risk mitigation and policy alignment and sustainable long-term planning.

An improved method to estimate the similarity between LR-type fuzzy numbers

This paper presents an improved method for estimating the similarity between LR-type fuzzy numbers. By considering different left and right shape functions, we can conveniently compute the similarity between trapezoidal fuzzy numbers, Gaussian fuzzy numbers and normal fuzzy numbers. An algorithm for computing the proposed similarity measure between LR-type fuzzy numbers and a numerical method for approximating it are provided. Some of its properties are also examined. The performance of the proposed method is compared with some existing similarity measures through using different sets of LR-type fuzzy numbers. It is shown that the similarity between LR-type fuzzy numbers with parameters from any real number belonged to the real number set R is considered in the proposed method. The given similarity measure is scale-invariant and takes into account the shape of LR-type fuzzy numbers, which can overcome the shortcomings of existing methods.

APPROXIMATION OF FUZZY NUMBERS BY FAVARD-SZASZMIRAKYAN OPERATORS OF MAX-PRODUCT

– In this study, the definition of the generalized Favard-Szasz-Mirakyan operators of max-productkind is extended to an arbitrary compact interval, by proving that their order of uniform approximation isthe same as in the particular case of the unit interval. We will show that if fuzzy numbers are presented inparametric form, the generalized Favard-Szasz-Mirakyan maximum product operator produces a set offuzzy numbers such that it is approximately the same as the uncertainty and value of the fuzzy number.

Quasilineability and topological properties of the set of fuzzy numbers

In this paper we show that the cardinality of the set of fuzzy numbers coincides with that of the real numbers. We also show that the set of triangular fuzzy numbers is nowhere dense within the set of fuzzy numbers (with a suitable distance) and that the set of real numbers is also nowhere dense within the set of triangular fuzzy numbers. In addition, we introduce the concept of quasilineability and study the set of bounded fuzzy number sequences that do not have a lower limit and that of monotonic decreasing, bounded with respect a partial ordering and not convergent.

Nonlinear distance measures under the framework of Pythagorean fuzzy sets with applications in problems of pattern recognition, medical diagnosis, and COVID-19 medicine selection

BackgroundThe concept of Pythagorean fuzzy sets (PFSs) is an utmost valuable mathematical framework, which handles the ambiguity generally arising in decision-making problems. Three parameters, namely membership degree, non-membership degree, and indeterminate (hesitancy) degree, characterize a PFS, where the sum of the square of each of the parameters equals one. PFSs have the unique ability to handle indeterminate or inconsistent information at ease, and which demonstrates its wider scope of applicability over intuitionistic fuzzy sets.ResultsIn the present article, we opt to define two nonlinear distances, namely generalized chordal distance and non-Archimedean chordal distance for PFSs. Most of the established measures possess linearity, and we cannot incorporate them to approximate the nonlinear nature of information as it might lead to counter-intuitive results. Moreover, the concept of non-Archimedean normed space theory plays a significant role in numerous research domains. The proficiency of our proposed measures to overcome the impediments of the existing measures is demonstrated utilizing twelve different sets of fuzzy numbers, supported by a diligent comparative analysis. Numerical examples of pattern recognition and medical diagnosis have been considered where we depict the validity and applicability of our newly constructed distances. In addition, we also demonstrate a problem of suitable medicine selection for COVID-19 so that the transmission rate of the prevailing viral pandemic could be minimized and more lives could be saved.ConclusionsAlthough the issues concerning the COVID-19 pandemic are very much challenging, yet it is the current need of the hour to save the human race. Furthermore, the justifiable structure of our proposed distances and also their feasible nature suggest that their applications are not only limited to some specific research domains, but decision-makers from other spheres as well shall hugely benefit from them and possibly come up with some further extensions of the ideas.

Scenario derivation and consequence evaluation of dust explosion accident based on dynamic Bayesian network

In order to clarify the correlation between the evolution path of dust explosion accidents and emergency decision-making, and to accurately predict the disaster damage levels of various disaster bearing bodies. This paper extracts 56 key scenario elements from four aspects, namely state, answer, goal and environment, based on the analysis of typical dust explosion accident cases. At the same time, a general scenario evolution path of dust explosion accident is constructed. Using fuzzy number set theory and dynamic Bayes joint probability model, the accurate solution of scenario state probability was realized. With the help of accident cases and dynamic Bayes approach, the dust explosion consequence prediction index system and evaluation criteria were constructed, covering factors such as dust explosion intensity, casualties, direct economic losses, equipment damage, building damage, environmental damage and other factors. A polyethylene wax dust explosion accident in a city of China was used to verify the dust explosion accident scenario evolution model and consequences prediction model. The predicted results were in good agreement with the actual damage of various carriers of the accident, which indicated that the model could be used for dust explosion accident prediction and disaster loss prediction. The research results provided reference and technical support for the prediction of dust explosion accident evolution direction, emergency aid measures decision and deployment, disaster damage prediction and evaluation.

