A developed distance method for ranking generalized fuzzy numbers

Solving fuzzy linear programming (FLP) requires the employment of a consistent ranking of fuzzy numbers. Ineffective fuzzy number ranking would lead to a flawed and erroneous solving approach. This paper presents a comprehensive and extensive review on fuzzy number ranking methods. Ranking techniques are categorised into six classes based on their characteristics. They include centroid methods, distance methods, area methods, lexicographical methods, methods based on decision maker’s viewpoint, and methods based on left and right spreads. A survey on solving approaches to FLP is also reported. We then point out errors in several existing methods that are relevant to the ranking of fuzzy numbers and thence suggest an effective method to solve FLP. Consequently, FLP problems are converted into non-fuzzy single (or multiple) objective linear programming based on a consistent centroid-based ranking of fuzzy numbers. Solutions of FLP are then obtained by solving corresponding crisp single (or multiple) objective programming problems by conventional methods.

Fuzzy numbers represent ambiguous numeric values; therefore, it is difficult to rank them according to their magnitude. In making decisions, it is important to rank the fuzzy numbers. Since fuzzy numbers in a fuzzy environment often measure alternatives, a comparison of these fuzzy numbers is, in fact, a comparison of alternatives. There are a lot of different types of methods for ranking fuzzy numbers that exist in the literature. Still, there is no single method superior to all others regarding discrimination and consistency. This paper proposes a method for ranking fuzzy numbers with a unified integral value that multiplies two discriminatory components, the mode area integral and a linear sum of the absolute values of the integrals of the left and the right limits of alpha-cut of the normalized form of a fuzzy number. The method can rank two or more fuzzy numbers simultaneously, regardless of their linear or nonlinear membership functions. Furthermore, the unified integral value consistently ranks fuzzy numbers and their images and symmetric fuzzy numbers with the same altitude. Various types of fuzzy numbers are used in examples for comparative studies and investigations.

Ranking fuzzy numbers with integral value

Ranking generalized fuzzy numbers based on centroid and rank index

Ranking fuzzy numbers is an important aspect of decision making in a fuzzy environment. In fuzzy decision making problems, fuzzy numbers must be ranked before an action is taken by a decision maker. This article is about ranking Fuzzy numbers and describes a ranking method for ordering LR fuzzy numbers based on the area of fuzzy numbers. This method is simple in evaluation and can rank various types of LR fuzzy numbers and also crisp numbers which are considered to be a special class of fuzzy numbers.

The ranking of fuzzy numbers has an important position in fuzzy application models and systems. In this paper, a two-dimensional approach is proposed by the Monte Carlo simulation technique, and the simulation formula of the possibility degree is established. The algorithms for ranking interval numbers, triangular fuzzy numbers and trapezoidal fuzzy numbers are elaborated on. Moreover, the axiomatic properties of the possibility degree formulae are investigated. It is found that the simulation formula of the possibility degree is reasonable for the satisfaction of the considered axiomatic properties. Finally, by considering a set of fuzzy numbers and the effect of a viewpoint, some comparisons are reported by carrying out numerical examples. The observations reveal that the proposed simulation method is effective to reflect the uncertainty of fuzzy numbers and coherent with the human intuition.

Ranking Fuzzy Numbers with an Area Method Using Circumcenter of Centroids

L.A.Zadeh introduced the concept of fuzzy set theory as the generalization of classical set theory in 1965 and further it has been generalized to intuitionistic fuzzy sets (IFSs) by Atanassov in 1983 to model information by the membership, nonmembership and hesitancy degree more accurately than the theory of fuzzy logic. The notions of intuitionistic fuzzy numbers in different contexts were studied in literature and applied in real life applications. Problems in different fields involving qualitative, quantitative and uncertain information can be modeled better using intutionistic fuzzy numbers introduced in [17] which generalizes intuitionistic fuzzy values [1,7,17], interval valued intuitionistic fuzzy number (IVIFN) [10] than with usual IFNs [5,11,21]. Ranking of fuzzy numbers have started in early seventies in the last century and a complete ranking on the class of fuzzy numbers have achieved by W.Wang and Z.Wang only on 2014. A complete ranking on the class of IVIFNs, using axiomatic set of membership, non membership, vague and precise score functions has been introduced and studied by Geetha et al.[10]. In this paper, a total ordering on the class of IFNs [17] using double upper dense sequence in the interval [0; 1] which generalizes the total ordering on fuzzy numbers (FNs) is proposed and illustrated with examples. Examples are given to show the proposed method on this type of IFN is better than existing methods and this paper will give the better understanding over this new type of IFNs.

