Ranking fuzzy numbers with integral value

Ranking Fuzzy Numbers with an Area Method Using Circumcenter of Centroids

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.

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.

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

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.

Fuzzy logic is one of the effective tools to handle uncertainty and vagueness in engineering and mathematics. One major part of fuzzy logic is ranking fuzzy numbers. In many fuzzy program systems, ranking fuzzy numbers has a remarkable role in decision making and data analysis. Despite the fact that a variety of methods exists for ranking fuzzy numbers, no one can rank fuzzy numbers perfectly in all cases and situations. In this paper, a new method for ranking fuzzy numbers based on the left and right using distance method and ?-cut has been presented. To achieve this, a fuzzy distance measure between two generalized fuzzy numbers is proposed. The new measure is expanded with the help of the fuzzy ambiguity measure. The calculation of this method is derived from generalized trapezoidal fuzzy numbers and distance method concepts. Furthermore, a comparison of generalized fuzzy numbers between the proposed method and other resembled methods is provided.

Multi-criteria decision making method is used in such cases having multiple targets. Multi-objective linear programming problems, which deal with uncertain measurements for both objectives and constraints, are solved by fuzzy multi-objective linear programming methods. Analytic hierarchy process (AHP), one of the multi-criteria decision-making methods based on weighting of objectives, is designed to solve complex problems. In this study, multiple criteria decision making problems and decision makers’ opinions on these problems as well as their solution processes are discussed. Firstly, a fuzzy multi objective linear programming problem is solved using the approaches of Zimmerman and Hybrid. In the literature, these methods are applied using triangular fuzzy numbers based on decision makers’ opinions only in cases where the objective function is minimization. In this study, as an alternative to the triangular fuzzy numbers, trapezoidal fuzzy numbers are defined, and both minimization and maximization of objective functions are examined. Finally, a solution algorithm for a fuzzy multi-objective linear programming problem is proposed under some given conditions. The algorithm is applied to a sample problem that involves modeling of a congress organizing which aims to place attendees to accommodations under specific objectives and constraints. In the evaluation phase of the algorithm, targets are weighted by using triangular and trapezoidal fuzzy numbers. At the end, the results of the proposed algorithm are compared with the results of other methods in the literature.

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.

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.

System of simultaneous linear equations plays a vital role in mathematics, Operations Research, Statistics, Physics, Engineering and Social Sciences etc. In many applications at least some of the system's parameters and measurements are represented by fuzzy numbers rather than crisp numbers. Therefore it is imperative to develop mathematical models and numerical procedures to solve such a fuzzy linear system. The general model of a fuzzy linear system whose coefficient matrix is crisp and the right hand side column is an arbitrary fuzzy vector. In the fully fuzzy linear system all the parameters are considered to be fuzzy numbers. Since triangular fuzzy numbers is a special case of trapezoidal fuzzy numbers, hence in this paper we considered fully fuzzy linear system with trapezoidal fuzzy numbers. LU decomposition method for a crisp matrix is well known in solving linear system of equations. We discuss LU decomposition of the coefficient matrix of the fully fuzzy linear system, in which the coefficients are trapezoidal fuzzy numbers.

Ranking generalized fuzzy numbers based on centroid and rank index

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.

In conventional transportation problem (TP), supplies, demands and costs are always certain. In this paper, the author tried to categories the TP under the mixture of certain and uncertain environment and formulates the problem and utilizes the crisp numbers, triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers (TrFNs) to solve the TP. The existing ranking procedure of Liou and Wang is used to transform the type-1 and type-3 fuzzy transportation problem (FTP) into a crisp one so that the conventional method may be applied to solve the TP. The solution procedure differs from TP to type-1 and type-3 FTP in allocation step only. Therefore, the new method called PSK method and new multiplication operation on TrFN is proposed to find the mixed optimal solution in terms of crisp numbers, TFNs and TrFNs. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful. The effectiveness of the proposed method is illustrated by means of a numerical example.

A fuzzy number is an important concept for handling vague, imprecise and ambiguous cases in daily life. We propose to study different types of fuzzy numbers with examples and some results on fuzzy numbers. In biological science, the growth/decay of bacteria is a very important factor in knowing the symptoms/diseases in humans and animals. The decagonal fuzzy number has proved important in studying bacteria, particularly the growth rate in bacteria, which consists of 10 points, so it is not easy to handle this situation by applying triangular or trapezoidal fuzzy numbers. Therefore, we plan to study decagonal fuzzy numbers in detail here.

In this paper, we forecasted the future of a business by using fuzzy trapezoidal numbers and the fuzzy Delphi method. This result is compared with another result obtained using the fuzzy triangular numbers and Delphi method. At last, we see that our method is more general than others. J. Bangladesh Acad. Sci. 45(2); 147-154: December 2021