Fuzzy risk analysis using area and height based similarity measure on generalized trapezoidal fuzzy numbers and its application

In this paper, we have proposed a new method of similarity measure associating the geometric distance, area and height of generalized trapezoidal fuzzy numbers. Some properties regarding the proposed new method of similarity measure have been derived. To illustrate the effectiveness of this method, it is compared with existing techniques taking thirty two different sets of generalized trapezoidal fuzzy numbers. Moreover, the proposed method has been used for calculating the fuzzy risk analysis in a production system in which different parameters are represented by linguistic trapezoidal fuzzy numbers.     

An improved similarity measure for generalized fuzzy numbers and its application to fuzzy risk analysis

This paper presents an improved method to compute the degree of similarity between generalized trapezoidal fuzzy numbers. The proposed similarity measure contains many features of fuzzy numbers such as geometric distance, center of gravity (COG), area, perimeter, and height. The previous methods are criticized via presenting some examples. In addition, the performance of the proposed methods is compared by the existing similarity measures using twenty different sets of generalized trapezoidal fuzzy numbers. Furthermore, the proposed method is used for fuzzy risk analysis based on similarity measures. Finally, an example is introduced to illustrate the fuzzy risk analysis.     

基于包含度的直觉模糊相似度量方法

针对现有文献中涉及到直觉模糊相似度量的计算公式大多是基于直觉模糊距离测度的现状,提出一种基于包含度的直觉模糊相似度量方法。借助模糊蕴涵算子和集合基数,建立了IFS包含度的一系列具体公式,揭示了IFS包含度与相似度的关系,提出了一种基于包含度的IFS相似度量方法,并以具体算例表明该方法不仅可以解决中部分存在的问题,而且为研究IFS相似度量理论开辟一个新的途径。     

Fuzzy risk analysis using similarity measure of interval-valued fuzzy numbers and its application in poultry farming

Similarity measure plays an important role in the decision-making process under an uncertain environment where parameters involved are linguistics terms. Mostly, similarity measure is discussed on generalized fuzzy numbers. However, a few efforts have been made to study this measure on interval-valued fuzzy numbers. Sometimes, methods involving interval-valued fuzzy numbers depict limitations and drawbacks. Moreover, some of the methods are just confined to interval-valued fuzzy numbers. Hence, these methods fail when similarity has to be determined between crisp-valued fuzzy numbers and interval-valued fuzzy numbers. Hence, a new method of similarity measure has been developed based on the concepts of geometric distance, heights and the radius of gyration of the interval-valued fuzzy numbers. Although the method is being discussed on interval-valued fuzzy numbers, yet it is not just confined to such numbers. This method can be applied efficiently to generalized fuzzy numbers too. The method seems to out-perform in many situations and overcome the drawbacks and limitations of existing methods. A few sets of fuzzy numbers are considered for a comparative study and draw out the out-performance of the proposed method. A real-life problem of risk analysis in poultry farming has been discussed using the proposed similarity measure.     

A new method for solving dual hesitant fuzzy assignment problems with restrictions based on similarity measure

Zhu et al. (2012) proposed dual hesitant fuzzy set as an extension of hesitant fuzzy sets which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. Dual hesitant fuzzy sets consist of two parts, that is, the membership and nonmembership degrees, which are represented by two sets of possible values. Therefore, in accordance with the practical demand these sets are more flexible, and provides much more information about the situation. In this paper, the axiom definition of a similarity measure between dual hesitant fuzzy sets is introduced. A new similarity measure considering membership and nonmembership degrees of dual hesitant fuzzy sets has been presented and also it is shown that the corresponding distance measures can be obtained from the proposed similarity measures. To check the effectiveness, the proposed similarity measure is applied in a bidirectional approximate reasoning systems. Mathematical formulation of dual hesitant fuzzy assignment problem with restrictions is presented. Two algorithms based on the proposed similarity measure, are developed to finds the optimal solution of dual hesitant fuzzy assignment problem with restrictions. Finally, the proposed method is illustrated by numerical examples.     

A new approach for fuzzy risk analysis based on similarity measures of generalized fuzzy numbers

In this paper, we present a new method for fuzzy risk analysis based on similarity measures between generalized fuzzy numbers. First, we present a new similarity measure between generalized fuzzy numbers. It combines the concepts of geometric distance, the perimeter and the height of generalized fuzzy numbers for calculating the degree of similarity between generalized fuzzy numbers. We also prove some properties of the proposed similarity measure. We make an experiment to use 15 sets of generalized fuzzy numbers to compare the experimental results of the proposed method with the existing similarity measures. The proposed method can overcome the drawbacks of the existing similarity measures. Based on the proposed similarity measure between generalized fuzzy numbers, we present a new fuzzy risk analysis algorithm for dealing with fuzzy risk analysis problems, where the values of the evaluating items are represented by generalized fuzzy numbers. The proposed method provides a useful way to deal with fuzzy risk analysis problems.     

Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship

In this paper, we introduce an axiomatic definition of an interval-valued fuzzy sets’ inclusion measure which is different from Bustince’s [H. Bustince, Indicator of inclusion grade for interval-valued fuzzy sets, Applications to approximate reasoning based on interval-valued fuzzy sets, International Journal of Approximate Reasoning, 23 (2000) 137-209]. The relationship among the normalized distance, the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets is investigated in detail. Furthermore, six theorems are proposed showing how the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets can be deduced by the interval-valued fuzzy sets’ normalized distance based on their axiomatic definitions. Some formulas have also been put forward to calculate the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets.     

An efficient similarity measure for intuitionistic fuzzy sets

We introduce a new methodology for measuring the degree of similarity between two intuitionistic fuzzy sets. The new method is developed on the basis of a distance defined on an interval by the use of convex combination of endpoints and also focusing on the property of min and max operators. It is shown that among the existing methods, the proposed method meets all the well-known properties of a similarity measure and has no counter-intuitive examples. The validity and applicability of the proposed similarity measure is illustrated with two examples known as pattern recognition and medical diagnosis.     

Fuzzy risk analysis based on similarity measures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operators

In this paper, we present a new method for fuzzy risk analysis based on a new similarity measure between interval-valued fuzzy numbers and new interval-valued fuzzy number arithmetic operators. First, we present a new similarity measure between interval-valued fuzzy numbers. The proposed similarity measure considers the similarity of the gravities on the X-axis between upper fuzzy numbers, the difference of the spreads between upper fuzzy numbers, the heights of the upper fuzzy numbers, the degree of similarity on the X-axis between interval-valued fuzzy numbers, and the gravities on the Y-axis between interval-valued fuzzy numbers. We also present three properties of the proposed similarity measure between interval-valued fuzzy numbers. Then, we present new interval-valued fuzzy number arithmetic operators. Finally, we apply the proposed similarity measure between interval-valued fuzzy numbers and the proposed interval-valued fuzzy number arithmetic operators to propose a fuzzy risk analysis algorithm to deal with fuzzy risk analysis problems. The proposed method provides a useful way for handling fuzzy risk analysis problems based on interval-valued fuzzy numbers.     

Analyzing fuzzy risk based on similarity measures between interval-valued fuzzy numbers

In this paper, we present a new method for handling fuzzy risk analysis problems based on the proposed new similarity measure between interval-valued fuzzy numbers. First, we present a new similarity measure between interval-valued fuzzy numbers. It considers the degrees of closeness between interval-valued fuzzy numbers on the X-axis and the degrees of differences between the shapes of the interval-valued fuzzy numbers on the X-axis and the Y-axis, respectively. We also prove three properties of the proposed similarity measure. Then, we make an experiment to compare the experimental results of the proposed method with the existing similarity measures between interval-valued fuzzy numbers. The proposed method can overcome the drawbacks of the existing methods. Finally, based on the proposed similarity measure between interval-valued fuzzy numbers, we present a new fuzzy risk analysis algorithm for dealing with fuzzy risk analysis problems. Because the proposed method allows the evaluating values of sub-components to be represented by interval-valued fuzzy numbers, it is more flexible than Chen and Chen’s method (2003).     

Fuzzy absolute error distance measure based on a generalised difference operation

The main goal of this paper is to introduce a notion of fuzzy absolute error distance measure between two fuzzy numbers. For this purpose, a notion of generalised difference operation between two fuzzy numbers and absolute value of a fuzzy number was first introduced. The proposed methods were conducted on the basis of α-values of fuzzy numbers. Main properties of the proposed fuzzy distance measure was also verified in the space of fuzzy numbers. The proposed fuzzy distance measure evaluates the fuzzy distance between the two fuzzy numbers as a fuzzy number. Notably, the main advantage of such generalised difference operation is that it always exists. Therefore, it improves the shortcoming of a well-known generalised difference operation called Hakuhara difference. Some of the main properties of the proposed fuzzy absolute error distance measure including robustness were also studied in the space of fuzzy numbers. Several fuzzy distance measures, especially fuzzy absolute error distance, have been proposed so far. However, none of them save all reasonable properties required for an absolute error distance measure in fuzzy environment. Shortcomings relevant to other methods and advantages of the proposed method were also discussed.     

An improved fuzzy risk analysis based on a new similarity measures of generalized fuzzy numbers

This paper presents a novel method of fuzzy risk analysis based on a new similarity measure of generalized fuzzy numbers. This similarity measure considers many features of generalized fuzzy numbers such as the area, perimeter, height and geometric distance of these kinds of fuzzy numbers. Using some sets of generalized fuzzy numbers, we show the power of this similarity measurement method to overcome the drawbacks that other methods are suffering. Applying the proposed method, we present an improved fuzzy risk analysis method which develops the capability of fuzzy risk analysis methods to deal with sophisticated problems. In the proposed method we use new factors such as probability of failure detection and economic disbenefits of failure occurrence which have not been used in fuzzy risk analysis methods before.     

