On the Issue of Comparison of 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.

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.

Journal of Mahani Mathematical Research Center, 2013

In a previous work, we introduced particular fuzzy numbers and discussed some of their properties. In this paper we use the comparison method introduced by Dorohonceanu and Marin[5] to compare between these fuzzy numbers.

Journal of Fuzzy Set Valued Analysis, 2014

This study presents an approximate approach for ranking fuzzy numbers based on the centroid point of a fuzzy number and its area. The total approximate is determined by convex combining of fuzzy number's relative and its area that based on decision maker's optimistic perspectives. The proposed approach is simple in terms of computational efforts and is efficient in ranking a large quantity of fuzzy numbers. By a group of examples in [3] demonstrate the accuracy and applicability of the proposed approach. Finally by this approach, a new distance is introduced between two fuzzy numbers.

Mathematical Problems in Engineering, 2013

This study presents an approximate approach for ranking fuzzy numbers based on the centroid point of a fuzzy number and its area. The total approximate is determined by convex combining of fuzzy number's relative and its area that is based on decision maker's optimistic perspectives. The proposed approach is simple in terms of computational efforts and is efficient in ranking a large quantity of fuzzy numbers. A group of examples by demonstrate the accuracy and applicability of the proposed approach. Finally by this approach, a new measure is introduced between two 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

Springer eBooks, 2019

In this chapter, preliminaries related to fuzzy numbers have been discussed. Fuzzy numbers and fuzzy arithmetic may be considered as an extension of classical real numbers and its arithmetic. As such, we may understand fuzzy arithmetic as basics for handling fuzzy eigenvalue problems, nonlinear equations, system of nonlinear equations (Abbasbandy and Asady 2004), differential equations (Chakraverty et al. 2016), etc. There exist different types of fuzzy numbers as discussed in Hanss (2005), but for the sake of completeness of the chapter, triangular, trapezoidal, and Gaussian fuzzy numbers based on the membership functions have only been included here. Further, the conversions of these fuzzy numbers to fuzzy intervals with respect to the concept of intervals (Chap. 1) are incorporated. In this regard, the interval arithmetic mentioned in Chap. 1 has been further extended to fuzzy intervals in Sect. 3.4. 3.1 Preliminaries of Fuzzy Numbers A convex fuzzy setà is a fuzzy set having membership function μÃ(x), satisfying μÃ(λx 1 + (1 − λ)x 2) ≥ min(μÃ(x 1), μÃ(x 2)), (3.1) where x 1 , x 2 ∈ X and λ ∈ [0, 1]. Figure 3.1 depicts convex and non-convex fuzzy sets. Convex fuzzy sets defined with respect to universal set (set of all real numbers) may be interpreted as fuzzy numbers. In this respect, the classical definition of fuzzy number is given below. Fuzzy number: A fuzzy setà is referred as a fuzzy numberã if the following properties are satisfied:

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.

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.

CERN European Organization for Nuclear Research - Zenodo, 2022

In multi criteria decision making, if the judgments are expressed as fuzzy numbers, the normalization process is necessary in obtaining the rank of alternatives. Extent analysis method on Fuzzy Analytical Hierarchy Process is a popular decision making method introduced by D.Y. Chang(1996). This method was applied on various applications to obtain the rank of alternatives. The normalization method is to normalize Triangular Fuzzy Numbers, it has some flaws according to Wang and Elhag (2006). This paper presents the modified extent analysis method with accurate normalization procedure and an extent analysis method was introduced to find the weights of alternatives when the judgments are provided as Trapezoidal Fuzzy Numbers.