Complex Decision Modeling Framework with Fairly Operators and Quaternion Numbers under Intuitionistic Fuzzy Rough Context

The main goal of informal computing is to overcome the limitations of hypersensitivity to defects and uncertainty while maintaining a balance between high accuracy,accessibility,and cost-effectiveness.This paper investigates the potential applications of intuitionistic fuzzy sets(IFS)with rough sets in the context of sparse data.When it comes to capture uncertain information emanating fromboth upper and lower approximations,these intuitionistic fuzzy rough numbers(IFRNs)are superior to intuitionistic fuzzy sets and pythagorean fuzzy sets,respectively.We use rough sets in conjunction with IFSs to develop several fairly aggregation operators and analyze their underlying properties.We present numerous impartial laws that incorporate the idea of proportionate dispersion in order to ensure that the membership and non-membership activities of IFRNs are treated equally within these principles.These operations lead to the development of the intuitionistic fuzzy rough weighted fairly aggregation operator(IFRWFA)and intuitionistic fuzzy rough ordered weighted fairly aggregation operator(IFRFOWA).These operators successfully adjust to membership and non-membership categories with fairness and subtlety.We highlight the unique qualities of these suggested aggregation operators and investigate their use in the multiattribute decision-making field.We use the intuitionistic fuzzy rough environment’s architecture to create a novel strategy in situation involving several decision-makers and non-weighted data.Additionally,we developed a novel technique by combining the IFSs with quaternion numbers.We establish a unique connection between alternatives and qualities by using intuitionistic fuzzy quaternion numbers(IFQNs).With the help of this framework,we can simulate uncertainty in real-world situations and address a number of decision-making problems.Using the examples we have released,we offer a sophisticated and systematically constructed illustrative scenario that is intricately woven with the complexity ofmedical evaluation in order to thoroughly assess the relevance and efficacy of the suggested methodology.

Cardinal Numbers of Fuzzy Sets

Up to now, the study on the cardinal number of fuzzy sets has advanced at on pace since it is very hard to give it an appropriate definition. Althrough for it in [1], it is with some harsh terms and is not reasonable as we point out in this paper. In the paper, we give a general definition of fuzzy cardinal numbers. Based on this definition, we not only obtain a large part of results with re spect to cardinal numbers, but also give a few of new properties of fuzzy cardinal numbers.

A novel similarity measure between intuitionistic fuzzy sets based on the mid points of transformed triangular fuzzy numbers with applications to pattern recognition and medical diagnosis

Similarity measure is an essential tool to compare and determine the degree of similarity between intuitionistic fuzzy sets (IFSs). In this paper, a new similarity measure between intuitionistic fuzzy sets based on the mid points of transformed triangular fuzzy numbers is proposed. The proposed similarity measure provides reasonable results not only for the sets available in the literature but also gives very reasonable results, especially for fuzzy sets as well as for most intuitionistic fuzzy sets. To provide supportive evidence, the proposed similarity measure is tested on certain sets available in literature and is also applied to pattern recognition and medical diagnosis problems. It is observed that the proposed similarity measure provides a very intuitive quantification.

A METHOD OF SOLVING THE FUZZY FINITE ELEMENT EQUATIONS IN MONOSOURCE FUZZY NUMBERS

For same cases the rules of monosource fuzzy numbers con be used into the solution of fuzzy stochastic finite element equations in engineering. This method can reduce the computing quantity of the solution. It can be proved that the amount of the solution is nearly as much as that with the general stochastic finite element method (SFEM). In addition, a new method to appreciate the structural fuzzy failure probability is presented for the needs of the modem engineering design.

Ranking Method for Complementary Judgment Matrixes with Fuzzy Numbers Based on Hausdorff Metric Distance

A method for ranking complementary judgment matrixes with traspezoidal fuzzy numbers based on Hausdorff metric distance and fuzzy compromise decision approach is proposed. With regard to fuzzy number complementary judgment matrixes given by a decider group whose members have various weights, the expert＇s information was aggregated first by means of simple weight average（SWA） method and Bonissone calculational method. Hence a matrix including all the experts＇ preference information was got. Then the matrix＇ column members were added up and the fuzzy evaluation values of the alternatives were got. Lastly, the Hausdorff metric distance and fuzzy compromise decision approach were used to rank the fuzzy evaluation values and then the ranking values of all the alternatives were got. Because exact numbers and triangular fuzzy numbers could all be transformed into trapezoidal fuzzy numbers, the method developed can rank complementary judgment matrixes with trapezoidal fuzzy numbers, triangular fuzzy numbers and exact numbers as well. An illustrative example is also given to verify the developed method and to demonstrate its feasibility and practicality.

The Pointwise Characterizations of Fuzzy Mappings and Fuzzy Cardinal Numbers

In this paper,the pointwise characterizations of fuzzy mappings are given. Based of this definition,we give a few of new properties of fuzzy cardinal numbers.

