Operations Of Fuzzy Numbers Research Articles

Examining multiplication and division on fuzzy numbers using composite tables

A fuzzy number is a concept that represents a real number abstractly. It doesn’t assign a single value, but instead a range of uncertain values, each with its own weight between 0 and 1. It is essential to comprehend the arithmetic operations of fuzzy numbers in the real number of fuzzy mathematics. One of the fundamental concepts that should not be ignored is the explanation of the interval operation. In this paper, a composition table is constructed to demonstrate the multiplication and division of fuzzy numbers. Fuzzy arithmetic operations are commonly utilized to solve mathematical equations that involve fuzzy numbers. In this paper, we explore the relationship between multiplication and division operations and fuzzy numbers. We also demonstrate that the shape of their membership functions heavily influences the outcome of our calculations when using fuzzy numbers.

Group decision making based on linguistic interval-valued Atanassov intuitionistic fuzzy Yager weighted arithmetic aggregation operator of linguistic interval-valued Atanassov intuitionistic fuzzy numbers

In this paper, we propose a noval addition operation (AOP) and a novel scalar multiplication operation (SMOP) of linguistic interval-valued Atanassov intuitionistic fuzzy numbers (LIVAIFNs) based on Yager’s t-conorm and t-norm. The proposed AOP and the proposed SMOP of LIVAIFNs can conquer the shortcomings of the existing AOP and the existing SMOP of LIVAIFNs. Based on the proposed AOP and the proposed SMOP of LIVAIFNs, we propose the linguistic interval-valued Atanassov intuitionistic fuzzy Yager weighted arithmetic (LIVAIYFWA) aggregation operator (AO) of LIVAIFNs. We also prove some properties of the proposed LIVAIFYWA AO of LIVAIFNs. Moreover, by using the proposed LIVAIFYWA AO of LIVAIFNs, we propose a noval group decision making (GDM) method. The proposed GDM method can conquer the drawbacks of the existing GDM methods in the context of LIVAIFNs.

The Inverse and General Inverse of Trapezoidal Fuzzy Numbers with Modified Elementary Row Operations

Trapezoidal positive/negative fuzzy numbers have no single definition; instead, various authors define them in relation to different concepts. This means that arithmetic operations for trapezoidal fuzzy numbers also differ. For the operations of addition, subtraction, and scalar multiplication, there are not many differences; for multiplication, however, there are many differences. In general, multiplication is divided into various cases. For the inverse operation, there is not much to define; in general, for any trapezoidal fuzzy number u~, u~⊗1u~=i~=(1,1,0,0) does not necessarily apply. As a result of the different arithmetic operations for multiplication and division employed by various authors, several researchers have tackled the same problem and reached different solutions, meaning that the application will also produce different results. To date, many authors have proposed various alternatives for the algebra of the trapezoidal fuzzy number. In this paper, using the parametric form approach to trapezoidal fuzzy numbers, an alternative to multiplication with only one formula is constructed for various cases. Furthermore, based on the definition of multiplication for any trapezoidal fuzzy number, u~ is constructed 1u~ so that u~⊗1u~=i~=(1,1,0,0). Based on these conditions, we show that various properties that apply to real numbers also apply to any trapezoidal fuzzy number. Furthermore, we modify the elementary row operational steps for the trapezoidal fuzzy number matrix, which can be used to determine the inverse of a trapezoidal fuzzy number matrix with the order m×m. We also give the steps and examples necessary to determine the general inverse for a trapezoidal fuzzy number matrix of the order m×n with m ≠n. This ability to easily determine the inverse and general inverse of a trapezoidal fuzzy number matrix has a number of applications, such as solving fully trapezoidal fuzzy number linear systems and fuzzy transportation problems, especially in applications in fields outside of mathematics; for example, the application of triangular fuzzy numbers in medical problems is a topic currently receiving a significant amount of attention.

