© Solving the fuzzy 1-center problem with an innovative ranking method for fuzzy numbers

Ranking of fuzzy numbers has become an important research direction for decision-making problems due to its role to find the best objects under uncertainty. In this paper, we propose a new approach to perform multiple attribute group decision-making (MAGDM) problems using the ranking of interval type-2 fuzzy sets. Initially, a new ranking method for interval type-2 fuzzy numbers based on centroid and rank index has been proposed. Next, we present a comparative study to analyze the ranking values of the proposed method with the existing approaches, where we explore the necessity of the proposed ranking method. After that, a new MAGDM approach has been developed using the proposed ranking procedure to solve uncertain MAGDM problems. Finally, the applicability of the proposed approach has been illustrated using two numerical examples and a case study related to car-sharing problems. The proposed study exhibits a useful way to solve fuzzy MAGDM problems with much efficient manner since it applies interval type-2 fuzzy sets compared to type-1 fuzzy sets to signify the evaluating values and weights of the attributes.

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.

This work summarizes some ranking methods for fuzzy numbers. A point view of reference system is proposed as a general way to establish ranking methods on sets of fuzzy numbers. Four goodness criteria of ranking methods are established, which can be applied further to decision-making on environment with uncertainty. The idea of ranking fuzzy numbers can also be used for ranking stock performance. As an application, a set of real NASDAQ data is adopted in an example to show the details of the ranking method by lexicography that is a reference system as well.

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.

Linear programming problems and ranking of fuzzy numbers

Ranking Fuzzy Numbers with an Area Method Using Circumcenter of Centroids

Coevolutionary makespan optimisation through different ranking methods for the fuzzy flexible job shop

The estimation of various components in science always faces with errors and inaccuracies that are presented in the form of an uncertainty in identifying the considered components. This uncertainty can be represented in different shapes, one of which is the representation of estimated components in the form of fuzzy numbers. In a situation where a shortage of information or ambiguity in the studied system can be seen, fuzzy numbers can be used to express the ambiguity available in the components. In this study, we will introduce new approach for the formation of type-2 fuzzy numbers in which the mean and standard deviations of data are used to determine the uncertainty zone. Finally, with proposed type-2 fuzzy numbers, in proportion to any situation we have, we face with the claims related to the mean and assess with the subject of the statistical hypothesis test with fuzzy approach.

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.

We review type-1 and type-2 fuzzy numbers in this chapter, and propose one way of perceiving the concept of fuzzy numbers by comparing with that of round numbers. There are some definitions of fuzzy numbers, but we particularly adopt the definition often used in fuzzy analysis. Thereby, we emphasize that fuzzy number theory can be reduced to an argument for interval analysis. Moreover, we explain type-2 fuzzy sets and list two specific type-2 fuzzy numbers, one is a (triangular) perfect quasi-type-2 fuzzy number and the other is a triangular shaped type-2 fuzzy number. Finally, we mention the importance and utility of using type-2 fuzzy numbers.

Network analysis is a technique which determines the various sequences of activities concerning a project and the project completion time. The popular methods of this technique which is widely used are the critical path method and program evaluation and review techniques. The aim of this paper is to present an analytical method for measuring the criticality in an (Atanassov) intuitionistic fuzzy project network. Vague parameters in the project network are represented by (Atanassov) intuitionistic trapezoidal fuzzy numbers. A metric distance ranking method for (Atanassov) intuitionistic fuzzy numbers to a critical path method is proposed. (Atanassov) Intuitionistic fuzzy critical length of the project network is found without converting the (Atanassov) intuitionistic fuzzy activity times to classical numbers. The fuzzified conversion of the problem has been discussed with the numerical example. We also apply four different ranking procedures and we compare it with metric distance ranking method. Comparison reveals that the proposed ranking method is better than other raking procedures.

In risk assessment, generally model parameters are affected by uncertainty arises due to vagueness, imprecision, lack of data, small sample sizes etc. Fuzzy set theory and Dempster-Shafer theory (In short DST) of evidence should be explored to handle this type of uncertainty. Representation of parameters of risk assessment models may be Dempster-Shafer structure (in short DSS) and fuzzy numbers. To deal with such situations, it is important to device new techniques. This paper presents two algorithms to combine Dempster-Shafer structure with generalized/normal fuzzy focal elements, generalized/normal fuzzy numbers within the same framework. Sampling technique for evidence theory and alpha-cut for fuzzy numbers are considered to execute the algorithms. Finally, results are obtained in the form of fuzzy numbers (normal/generalized) at different fractiles.

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.

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.