Dynamic Analysis of Customer Demand Based on Intuitionistic Fuzzy Number in Product Planning

To address the fuzziness and variability in determining customer demand importance,a dynamic analysis method based on intuitionistic fuzzy numbers is proposed.First,selected customers use intuitionistic fuzzy numbers to represent the importance of each demand.Then,the preference information is aggregated using customer weights and time period weights through the intuitionistic fuzzy ordered weighted average operator,yielding a dynamic vector of the subjective importance of the demand index.Finally,the feasibility of the proposed method is demonstrated through an application example of a vibrating sorting screen.

A NEW TECHNIQUE FOR COMPAIRING FUZZY NUMBERS USING AN INDEX OF OPTIMISM

Fuzzy numbers are convenient for representing imprecise numerical quantities in a vague environment, and their comparison or ranking is very important for application purposes. Despite many methods suggested in the literature, there is no single measure that is universally applicable to a wide variety of situations. This paper suggested a new method for comparing fuzzy numbers based on the combination of maximizing possibility and minimizing possibility using an index of optimism in [0,1] reflecting the decision makers’ risk taking attitude. The method is simple, but has many comparative advantages.

SOME CLASSES OF DIFFERENCE SEQUENCES OF FUZZY NUMBERS DEFINED BY A SEQUENCE OF MODULI

In this article, we introduce and examine some properties of new difference sequence spaces of fuzzy numbers defined using a sequence of modulus functions.

Analysis of Helical Gear System Dynamic Response Based on Fuzzy Numbers

A non-linear dynamic model with the single degree of freedom of a helical gear pair introducing frzzy numbers is developed. In this proposed model, time-variant mesh stiffness, which is a non-linear parameter, mesh damping and composite error of a pair of meshing tooth of the gear pair are all included. Mesh stiffness is calculated by expressing Bo (r) as a Fourier series. Ⅱshape function is introduced as the membership function to characterize the fuzziness of the error. Fuzzy displacement dynamic response of the geared system at A- level, which is a closed interval, is ohtained by removing the fuzziness of the fuzzy differential equations and using Runge-Kutta numerical method. In fact, the fuzzy dynamic response and dynamic loading factor are aH the interval functions related λ. The result obtained here can be used to the fuzzy dynamic optimization design course of the helical gear system. The main advantage of this method is to introduce the concept of fuzzy number for the first time to the analysis of the gear system dynamics.

Approximate Solution of Fuzzy Matrix Equations with LR Fuzzy Numbers

In the paper, a class of fuzzy matrix equations AX=B where A is an m × n crisp matrix and is an m × p arbitrary LR fuzzy numbers matrix, is investigated. We convert the fuzzy matrix equation into two crisp matrix equations. Then the fuzzy approximate solution of the fuzzy matrix equation is obtained by solving two crisp matrix equations. The existence condition of the strong LR fuzzy solution to the fuzzy matrix equation is also discussed. Some examples are given to illustrate the proposed method. Our results enrich the fuzzy linear systems theory.

On Weighted Possibilistic Mean,Variance and Correlation of Interval-valued Fuzzy Numbers

In this paper, the concept of weighted possibilistic mean of interval- valued fuzzy number is first introduced. Further, the notions of weighted possibilistic variance, covariance and correlation of interval-valued fuzzy numbers are presented. Meantime, some important properties of them and relationships between them are studied.

Regression Analysis of a Kind of Trapezoidal Fuzzy Numbers Based on a Shape Preserving Operator

Fuzzy regression provides more approaches for us to deal with imprecise or vague problems. Traditional fuzzy regression is established on triangular fuzzy numbers, which can be represented by trapezoidal numbers. The independent variables, coefficients of independent variables and dependent variable in the regression model are fuzzy numbers in different times and TW, the shape preserving operator, is the only T-norm which induces a shape preserving multiplication of LL-type of fuzzy numbers. So, in this paper, we propose a new fuzzy regression model based on LL-type of trapezoidal fuzzy numbers and TW. Firstly, we introduce the basic fuzzy set theories, the basic arithmetic propositions of the shape preserving operator and a new distance measure between trapezoidal numbers. Secondly, we investigate the specific model algorithms for FIFCFO model (fuzzy input-fuzzy coefficient-fuzzy output model) and introduce three advantages of fit criteria, Error Index, Similarity Measure and Distance Criterion. Thirdly, we use a design set and two reference sets to make a comparison between our proposed model and the reference models and determine their goodness with the above three criteria. Finally, we draw the conclusion that our proposed model is reasonable and has better prediction accuracy, but short of robust, comparing to the reference models by the three goodness of fit criteria. So, we can expand our traditional fuzzy regression model to our proposed new model.

