Comparison of clusters from fuzzy numbers

Fuzzy Sets and Systems, 2001

A key issue in operationalizing fuzzy set theory (particularly in decision analysis) is how to compare fuzzy numbers. In this paper, the case of L-R fuzzy numbers, i.e. the most general form of fuzzy numbers, is considered. In particular here, L-R fuzzy numbers represented by continuous, convex membership functions allowing also deÿnite integration is taken into consideration, normality is not required. Traditional comparison methods are generally limited to the use of triangular fuzzy numbers, and often the shape of the membership function is not taken into account or only a part of it is used (leading to a loss of information). Most of the approaches one can ÿnd in the literature are characterised by the use of -cuts and credibility levels, the use of areas for comparing fuzzy numbers has been proposed only recently. In particular, in the so-called NAIADE method a new semantic distance able to compare crisp numbers, fuzzy numbers and density functions has been developed. The basic idea underlying this paper is that, if only L-R fuzzy numbers are considered, other methodologies for comparing fuzzy numbers can be developed. Three indices based on the use of areas are studied, i.e. the expected value, the variance (with its decomposition into positive and negative semivariances) and the degree of coincidence of two fuzzy numbers. A justiÿcation of the use of these indices and a ÿrst tentative of axiomatisation is given. A short discussion on the issue of possible aggregation conventions of these indices is presented, and an empirical example is examined too.

2014

Abstract. The fuzzy set theory has been applied in many field s such as management, engineering etc. In modern management applications ra king using fuzzy numbers are the most important aspect in decision making proces s. This paper proposes a new method on the incentre of centroids and uses of Euclidean distance to ranking generalized hexagonal fuzzy numbers. We have used a ranking met hod for ordering fuzzy numbers based on areas and weights of generalized fuzzy num bers.

Mathematical Optimization Theory and Operations Research, 2019

In the class of decision-making problems with fuzzy information concerning criterion values, the problem of comparing fuzzy numbers is relevant. There are various approaches to solving it. They are determined by the specific character of the problem under consideration. This paper proposes one approach to comparing fuzzy numbers. The proposed approach is as follows. At first, a rule is constructed for comparing a real number with a level set of a fuzzy number. Then, with the help of a procedure for constructing the exact lower approximation for the collection of sets, a fuzzy set is constructed. This fuzzy set determine the rule for comparing a real number with a fuzzy number. Using this rule and the approach based on separating two fuzzy numbers with a real number, the procedure is chosen for comparing two fuzzy numbers. As an example, fuzzy numbers with trapezoidal membership functions are considered, and the geometric interpretation of the results being given.

2020

Abstract: Ranking fuzzy numbers plays an important role in a fuzzy decision making process. However, fuzzy numbers may not be easily ordered into one sequence due to the overlap between fuzzy numbers. A new approach is introduced to detect the overlapped fuzzy numbers based on the concept of similarity measure incorporating the preference of the decision maker into the fuzzy ranking process. Numerical examples and comparisons with other method are straight forward and are practically capable of comparing similar fuzzy numbers. The proposed method is an absolute Ranking and no pair wise comparison of fuzzy numbers is necessary. Furthermore, through some examples discussed in this work, it is proved that the proposed method possesses several good characteristics as compared to the other comparable methods examined in this work.

Fuzzy Information and Engineering, 2013

This paper proposes a new method for ranking fuzzy numbers based on the area between circumcenter of centroids of a fuzzy number and the origin. The proposed method not only uses an index of optimism, which reflects the decision maker's optimistic attitude but also makes use of an index of modality which represents the importance of mode and spreads. This method ranks various types of fuzzy numbers which includes normal, generalized trapezoidal and triangular fuzzy numbers along with crisp numbers which are a special case of fuzzy numbers. Some numerical examples are presented to illustrate the validity and advantages of the proposed method.

International Journal of Fuzzy Systems

Ranking fuzzy numbers plays a very important role in the decision process, data analysis, and applications. The last few decades have seen a large number of methods investigated for ranking fuzzy numbers. The most commonly used approach for ranking fuzzy numbers is ranking indices based on the centroids of fuzzy numbers. However, there are some weaknesses associated with these indices. This paper reviews several fuzzy number ranking methods based on centroid indices and proposes a new centroid-index ranking method that is capable of effectively ranking various types of fuzzy numbers. The contents herein present several comparative examples demonstrating the usage and advantages of the proposed centroid-index ranking method for fuzzy numbers.

Computers & Mathematics with Applications, 2009

Ranking of fuzzy numbers Trapezoidal fuzzy number Parametric form of fuzzy number Magnitude of fuzzy number a b s t r a c t

Bulletin of Electrical Engineering and Informatics, 2019

Similarity measure between two fuzzy sets is an important tool for comparing various characteristics of the fuzzy sets. It is a preferred approach as compared to distance methods as the defuzzification process in obtaining the distance between fuzzy sets will incur loss of information. Many similarity measures have been introduced but most of them are not capable to discriminate certain type of fuzzy numbers. In this paper, an improvised similarity measure for generalized fuzzy numbers that incorporate several essential features is proposed. The features under consideration are geometric mean averaging, Hausdorff distance, distance between elements, distance between center of gravity and the Jaccard index. The new similarity measure is validated using some benchmark sample sets. The proposed similarity measure is found to be consistent with other existing methods with an advantage of able to solve some discriminant problems that other methods cannot. Analysis of the advantages of th...