Chebyshev Distance Entropy Combined with TODIM Method in Lq-ROFS Multiattribute Group Decision Making

Linguistic q-rung orthopair fuzzy numbers (Lq-ROFNs) are composed of a set of q-rung orthopair fuzzy numbers (q-ROFNs) with membership and nonmembership degrees of linguistic variables, which can be regarded as an extension of linguistic intuitionistic fuzzy numbers (LIFNs) and linguistic Pythagorean fuzzy numbers (LPFNs). Compared with LIFNs and LPFNs, Lq-ROFNs have a wider range of applicability because the value of q can have a wider range, thus allowing to handle more multiattribute group decision making (MAGDM) problems. Based on Lq-ROFS, this paper first proposes the Chebyshev distance metric, then develops the Chebyshev distance entropy for deriving objective solution of decision makers’ (DMs’) weight vector and attribute weight vector by combining the metric and entropy measure. The TODIM method has been widely valued by the society and has achieved good results in many MAGDM problems. Subsequently, the TODIM decision method in Lq-ROFS is presented and combined with the Chebyshev distance entropy model as a new decision method (CDE-TODIM) to solve the MAGDM problems. This method process is objective and direct and takes into account the decision maker’s preferences, and the decision results are more persuasive. Finally, the effectiveness and rationality of this MAGDM method are illustrated by a case as well as comparisons.

Multiple-Attribute Decision Making Based on Intuitionistic Hesitant Fuzzy Connection Set Environment

The intuitionistic hesitant fuzzy set (IHFS) is an enriched version of hesitant fuzzy sets (HFSs) that can cover both fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). By assigning membership and non-membership grades as subsets of [0, 1], the IHFS can model and handle situations more proficiently. Another related theory is the theory of set pair analysis (SPA), which considers both certainties and uncertainties as a cohesive system and represents them from three aspects: identity, discrepancy, and contrary. In this article, we explore the suitability of combining the IHFS and SPA theories in multi-attribute decision making (MADM) and present the hybrid model named intuitionistic hesitant fuzzy connection number set (IHCS). To facilitate the design of a novel MADM algorithm, we first develop several averaging and geometric aggregation operators on IHCS. Finally, we highlight the benefits of our proposed work, including a comparative examination of the recommended models with a few current models to demonstrate the practicality of an ideal decision in practice. Additionally, we provide a graphical interpretation of the devised attempt to exhibit the consistency and efficiency of our approach.

An Average Based Method for Finding the Basic Feasible Solution for the Fuzzy Transportation Problems

Fuzzy transport problems are another special type of transport problem (TP). In a transportation problem, what is primarily considered is how to carry out the relevant process while reduce the total cost of the transporting the goods to different destinations. This objective is also valid for fuzzy TP. However, the supply quantity, demand, and unit cost values cannot be determined precisely, and those values are represented by "fuzzy number sets." There, the relevant solution value is obtained as a basic solution or an optimal solution. Thus, various researchers have proposed various algorithms to obtain an efficient initial solution or an optimal solution (OS) to fuzzy transportation problems. Accordingly, in this research article, we have presented another method to obtain an basic feasible solution (BFS) value for fuzzy transportation problems. It is prepared by creating a new value for each cell based on Yager's robust ranking method. In obtaining these values, the average of the crisp values of the columns and rows of the relevant column or row was basically considered. After that, the algorithm was used to solve mathematical problems. In here, the proposed method is primarily considered for triangular and trapezoidal fuzzy transportation problems. Also, the basic solution obtained from those solutions was that algorithm and the current approach are compared, and the efficiency and correctness of the proposed method were tested. Based on the analysis of the obtained data, the new method can be shown as an easy method to understand the efficiency that can be used to obtain the BFS to fuzzy transportation problems.