Special Discrete Fuzzy Numbers on Countable Sets and Their Applications

There are some drawbacks to arithmetic and logic operations of general discrete fuzzy numbers, which limit their application. For example, the result of the addition operation of general discrete fuzzy numbers defined by the Zadeh’s extension principle may not satisfy the condition of becoming a discrete fuzzy number. In order to solve these problems, special discrete fuzzy numbers on countable sets are investigated in this paper. Since the representation theorem of fuzzy numbers is the basic tool of fuzzy analysis, two kinds of representation theorems of special discrete fuzzy numbers on countable sets are studied first. Then, the metrics of special discrete fuzzy numbers on countable sets are defined, and the relationship between these metrics and the uniform Hausdorff metric (i.e., supremum metric) of general fuzzy numbers is discussed. In addition, the triangular norm and triangular conorm operations (t-norm and t-conorm for short) of special discrete fuzzy numbers on countable sets are presented, and the properties of these two operators are proven. We also prove that these two operators satisfy the basic conditions for closure of operation and present some examples. Finally, the applications of special discrete fuzzy numbers on countable sets in image fusion and aggregation of subjective evaluation are proposed.

Arithmetic Operations of Trigonal Fuzzy Numbers Using Alpha Cuts: A Flexible Approach to Handling Uncertainty

Arithmetic Operations of Trigonal Fuzzy Numbers Using Alpha Cuts: A Flexible Approach to Handling Uncertainty

Version [1.1]- [pyFDM: A python library for uncertainty decision analysis methods

This paper describes an extension of the pyFDM library by extending the functionalities of the programming package with new Fuzzy Multi-Criteria Decision Analysis methods such as Combined Compromise Solution (CoCoSo), Stable Preference Ordering Towards Ideal Solution (SPOTIS), Weighted Aggregated Sum Product ASSessment (WASPAS), Weighted Product Model (WPM), and Weighted Sum Model (WSM). Furthermore, the proposed update includes new normalization techniques, new distance metrics, and defuzzification methods to allow users to examine the greater variety of research approaches in a Triangular Fuzzy environment. Furthermore, the update also extends with the implementations allowing for fuzzy calculations outside the multi-criteria field, which is allowed by the implementation of selected mathematical operations for Triangular Fuzzy Numbers. The opportunity to visualize the data within the fuzzy extension considered is also included. The extension of the pyFDM package broadens the scope of its usage in research approaches, making it a more suitable tool for multi-criteria decision analysis within the fuzzy environment.

A Brief Analysis and Interpretation on Arithmetic Operations of Fuzzy Numbers

Fuzzy set theory is a generalized form of crisp set theory where elements are binary inclusion forms. In fuzzy set, it differs with degree of membership for every element in the set. There are several strategies for arithmetic operations on fuzzy numbers. Previous studies show that there are many approaches, such as the α−cut technique, extension principle, vertex method, etc., to execute arithmetic operations on fuzzy numbers. In this study we perform details analysis and interpretation on arithmetic operations based on the α−cut method in a new way.

A generalized and unified approach to the approximation of fuzzy numbers and its arithmetic and characteristics

In this paper we propose a general method for the approximation of an arbitrary fuzzy number. This method, which is constructive, recovers and properly extends some well-known approximations such as those obtained in terms of polygonal fuzzy numbers or simple fuzzy numbers. We prove the convergence of the general method and study the properties of the approximation operator, such as its compatibility with arithmetic operations of fuzzy numbers and with some of their important characteristics. In addition to this, we illustrate the method with some particularly interesting cases by providing algorithms, of great simplicity for practical use and apply them to some numerical examples. Furthermore, the approximations we construct are particularly simple from the point of view of fuzzy arithmetic and preserve some of their most important characteristics.

On n-polygonal interval-valued fuzzy numbers and application in e-commerce risk assessment

The polygonal interval-valued fuzzy number is constructed based on the polygonal fuzzy number and the interval-valued fuzzy number. Its main feature is that the linear operation of finite ordered points reduces the complexity of traditional interval-valued fuzzy number operations. This research presents a generalized distance formula between two polygonal interval-valued fuzzy numbers and explores topological properties under the distance of polygonal interval-valued fuzzy numbers. In addition, we adopt the TOPSIS (technique for order preference by similarity to an ideal solution) and prospect theory approach for the multi-attribute decision-making problem. The information of attributes describes with polygonal interval-valued fuzzy numbers, and we then implement optimized ranking on the alternatives according to the profit and loss ratio. Finally, we verify the effectiveness and practicability of the decision-making method and fuzzy numbers at polygonal interval-valued fuzzy numbers in e-commerce risk assessment.

