Arithmetic Operations Of Fuzzy Numbers Research Articles

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

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.

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.

Similarity measure of the interval-valued fuzzy numbers and its application in risk analysis in paddy cultivation

Generally, in a risk analysis problems, data are collected from experts in terms of linguistic variables, which are then observed as fuzzy numbers. Eventually, the total risk is calculated, and obtained as fuzzy numbers, using the arithmetic operations of fuzzy numbers. Finally, the risk so obtained has to be expressed in terms of linguistic variables, so that it becomes communicable to the general audience. One such tool in fuzzy mathematics is the similarity measure. In this paper, a new method of determining the similarity between two interval-valued fuzzy numbers has been developed. Generally, various parameters were used in deriving a similarity measure. However, the parameters that describe the shape of an interval-valued fuzzy numbers was never a concern in the earlier studies. Hence, in this study, these parameters are incorporated in the similarity measure. Further, a few reasonable properties are being developed, which will be considered as standard in developing a similarity measure. Finally, a risk analysis problem in paddy cultivation is discussed. Using the proposed similarity measure, the final risk is expressed in terms of a linguistic variable. The final risk so obtained for paddy cultivation, under the assumed scenario, is low.

Reliability Range Through Upgraded Operation with Trapezoidal Fuzzy Number

This work presents an application of upgraded arithmetic operations on trapezoidal fuzzy number to extend the reliability from point estimation to interval estimation. A realistic example of an accidental case has been taken to show the desired results. As we know that uncertainty is a disregardable phenomenon and in this situation crisp reliability may not be exact. To get a better hold on this situation, we introduce a method of trapezoidal fuzzy number and upgraded arithmetic operations. In this paper, the reliability of each factor is represented by the trapezoidal fuzzy number and the final reliability is obtained in the form of trapezoidal fuzzy number. Hence, the dejection of fuzzy number is preserved and the crisp region is increased to illustrate the highest membership grade as an interval. So it can help the decision maker to analyse the behaviour of the system and take effective decision accordingly to reduce the chance of mishappening.

Основні арифметичні операції з нечіткими числами та новітні підходи до застосування теорії нечітких множин при класифікуванні космічних зображень

Classification in remote sensing is a very difficult procedure, because it involves a lot of steps and data preprocessing. Fuzzy Set Theory plays a very important role in classification problems, because the fuzzy approach can capture the structure of the image. Most concepts are fuzzy in nature. Fuzzy sets allow to deal with uncertain and imprecise data. Many classification problems are formalized by using fuzzy concepts, because crisp classes represent an oversimplification of reality, leading to wrong results of classification. Fuzzy Set Theory is an important mathematical tool to process complex and fuzzy da-ta. This theory is suitable for high resolution remote sensing image classification. Fuzzy sets and fuzzy numbers are used to determine basic probability assignment. Fuzzy numbers are used for detection of the optimal number of clusters in Fuzzy Clustering Methods. Image is modeled as a fuzzy graph, when we represent the dissimilitude between pixels in some classification tasks. Fuzzy sets are also applied in different tasks of processing digital optical images. It was noted, that fuzzy sets play an important role in analysis of results of classification, when different agreement measures between the reference data and final classification are considered. In this work arithmetic operations of fuzzy numbers using alpha-cut method were considered. Addition, subtraction, multiplication, division of fuzzy numbers and square root of fuzzy number were described in this paper. Moreover, it was illustrated examples with different arithmetic operations of fuzzy numbers. Fuzzy Set Theory and fuzzy numbers can be applied for analysis and classification of hyperspectral satellite images, solving ecological tasks, vegetation clas-sification, in remote searching for minerals.

Evaluating the Efficiency of School Preceptors by Fuzzy Risk Analysis

This paper presents a new method for evaluating the efficiency of school preceptors based on fuzzy number arithmetic operations. It uses fuzzy numbers to represent fuzzy grades. The fuzzy weights of criteria are automatically generated from the opinions of evaluators. The simplified fuzzy number arithmetic operations are used for calculating the average of fuzzy numbers. It can evaluate the efficiency of school preceptors in a more flexible and more intelligent manner.

A Multiple Criteria Decision Method for Selecting Maritime Search & Rescue Scheme

The evaluation and selection of Search And Rescue (SAR) schemes is one of the most important decision issues for maritime SAR. Owing to vague concept frequently represented in decision data, a group fuzzy multi-criteria decision-making approach is proposed to solve the SAR scheme optimal selection problem. In the proposed method, the experts’ opinions are described by trapezoidal fuzzy numbers, and the fuzzy Delphi method is adopted to adjust each expert’s opinion to achieve the consensus condition. By using the logarithmic least square method and trapezoidal fuzzy number arithmetic operations to estimate the normalized fuzzy priority weights, Analytic Hierarchy Process (AHP) is presented and consequently the consistency check of fuzzy judgment matrix is avoided. The results of simulation show that the method is flexible and credible and provides references in intelligent SAR decision-making.

Solution of the Fuzzy Equation <i>A + X = B</i> Using the Method of Superimposition

Fuzzy equations were solved by using different standard methods. One of the well-known methods is the method of -cut. The method of superimposition of sets has been used to define arithmetic operations of fuzzy numbers. In this article, it has been shown that the fuzzy equation AX B   , where A, X, B are fuzzy numbers can be solved by using the method of superimposition of sets. It has also been shown that the method gives same result as the method of -cut.

Appraising the performance of high school teachers based on fuzzy number arithmetic operations

This paper presents a new method for appraising the performance of high school teachers based on fuzzy number arithmetic operations. It uses fuzzy numbers to represent fuzzy grades. The fuzzy weights of criteria are automatically generated from the opinions of evaluators. The simplified fuzzy number arithmetic operations are used for calculating the average of fuzzy numbers. It can appraise the performance of high school teachers in a more flexible and more intelligent manner.

