A short note on methods of ranking fuzzy numbers in risk analysis problems

The numerous studies on comparing and ranking fuzzy numbers clear that this task is still young. However, the observation that many papers do not hesitate to use some incorrect ranking methods in risk analysis problems, encouraged the author to point out the shortcoming of some of these methods. In this note, we review briefly the methods for ranking fuzzy number in risk analysis

Ranking fuzzy numbers by volume of solid of revolution of membership function about axis of support

It is admissible that fuzzy numbers (FNs) are apt for representing imprecise or vague data in real-world problems. While using FNs in decision-making problems, selecting the best alternative among available alternatives is challenging, and therefore, ranking FNs is essential. We can find different studies in the literature, but to our knowledge, no one attempted to rank FNs using the concept of volume. This paper proposes a new method for ranking generalized fuzzy numbers (GFNs) using the volume of the solid obtained by revolving its membership function (MF) about the x-axis. We calculate the volumes of positive and negative sides along with the centroid of a generalized fuzzy number(GFN) to define the fuzzy number(FN) score. This score represents the defuzzified value of FN, is used to select the best alternative, and overcomes the limitations in some existing methods like ranking FNs having the same centroid, crisp numbers, symmetric fuzzy numbers, and FNs with the same core

Fuzzy Risk Analysis for a Production System Based on the Nagel Point of a Triangle

Ordering and ranking fuzzy numbers and their comparisons play a significant role in decision-making problems such as social and economic systems, forecasting, optimization, and risk analysis problems. In this paper, a new method for ordering triangular fuzzy numbers using the Nagel point of a triangle is presented. With the aid of the proposed method, reasonable properties of ordering fuzzy numbers are verified. Certain comparative examples are given to illustrate the advantages of the new method. Many papers have been devoted to studies on fuzzy ranking methods, but some of these studies have certain shortcomings. The proposed method overcomes the drawbacks of the existing methods in the literature. The suggested method can order triangular fuzzy numbers as well as crisp numbers and fuzzy numbers with the same centroid point. An application to the fuzzy risk analysis problem is given, based on the suggested ordering approach

An improvised similarity measure for generalized fuzzy numbers

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 the improvised similarity measure is presented and discussed. The proposed similarity measure can be incorporated in decision making procedure with fuzzy environment for ranking purposes

A new lexicographical approach for ranking fuzzy numbers

In the literature many ranking methods have been proposed for comparing the fuzzy numbers, most of them suffer from plenty of shortcomings such as complex calculations, inconsistency with human intuition. To overcome such shortcomings, a new ranking method is proposed for L-R flat fuzzy numbers which is based on the lexicographical ordering approach. It is shown that proposed ranking method satisfies all the reasonable properties of the ordering fuzzy quantities proposed by Wang & Kerre (Fuzzy Sets and Systems 118(2001) 375-385). Finally a comprehensive comparison is done between the existing ranking methods with the proposed one to demonstrate the effectiveness of the proposed ranking method. Keywords: Ranking method, L-R flat fuzzy number