A Quadratic Programming with Triangular Fuzzy Numbers

Quadratic Programming (QP) is a mathematical modeling technique designed to optimize the usage of limited resources and has been widely applied to solve real world problems. In conventional quadratic programming model the parameters are known constants. However in many practical situations, it is not reasonable to require that the constraints or the objective function in quadratic programming problems be specified in precise, crisp terms. In such situations, it is desirable to use some type of Fuzzy Quadratic Programming (FQP) problem. In this paper a new approach is proposed to derive the fuzzy objective value of fuzzy quadratic programming problem, where the constraints coefficients and the right-hand sides are all triangular fuzzy numbers. The proposed method is solved using MATLABTM toolbox and the numerical results are presented.

Understanding the financial innovation priorities for renewable energy investors via QFD-based picture fuzzy and rough numbers

This study evaluates financial innovation priorities for renewable energy investors by generating a novel hybrid fuzzy decision-making model.First,SERVQUAL-based customer needs for financial innovation are weighted with decision-making trial and evaluation laboratory based on picture fuzzy sets.Second,the financial innovation priorities are ranked by technique for order preference by similarity to ideal solutions based on picture fuzzy rough sets.In this process,Theory of the solution of inventive problems-based technical characteristics for financial services,the process for innovative services,and competencies for financial innovation are considered using quality function deployment phases.In addition,the Vise Kriterijumska Optimizacija I Kompromisno Resenje method is also considered for an alternative ranking.Similarly,sensitivity analysis is also performed by considering five different cases.It is determined that the ranking priorities based on the proposed model are almost identical,demonstrating the proposed model’s validity and reliability.Assurance is the most crucial factor for the customer needs regarding the financial innovation priorities for renewable energy investors.Concerning the financial innovation priorities,the product is the essential priority for financial innovation;hence,it is recommended that companies engage qualified employees to effectively design the financial innovation for renewable energy investors.Additionally,necessary training should be given to the employees who currently work in the company,which can increase the renewable energy investors’trust in the innovative financial products.Companies should mainly focus on the product to provide better financial innovation to attract renewable energy investors.An effectively designed financial innovation product can help solve the financing problem of renewable energy investors.

Weighted (<i>λ</i>, <i>μ</i>)-Ideal Statistical Convergence and Strongly Weighted (<i>λ</i>, <i>μ</i>)-Ideal Convergence of Double Sequences of Fuzzy Numbers

The paper aims to investigate different types of weighted ideal statistical convergence and strongly weighted ideal convergence of double sequences of fuzzy numbers. Relations connecting ideal statistical convergence and strongly ideal convergence have been investigated in the environment of the newly defined classes of double sequences of fuzzy numbers. At the same time, we have examined relevant inclusion relations concerning weighted (λ, μ)-ideal statistical convergence and strongly weighted (λ, μ)-ideal convergence of double sequences of fuzzy numbers. Also, some properties of these new sequence spaces are investigated.

Optimal Solution of Fuzzy Transportation Problem Using Octagonal Fuzzy Numbers

In this paper a fuzzy transportation problem under a fuzzy environment is solved using octagonal fuzzy numbers.The transportation problem is significant and has been widely studied in the field of applied mathematics to solve a system of linear equations in many applications in science.Systems of concurrent linear equations play a vital major role in operational research.The main perspective of this research paper is to find out the minimum amount of transportation cost of some supplies through a capacitated network formerly the availability and the demand notes are octagonal fuzzy numbers.Octagonal fuzzy numbers are used and showed a membership function.To illustrate this method,a fuzzy transportation problem is solved by using octagonal fuzzy numbers using the ranking technique.It is shown that it is the best optimal solution and it is demonstrated with a numerical example.

Numerical Characteristics of Fuzzy Numbers Based on Credibility Measure

The numerical characteristics of fuzzy numbers include the optimistic value, pessimistic value, expected value, and the variance. We mainly provide the calculation formulae of several numerical characteristics of fuzzy numbers based on credibility measure. Especially, the variance of symmetric fuzzy numbers is formulated, and a super bound for the variance of fuzzy numbers is presented. Meanwhile, some conclusions relative to credibility measure, optimistic and pessimistic values are also given.

Strong Laws of Large Numbers for Fuzzy Set-Valued Random Variables in Gα Space

In this paper, we shall present the strong laws of large numbers for fuzzy set-valued random variables in the sense of d∞H . The results are based on the result of single-valued random variables obtained by Taylor [1] and set-valued random variables obtained by Li Guan [2].

Arithmetic Operations of Generalized Trapezoidal Picture Fuzzy Numbers by Vertex Method

In this article, we define the arithmetic operations of generalized trapezoidal picture fuzzy numbers by vertex method which is assembled on a combination of the (α, γ, β)-cut concept and standard interval analysis. Various related properties are explored. Finally, some computations of picture fuzzy functions over generalized picture fuzzy variables are illustrated by using our proposed technique.