Equivalence class in the set of fuzzy numbers and its application in decision-making problems

An equivalence relation is defined in the set of fuzzy numbers. In a particular equivalence class, arithmetic operations of fuzzy numbers are introduced. A fuzzy matrix with respect to a particular class and its associated crisp matrices are also introduced. The concept of equivalence class is applied in fuzzy decision-making problems and justified through a numerical example.

A NEW METHOD FOR HANDLING MULTICRITERIA FUZZY DECISION-MAKING PROBLEMS USING FN-IOWA OPERATORS

We use fuzzy numbers to extend the traditional induced ordered weight averaging (IOWA) operator to present the fuzzy-number IOWA (FN-IOWA) operator, wherein fuzzy numbers are used to describe the argument values and the weights of the FN-IOWA operator, and the aggregation results are obtained by using fuzzy-number arithmetic operations. We also present a new method for ranking fuzzy numbers. Based on the proposed FN-IOWA operator and the proposed ranking method of fuzzy numbers, we present a new algorithm to deal with multicriteria fuzzy decision-making problems. The proposed algorithm can deal with multicriteria fuzzy decision-making problems in a more intelligent and more flexible manner.

Evaluating weapon systems based on grey relational analysis and fuzzy arithmetic operations

In this paper, we propose a new and simple method for evaluating weapon systems by an analytic hierarchy process based on grey relational analysis and fuzzy arithmetic operations, which is a multiple criteria decision‐making approach in a fuzzy environment. Since the rank score matrix of the proposed method is generated by the grey relational analysis, it provides a systematic approach to describe the rank score of each criterion item. Also because the proposed method utilizes the simplified fuzzy arithmetic operations of fuzzy numbers rather than the fuzzy scales method or the complicated entropy weight calculations, its execution time is shorter than the previous methods. An example of tactical missile systems selection is utilized to illustrate the performance evaluation process of a weapon system. It can be seen that the simulation result is coincident with the conventional methods.

Operations of fuzzy numbers with step form membership function using function principle

The Function Principle was introduced in 1984 by Chen, to keep the original type of membership function and simplify the arithmetic operations of fuzzy numbers. This paper shows how to treat the operations of fuzzy numbers with step form by using Function Principle.

Fuzzy system reliability analysis by the use of Tω (the weakest t-norm) on fuzzy number arithmetic operations

In general, the sup-min convolution has been used for fuzzy arithmetic to analyze fuzzy system reliability, where the reliability of each system component is represented by fuzzy numbers. It is well known that T ω -based addition preserves the shape of L- R type fuzzy numbers. In this paper, we show T ω -based multiplication also preserves the shape of L- R type fuzzy numbers. We then apply T ω -based arithmetic operations to fuzzy system reliability analysis. In fact, we show that we can simplify fuzzy arithmetic operations and even get the exact solutions for L- R type fuzzy system reliability, while others [Singer, Fuzzy Sets Syst. 34 (1990) 145; Cheng and Mon, Fuzzy Sets Syst. 56 (1993) 29; Chen, Fuzzy Sets Syst. 64 (1994) 31] have got the approximate solutions using sup-min convolution for evaluating fuzzy system reliability.

NEW METHODS FOR SUBJECTIVE MENTAL WORKLOAD ASSESSMENT AND FUZZY RISK ANALYSIS

This paper presents new methods for subjective mental workload assess ment and fuzzy risk analysis in which simplified fuzzy number arithmetic operations are used for subjective mental workload assessment and fuzzy risk analysis The proposed methods can provide useful ways to handle subjective mental workload assessment problems and fuzzy risk analysis problems

NEW METHOD FOR FUZZY SYSTEM RELIABILITY ANALYSIS

This paper presents a new method for fuzzy system reliability analysis based on fuzzy time series and the alpha-cuts arithmetic operations of fuzzy numbers, where we allow the reliabilities of the components of a system at different times t (t= ...,0,1,2,...) to be represented by different membership functions. Because the proposed method allows the reliabilities of the components of a system at different times t to have different membership functions, it is more flexible than the ones presented in Chen 1994 , Cheng and Mon 1993 , and Singer 1990 . Furthermore, because the proposed method uses the simple alpha -cuts arithmetic operations of fuzzy numbers rather than the complicated nonlinear programming techniques mentioned in Mon and Cheng 1994 , it is simpler in calculating fuzzy system reliability than the one presented in that paper.

Evaluating weapon systems using fuzzy arithmetic operations

In this paper, we present a new method to deal with the performance evaluation of weapon systems using fuzzy arithmetic operations. An example of tactical missile systems selection is used to illustrate the performance evaluation process of weapon systems. Because the proposed methods uses simplified fuzzy arithmetic operations of fuzzy numbers rather than the complicated entropy weight calculations mentioned in [7], its execution is much faster than the one presented in [7].

Fuzzy system reliability analysis using fuzzy number arithmetic operations

This paper presents a new method to analyze fuzzy system reliability using fuzzy number arithmetic operations, where the reliability of each system component is represented by a triangular fuzzy number. Because the proposed method uses simplified fuzzy arithmetic operations of fuzzy numbers rather than the complicated interval arithmetic operations of fuzzy numbers mentioned by Cheng and Mon in Fuzzy Sets and Systems 56 (1993) 29–35 or the complicated extended algebraic operations of fuzzy numbers mentioned by Singer in Fuzzy Sets and Systems 34 (1990) 145–155, its execution is faster than the ones presented by Cheng and Mon, and Singer.