Differentiability and convexity of fuzzy mappings

Goetschel and Voxman [1] have introduced the notion of a derivative for fuzzy mappings of one variable in a manner different from the usual one. In this paper, we define a differentiable fuzzy mapping of several variables in ways that parallel the definition, proposed by Goetschel and Voxman [1], for a fuzzy mapping of one variable, and then study some basic differentiability properties of fuzzy mappings from the standpoint of convex analysis.     

基于指数型模糊数的多属性决策模型及其应用

针对模糊多属性决策问题,给出一种基于指数型模糊数的多属性决策模型。一方面,通过定义指数型模糊数的期望,以实现属性权重向量的解模糊化处理;另一方面,根据三元区间数理论和指数型模糊数的截集信息,定义指数型模糊数上一种新的距离度量,以计算各备选方案与正、负理想方案之间的距离。根据模糊理想点思想,基于指数型模糊数的期望和距离的定义,给出一种指数型模糊数上的Topsis多属性决策方法。将该模型应用于一个具体实例,其结果证实了该方法的有效性。     

Spectral clustering with fuzzy similarity measure

Spectral clustering algorithms have been successfully used in the field of pattern recognition and computer vision. The widely used similarity measure for spectral clustering is Gaussian kernel function which measures the similarity between data points. However, it is difficult for spectral clustering to choose the suitable scaling parameter in Gaussian kernel similarity measure. In this paper, utilizing the prototypes and partition matrix obtained by fuzzy c-means clustering algorithm, we develop a fuzzy similarity measure for spectral clustering (FSSC). Furthermore, we introduce the K-nearest neighbor sparse strategy into FSSC and apply the sparse FSSC to texture image segmentation. In our experiments, we firstly perform some experiments on artificial data to verify the efficiency of the proposed fuzzy similarity measure. Then we analyze the parameters sensitivity of our method. Finally, we take self-tuning spectral clustering and Nyström methods for baseline comparisons, and apply these three methods to the synthetic texture and remote sensing image segmentation. The experimental results show that the proposed method is significantly effective and stable.     

Application of practical fuzzy arithmetic to fuzzy internal model control

This paper applies a new fuzzy arithmetic of interval calculus and fuzzy quantities to automatic control. Practical results are obtained which overcome those based on the extension principle or α-cuts. The proposed approach is based on a different representation of fuzzy numbers, though most common arithmetic operators cannot be directly applied for designing a fuzzy controller due to the unjustified overestimation effect. To avoid this phenomenon, a procedure based on an “exact” resolution calculus is proposed, whose solutions allow creating a fuzzy internal model control scheme. The validity of the new method is illustrated by a real-time educational engineering application on classical control design: a coupled tanks system.     

改进的直觉模糊粗糙集相似性度量方法

针对现有的直觉模糊粗糙集相似性度量的问题,提出了一种改进的基于海明距离的直觉模糊粗糙集相似性度量方法。该方法考虑了犹豫度并引入加权参数,解决了相似性度量不精确的问题。首先给出了直觉模糊粗糙值间的相似性度量定义,并揭示其若干重要性质。在此基础上,提出了直觉模糊粗糙集间的相似性度量方法,并证明其具有同样性质。最后通过数值算例分析说明了该方法更合理、更有效。     

On cluster validity index for estimation of the optimal number of fuzzy clusters

A new cluster validity index is proposed that determines the optimal partition and optimal number of clusters for fuzzy partitions obtained from the fuzzy c-means algorithm. The proposed validity index exploits an overlap measure and a separation measure between clusters. The overlap measure, which indicates the degree of overlap between fuzzy clusters, is obtained by computing an inter-cluster overlap. The separation measure, which indicates the isolation distance between fuzzy clusters, is obtained by computing a distance between fuzzy clusters. A good fuzzy partition is expected to have a low degree of overlap and a larger separation distance. Testing of the proposed index and nine previously formulated indexes on well-known data sets showed the superior effectiveness and reliability of the proposed index in comparison to other indexes.     

Distance measure for Fermatean fuzzy linguistic term sets based on linguistic scale function: An illustration of the TODIM and TOPSIS methods

In this paper, we propose the Fermatean fuzzy linguistic term set (FFLTS) based on the linguistic scale function. A new similarity measure between FFLTSs is constructed, which not only includes the linguistic scale function but also combines the cosine similarity measure and Euclidean distance measure, and then the related properties of the similarity measure are proven. A corresponding distance measure is obtained according to the relationship between the distance measure and similarity measure. Furthermore, we extend the Tomada de Decisão Interativa Multicritério (TODIM) method and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method to the corresponding distance measure under the Fermatean fuzzy linguistic environment. The main advantages of the proposed methods are that they can not only transform linguistic information effectively in different decision environments but also improve the adaptability of FFLTS in decision-making problems. Finally, a numerical example is provided to illustrate the effectiveness and feasibility of the proposed methods, which are also compared with other existing methods. The sensitivity analysis of the parameters and the influence of linguistic scale function on the ranking results are also discussed.