Fractional derivatives of fuzzy complex-variable functions and application to fuzzy fractional complex differential equations

In this paper, through theory of fuzzy complex analysis, we introduce some representations of r-cuts of trivial fuzzy zero, nontrivial fuzzy zero and fuzzy complex numbers in polar form on complex plane and establish some of their basic properties. Based on this, the addition and scalar multiplication operations of fuzzy complex numbers are established and the calculus of fuzzy complex-variable functions is studied. Moreover, we introduce a notion of fuzzy Ortigueira fractional derivative of fuzzy complex-variable functions and develop a theory of fuzzy fractional calculus on complex plane including some geometric features. As applications, the existence and uniqueness theorem of solutions for nonlinear fuzzy fractional complex-variable differential equations is established and some examples are provided.

Research on Green Consumption Based on Visual Evaluation Method—Evidence from Stone Flooring Industry

Blind consumption discovered in the real application of flooring does not produce the visually anticipated impression, leading to additional time costs and stone flooring waste. Consumers cannot clearly articulate their visual imaging needs when purchasing stone flooring. Due to consumers’ lack of understanding of the visual imagery style of decorative stone flooring, manufacturers are unable to produce more visual styles of stone flooring in response to consumer demand, which leads to an unorganized production process and the wasting of stone resources. Additionally, manufacturers are unable to receive feedback on market demand, which makes communication links between sales teams difficult. A total of 40 adjectives were considered the most appropriate in a pool of 110 adjectives for the visual imagery evaluation of stone after ten interior design professionals with experience in decorative stone applications had narrowed the selection. Following this, a general consumer semantic difference method questionnaire survey and questionnaire data factor analysis statistics were used to create 10 sets of visual imagery adjectives for marble flooring, which were then divided into 10 different types of marble flooring. Following the computer simulation drawing with the 10 groups of visual imagery adjectives design questionnaire, the consumers completed the visual imagery evaluation questionnaire survey. They received a 304-question valid questionnaire, and using the triangular fuzzy number operation in fuzzy theory, they arrived at 10 marble floors in the visual imagery evaluation score. In order to clarify the current consumer demand for stone floor imagery, the high sales volume of stone flooring on the market for visual style division, which can guide consumers according to their visual needs for an efficient choice, can enhance the efficiency of communication between consumers and sellers. It can also help enterprises clarify the market consumer demand for orderly production to achieve the purpose of green consumption and to ensure the sustainable development of the decorative stone flooring market.

Performance Evaluation of Healthcare Supply Chain in Industry 4.0 with Linear Diophantine Fuzzy Sine-Trigonometric Aggregation Operations

The concept of linear Diophantine fuzzy set (LDFS) theory with its control parameters is a strong model for machine learning and data-driven multi-criteria decision making (MCDM). The sine-trigonometric function (STF) has two significant features, periodicity and symmetry about the origin that are very useful tools for information analysis. Keeping in view the characteristics of both STF and LDFS theory, this article introduces the sine-trigonometric operations for linear Diophantine fuzzy numbers (LDFNs). These operational laws lay a foundation for developing new linear Diophantine fuzzy sine-trigonometric aggregation operators (LDFSTAOs). The integration of Industry 4.0 technology into healthcare has the potential to revolutionize patient care. One of the most challenging tasks is the selection of efficient suppliers for the healthcare supply chain (HSC). The traditional suppliers are not efficient in accordance with Industry 4.0, with particular uncertainties. A new MCDM framework is presented based on LDFSTAOs to examine the HSC performance in industry 4.0. A credibility test, sensitivity analysis and comparative analysis are performed to express the novelty, reliability, and efficiency of the proposed methodology.