Fuzzy risk analysis based on ranking fuzzy numbers using [formula omitted]-cuts, belief features and signal/noise ratios

Ranking fuzzy numbers in decision-making, data analysis and application of fuzzy systems is important. Various methods for ranking fuzzy numbers have been proposed. These methods include: coefficient of variation (CV index), distance between fuzzy sets, centroid point and original point, and weighted mean value. In this article, a new method for ranking LR fuzzy numbers based on the distance (d LR ) is proposed in addition to expressing properties of the ranking function. At the end, we compare proposed method with previous methods.

Ranking fuzzy numbers is important for decision-making problems in a fuzzy environment since the measurements are imprecise in nature. In this paper one new method of ranking fuzzy numbers was proposed by using the angle of triangular and trapezoidal fuzzy number besides alpha-cuts of fuzzy number. Ranking-maker's preference is considered and membership functions of fuzzy numbers don't need to be known before ranking. The results of some preliminary numerical experiments show the discrimination is valid. This new way is simple and efficient for the calculation and comparison

The main objective of this study is to deal with the optimization of a fuzzy transportation problem having costs in fuzzy triangular numbers with demand and supplies as crisp numbers. It introduces a new ranking technique along with a unique approach to convert triangular fuzzy number to trapezoidal fuzzy number. Minimum demand supply algorithm is used to get optimal cost. This technique is compared with some existing methods. Two different ranking approaches based on trapezoidal fuzzy numbers are also compared. The new algorithms are found fast and better as compared to classical approaches.The main objective of this study is to deal with the optimization of a fuzzy transportation problem having costs in fuzzy triangular numbers with demand and supplies as crisp numbers. It introduces a new ranking technique along with a unique approach to convert triangular fuzzy number to trapezoidal fuzzy number. Minimum demand supply algorithm is used to get optimal cost. This technique is compared with some existing methods. Two different ranking approaches based on trapezoidal fuzzy numbers are also compared. The new algorithms are found fast and better as compared to classical approaches.

Based on the area between the curve of the membership function and the horizontal real axis, concepts of left and right wingspans are introduced. By them, a new index, called the w-center for fuzzy numbers is proposed. It is continuous with respect to the convergence of fuzzy number sequence. An intuitive and reasonable ranking method for fuzzy numbers based on their w-center is also established. This new ranking method is useful in fuzzy decision making and fuzzy data mining. The paper also points out that, in literature, some ranking methods based on the centroid of fuzzy numbers are not reasonable.

In this paper we propose a new approach for constructing a probabilistic risk assessment model of innovative project. The method is based on the task of comparison and ranking of fuzzy numbers that has an important role in more applications related to the decision analysis. In the literature there are many approaches to compare fuzzy numbers. The majority of these approaches are based on the quantitative measurements. For that, we propose a new method to calculate the risk level that we can accord to equality X <Y when X and Yare two generalized fuzzy numbers.

Ranking fuzzy numbers is an important issue in decision making analysis, optimisation, artificial intelligence and operations research. Therefore, many methods have been proposed for ranking fuzzy numbers in the literature. However, all of these methods have some limitations and shortcomings. Thus, in this paper, based on coordinates of the centre of incircle of triangles a novel ranking method is proposed to overcome the existing shortcomings. Since the incircle of each triangle is defined as unique, the coordinates of the incircle centre are used in the proposed ranking approach. The novel ranking method is proven to always guarantee consistency between the ranking of fuzzy quantities and that of their images. Unlike many existing methods, the proposed method not only can rank real numbers as well as normal and non-normal fuzzy numbers, but it also has some other noticeable mathematical properties that are proved in this paper. Moreover, several comparative examples are examined to show the superior validity, applicability, efficiency and simplicity of the proposed method in comparison with the existing methods. Finally, a petroleum distribution centre evaluation problem is given as a practical application to demonstrate the usage and applicability of the proposed ranking approach.