Communication in Mathematical Modeling and Applications, 2018

With no doubt, ranking the fuzzy numbers are extremely effective and useful in different scientific fields such as Artificial Intelligence, Economics, Engineering and decision-making units and etc. The fuzzy quantities must be ranked before their engagement in the cycle of the applied functionalities. In this article, We offer a valid and advanced method for ranking the fuzzy numbers based on the Distance Measure Meter. In addition to the Distance Measure, we define a particular condition of the generalized fuzzy numbers. Having discussed some examples in this regard, we touch upon the advantages of this new method.

2014

The fuzzy set theory has been applied in many fields such as management, engineering etc. In modern management applications ranking using fuzzy numbers are the most important aspect in decision making process. This paper proposes a new method on the incentre of centroids and uses of Euclidean distance to ranking generalized hexagonal fuzzy numbers. We have used a ranking method for ordering fuzzy numbers based on areas and weights of generalized fuzzy numbers.

Advances in Fuzzy Systems

Ranking fuzzy numbers are an important aspect of decision making in a fuzzy environment. Since their inception in 1965, many authors have proposed different methods for ranking fuzzy numbers. However, there is no method which gives a satisfactory result to all situations. Most of the methods proposed so far are nondiscriminating and counterintuitive. This paper proposes a new method for ranking fuzzy numbers based on the Circumcenter of Centroids and uses an index of optimism to reflect the decision maker's optimistic attitude and also an index of modality that represents the neutrality of the decision maker. This method ranks various types of fuzzy numbers which include normal, generalized trapezoidal, and triangular fuzzy numbers along with crisp numbers with the particularity that crisp numbers are to be considered particular cases of fuzzy numbers.

In this paper, we present a tool to help reduce the uncertainty presented in the resource selection problem when information is subjective in nature. The candidates and the "ideal" resource required by evaluators are modeled by fuzzy subsets whose elements are trapezoidal fuzzy numbers (TrFN). By modeling with TrFN the subjective variables used to determine the best among a set of resources, one should take into account in the decision-making process not only their expected value, but also the uncertainty that they express. A mean quadratic distance (MQD) function is defined to measure the separation between two TrFN. It allows us to consider the case when a TrFN is wholly or partially contained in another. Then, for each candidate a weighted mean asymmetric index (WMAI) evaluates the mean distance between the TrFNs for each of the variables and the corresponding TrFNs of the "ideal" candidate, allowing the decision-maker to choose among the candidates. We apply this index to the case of the selection of the product that is best suited for a "pilot test" to be carried out in some market segment.

Ranking fuzzy numbers plays a very important role in linguistic decision making and some other fuzzy application systems such as data analysis, artificial intelligence and socio economic systems. Various approaches have been proposed in the literature for the ranking of fuzzy numbers and most of the methods seem to suffer from drawbacks. In this paper a new method is proposed to rank fuzzy numbers. This method is based on the centroid of centroids of generalized trapezoidal fuzzy numbers and allows the participation of decision maker by using an index of optimism to reflect the decision maker’s optimistic attitude and also an index of modality that represents the importance of considering the areas of spreads by the decision maker. This method is relatively simple and easier in computation and ranks various types of fuzzy numbers along with crisp fuzzy numbers as special case of fuzzy numbers.

The article examines the use of fuzzy arithmetic in the problems of classification and cluster analysis based on the representation of data in the form of fuzzy gradations proposed by the author. The advantages of the proposed approach are analyzed. This approach allows us to expand the range of solvable problems, to increase the reliability of the distribution of objects by classes and reduce the ambiguity of the distribution of objects by clusters and levels of order. It makes possible to substantiate the choice of a measure of similarity between objects, to smooth the influence of errors associated with data inconsistency; at the same time, the complexity of analysis and calculations is significantly reduced. Examples are considered to illustrate the application of the proposed approach.

2012

Ordering fuzzy numbers plays an important role in approximate reasoning, optimization, forecasting, decision making, controlling, scheduling and other usage. This paper illustrates a ranking method for ordering fuzzy numbers based on Area, Mode, Spreads and Weights of generalized (non-normal)fuzzy numbers. The area used in this method is obtained from the non-normal trapezoidal fuzzy number, first by splitting the generalized trapezoidal fuzzy numbers into three plane figures and then calculating the Centroids of each plane figure followed by the Orthocentre of these Centroids and then finding the area of this Orthocentre from the original point. In this paper, we also apply mode and spreads in those cases where the discrimination is not possible. This method is simple in evaluation and can rank various types of fuzzy numbers and also crisp numbers which are considered to be a special case of fuzzy numbers.