The relationship of three difference operations for fuzzy numbers to three kinds of derivative

The difference operation for fuzzy number is an essential concept for the fuzzy set theory. There are several differences proposed: generalized difference, generalized Hukuhara difference and granule difference. Based on these differences, generalized differentiability, generalized Hukuhara differentiability and granule differentiability are also proposed, respectively. In this paper, the relations among these three kinds of differences and that of related three kinds of differentiability are clarified.

The Fuzzy Complex Linear Systems Based on a New Representation of Fuzzy Complex Numbers

Since the product of complex numbers and rectangular fuzzy complex numbers (RFCN) is not necessarily a RFCN in the former fuzzy complex linear system (FCLS), the scalar multiplication and addition operations of complex numbers and fuzzy complex numbers (FCN) based on a new representation of FCN are proposed. We also introduce a new method for solving FCLS, which can convert FCLS into two distinct linear systems. One is an n×n complex linear system, and the other is an (mn)×(mn) real linear system, where n is the number of unknown variables, and m is the number of substitutional cyclic sets composed of coefficients of FCLS. In particular, using this method to solve one-dimensional fuzzy linear systems, a (2n)×(2n) RLS is obtained, which is consistent with Friedman’s method. Finally, FCLS based on the RFCN as a special case are also investigated.

Interactive group decision making method based on probabilistic hesitant Pythagorean fuzzy information representation

Interactive group evaluation is a decision-making method to obtain group consensus by constantly modifying the initial weight of experts. Probabilistic hesitant Pythagorean fuzzy set (PrHPFS) is to be added the corresponding probability values for each membership degree and non-membership degree on the hesitant Pythagorean fuzzy set (HPFS). It is not only a generalization of HPFS and the Pythagorean fuzzy set (PFS), but also a more comprehensive and accurate reflection of the initial decision information given by experts. Especially, it can deal with the decision-making problem of multi-attribute fuzzy information in a wider area. In this paper, some basic definitions and related operations of the probabilistic hesitant Pythagorean fuzzy numbers (PrHPFNs) are first reviewed, and propose score function and accuracy function in PrHPFNs environment. Secondly, the concepts of Hamming distance measure, weighted distance measure and degree of similarity are put forward in PrHPFNs space, and the degree of similarity of two probabilistic hesitant Pythagorean fuzzy matrices (PrHPFMs) is suggested through the aggregation operator formula of PFNs. Finally, an interactive group decision-making method is designed based on the PrHPFM and the degree of similarity under the PrHPFNs environment, the effectiveness of the method is verified by an example, so as to overcome the hesitant psychological state of experts and achieve the consistent consensus evaluation of group preference.

A Branch and Bound Algorithm to Solve Travelling Salesman Problem (TSP) with Uncertain Parameters

The core of the theoretical Computing science and mathematics is computational complexity theory. It is usually concerned with the classification of computational problems in to P and NP problems by using their inherent challenges. There is no efficient algorithms for these problems. Travelling Salesman Problem is one of the most discussed problems in Combinatorial Mathematics. To deduct a Hamiltonian cycle in which the cost or time is minimum is the main objective of the TSP. There exist many algorithms to solve it. Since all the existing algorithms are not efficient to solve it, still many researchers are working to produce efficient algorithms. If the description of the parameters is vague, then fuzzy notions which include membership value are applied to model the parameters. Still the modeling does not give the exact representation of the vagueness. The Intuitionistic fuzzy set which includes non-membership value along with membership values in its domain is applied to model the parameters. The decision variables in the TSP, the cost, time or distance are modeled as intuitionistic fuzzy numbers, then the TSP is named as Intuitionistic fuzzy TSP (InFTSP). We develop the intuitionistic fuzzified version of littlewood's formula or branch and bound method to solve the Intuitionistic fuzzy TSP. This method is effective because it involves the simple arithmetic operation of Intuitionistic fuzzy numbers and ranking of intuitionistic fuzzy numbers. Ordering of intuitionistic fuzzy numbers is vital in optimization problems since it is equivalent to the ordering of alternatives. In this article, we used weighted arithmetic mean method to order the fuzzy numbers. Weighted arithmetic mean method satisfies linear property which is a very important characteristic of ranking function. Numerical examples are solved to validate the given algorithm and the results are discussed.