Study on intelligent fuzzy assessment of failure effects for marine power plants

Due to its advantages like zero carbon dioxide emissions, high power density, and long life cycle, power devices show great application prospects in civil cargo ships and icebreakers. The equipment composition and structure configuration of power plants are complex, with harsh operating conditions, which will yield to increased failure rates and serious security threats. Given that the relative importance relying on precise data, an intelligent assessment method of failure effects based on fuzzy logic is proposed. Marine power plant hierarchy division is descripted, suitable for the failure effect assessment process. The fuzzy sets and fuzzy numbers are used to get the fuzzy evaluations and relative weights. The alpha-level sets of fuzzy risk priority numbers (RPNs) are calculated with the proposed benchmark adjustment search strategy. Meanwhile, using the synthesized alpha-level set and centroid defuzzification, the fuzzy RPNs are defuzzied. The comparative analysis of examples reveals that the suggested assessment method can improve the confidence level of risk prioritization for marine power plants.

Interval Type-2 Fuzzy Differential Equations and Stability

This article is devoted to present a framework for studying type-2 fuzzy calculus. Some new concepts pertaining to interval type-2 fuzzy functions (IT2FF) and interval type-2 fuzzy differential equations (IT2FDE) with their stability analysis are introduced. A new definition of type-2 fuzzy derivative called type-2 granular derivative (T2gr-derivative) is defined to overcome the disadvantages of the previous definitions of type-2 fuzzy derivatives. Furthermore, the definitions of elementary arithmetic on the set of interval type-2 fuzzy numbers (IT2FNs) and the set of interval type-2 fuzzy complex numbers are presented. The notion of granular metric on the set of IT2FNs is also defined. Besides, the concepts of continuity, integral, and limit of IT2FFs are introduced. Using the concept of type-2 granular fuzzy Laplace transform and the approximations of T2gr-derivative proposed in this article, the exact and approximate solutions of interval T2FDE are presented. Then, some new concepts such as fuzzy node, fuzzy saddle spiral point, fuzzy star, and so forth lay the groundwork for the analysis of interval T2FDE stability. Several examples are given to illustrate the presented notions.

On fuzzy paranormed spaces

It is well-known that Paranormed spaces have more general properties than normed spaces. Until recently, the concept of paranorm has not been considered with the set of fuzzy numbers and there are no studies in literatüre from this point of view. So, our main aim in this work is to apply this notion to the set of fuzzy numbers and give the new concepts of fuzzy paranorm and fuzzy paranormed space. In addition, we discuss in detail the statistical convergence and statistically Cauchy sequence in a fuzzy paranormed space.

Multivariate fuzzy neural network interpolation operators and applications to image processing

In this paper, we introduce a novel family of multivariate fuzzy neural network interpolation operators activated by sigmoidal functions belonging to the new class of multivariate sigmoidal functions. To present an alternative way to the well-known shortcomings of the Hukuhara difference, we use a proper function defined on a set of fuzzy n-cell numbers. Moreover, we construct the Kantorovich variant of fuzzy NN interpolation operators, and also achieve approximation properties via Lp-type metric with respect to both modulus of continuity and Lp-modulus of continuity in fuzzy sense. Various special examples for the class of multivariate sigmoidal functions are presented. Also, we give some illustrative examples to demonstrate the approximation performances of all the above operators. Finally, we give a novel interpolation algorithm involving a multidimensional fuzzy inference system with applications in color image resizing and inpainting.

The topological structure of the set of fuzzy numbers with the supremum metric

We study the family of all fuzzy sets of the n-dimensional Euclidean space, which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in a non-degenerate convex subset Y, and prove that the following statements are equivalent: (i) The family of fuzzy sets with the topology induced by the supremum metric is homeomorphic to a non-separable Hilbert space whose weight is the cardinality of the set of all real numbers; (ii) the non-degenerate convex subset Y is topologically complete, or equivalently, it is a countable intersection of open sets in the n-dimensional Euclidean space.

The topological structure of the set of fuzzy numbers with Lp metric

For a non-degenerate convex subset Y of the n-dimensional Euclidean space Rn, let K(Y) be the family of all fuzzy sets in Rn, which are upper-semicontinuous, fuzzy convex, normal and with compact supports included in Y. We show that the space K(Y) with the topology of the Lp metric with p≥1, that is, for f,g∈K(Y),Lp(f,g)=(∫01(dH([f]α,[g]α))pdα)1p, is homeomorphic to the Hilbert cube Q=[−1,1]ω if and only if Y is compact; and the space K(Y) is homeomorphic to the pseudoboundary B(Q)=Q∖(−1,1)ω of the Hilbert cube Q=[−1,1]ω if and only if Y is non-compact and locally compact.