A new multi-criteria decision-making method utilizing power heronian operators with picture hesitant fuzzy information

In practice, picture hesitant fuzzy sets (PHFSs) combining the picture fuzzy sets (PFSs) and hesitant fuzzy sets (HFSs) are suitable to represent more complex multi-criteria decision-making (MCDM) information. The power heronian (PH) operators, which have the merits of power average (PA) and heronian mean (HM) operators, are extended to the environment of PHFSs in this article. First, some algebraic operations of picture hesitant fuzzy numbers (PHFNs), comparative functions and distance measure are introduced. Second, two novel operators, called as picture hesitant fuzzy weighted power heronian (PHFWPH) operator and picture hesitant fuzzy weighted geometric power heronian (PHFWGPH) operator, are defined. Meanwhile, some desirable characteristics and special instances of two operators are investigated as well. Third, a novel MCDM approach applying the proposed PH operators to handle PHFNs is explored. Lastly, to indicate the effectiveness of this novel method, an example regarding MCDM problem is conducted, as well as sensitivity and comparison analysis.

A method to solve strategy based decision making problems with logarithmic T-spherical fuzzy aggregation framework

T-Spherical fuzzy set (TSFS) is an improved extension in fuzzy set (FS) theory that takes into account four angles of the human judgment under uncertainty about a phenomenon that is membership degree (MD), abstinence degree (AD), non-membership degree (NMD), and refusal degree (RD). The purpose of this manuscript is to introduce and investigate logarithmic aggregation operators (LAOs) in the layout of TSFSs after observing the shortcomings of the previously existing AOs. First, we introduce the notions of logarithmic operations for T-spherical fuzzy numbers (TSFNs) and investigate some of their characteristics. The study is extended to develop T-spherical fuzzy (TSF) logarithmic AOs using the TSF logarithmic operations. The main theory includes the logarithmic TSF weighted averaging (LTSFWA) operator, and logarithmic TSF weighted geometric (LTSFWG) operator along with the conception of ordered weighted and hybrid AOs. An investigation about the validity of the logarithmic TSF AOs is established by using the induction method and examples are solved to examine the practicality of newly developed operators. Additionally, an algorithm for solving the problem of best production choice is developed using TSF information and logarithmic TSF AOs. An illustrative example is solved based on the proposed algorithm where the impact of the associated parameters is examined. We also did a comparative analysis to examine the advantages of the logarithmic TSF AOs.

Some properties of integration of real-valued function over a fuzzy interval

One of the most fundamental concepts in fuzzy set theory is the extension principle. It gives a generic way of dealing with fuzzy quantities by extending non-fuzzy mathematical concepts. There are a few examples, including the concept of fuzzy distance between fuzzy sets. The extension approach is then methodically applied to real algebra, with considerable development of fuzzy number operations. These operations are computationally appealing and generalized interval analysis. Although the set of real fuzzy numbers with extended addition or multiplication is no longer a group, it retains many structural qualities. The extension concept is demonstrated to be particularly beneficial for defining set-theoretic operations for higher fuzzy sets. We need some definitions related to our properties before we can create the properties of integration of a crisp real-valued function over a fuzzy interval. It is our goal in this article to develop and demonstrate certain characteristics of a real-valued function over a fuzzy interval in order to broaden the scope of the notion of integration of a real-valued function over a fuzzy interval. Some of these characteristics are linked to the operations of extended addition and extended subtraction, while others are not.

Solving Complex Fuzzy Linear Matrix Equations

In this paper, a kind of complex fuzzy linear matrix equation A X ˜ B = C ˜ , in which C ˜ is a complex fuzzy matrix and A and B are crisp matrices, is investigated by using a matrix method. The complex fuzzy matrix equation is extended into a crisp system of matrix equations by means of arithmetic operations of fuzzy numbers. Two brand new and simplified procedures for solving the original fuzzy equation are proposed and the correspondingly sufficient condition for strong fuzzy solution are analysed. Some examples are calculated in detail to illustrate our proposed method.