Ranking fuzzy numbers by distance minimization

In this paper, we proposed a defuzzification using minimizer of the distance between the two fuzzy numbers. Then, we obtain the nearest point with respect to a fuzzy numbers and by considering the nearest point, we can present a ranking method for the fuzzy numbers. Also we give two new properties for ordering. Theorems and remarks are proposed for existence and uniqueness of the nearest point. The method is illustrated by numerical examples and compared with other methods.     

A new parametric method for ranking fuzzy numbers

Ranking fuzzy numbers is important in decision-making, data analysis, artificial intelligence, economic systems and operations research. In this paper, to overcome the limitations of the existing studies and simplify the computational procedures an approach to ranking fuzzy numbers based on α$\alpha$-cuts is proposed. The approach is illustrated by numerical examples, showing that it overcomes several shortcomings such as the indiscriminative and counterintuitive behavior of existing fuzzy ranking approaches.     

A fast method of ranking alternatives using fuzzy numbers

We show a very fast method of ranking alternatives using fuzzy numbers discussed in [1] and [2].     

RM approach for ranking of generalized trapezoidal fuzzy numbers

Ranking of fuzzy numbers play an important role in decision making, optimization and forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, with the help of several counter examples, it is proved that ranking method proposed by Chen and Chen (Expert Systems with Applications 36 (3): 6833) is incorrect. The main aim of this paper is to propose a new approach for the ranking of generalized trapezoidal fuzzy numbers. The proposed ranking approach is based on rank and mode so it is named as an RM approach. The main advantage of the proposed approach is that the proposed approach provides the correct ordering of generalized and normal trapezoidal fuzzy numbers and also the proposed approach is very simple and easy to apply in the real life problems. It is shown that proposed ranking function satisfies all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Fuzzy Sets and Systems 118 (3): 375).     

A new method for ranking fuzzy numbers and its application to group decision making

In this paper, a new method for comparing fuzzy numbers based on a fuzzy probabilistic preference relation is introduced. The ranking order of fuzzy numbers with the weighted confidence level is derived from the pairwise comparison matrix based on 0.5-transitivity of the fuzzy probabilistic preference relation. The main difference between the proposed method and existing ones is that the comparison result between two fuzzy numbers is expressed as a fuzzy set instead of a crisp one. As such, the ranking order of n fuzzy numbers provides more information on the uncertainty level of the comparison. Illustrated by comparative examples, the proposed method overcomes certain unreasonable (due to the violation of the inequality properties) and indiscriminative problems exhibited by some existing methods. More importantly, the proposed method is able to provide decision makers with the probability of making errors when a crisp ranking order is obtained. The proposed method is also able to provide a probability-based explanation for conflicts among the comparison results provided by some existing methods using a proper ranking order, which ensures that ties of alternatives can be broken.     

Comparing and ranking fuzzy numbers using ideal solutions

This paper presents a new approach for comparing and ranking fuzzy numbers in a simple manner in decision making under uncertainty. The concept of ideal solutions is sensibly used, and a distance-based similarity measure between fuzzy numbers is appropriately adopted for effectively determining the overall performance of each fuzzy number in comparing and ranking fuzzy numbers. As a result, all the available information characterizing a fuzzy number is fully utilized, and both the absolute position and the relative position of fuzzy numbers are adequately considered, resulted in consistent rankings being produced in comparing and ranking fuzzy numbers. The approach is computationally simple and its underlying concepts are logically sound and comprehensible. A comparative study is conducted on the benchmark cases in the literature that shows the proposed approach compares favorably with other approaches examined.     

Comparison of fuzzy numbers using a fuzzy distance measure

A new approach for ranking fuzzy numbers based on a distance measure is introduced. A new class of distance measures for interval numbers that takes into account all the points in both intervals is developed first, and then it is used to formulate the distance measure for fuzzy numbers. The approach is illustrated by numerical examples, showing that it overcomes several shortcomings such as the indiscriminative and counterintuitive behavior of several existing fuzzy ranking approaches.     

A theoretical development on a fuzzy distance measure for fuzzy numbers

The objective of this paper is to introduce a fuzzy distance measure for generalized fuzzy numbers (GFN). It computes the fuzzy distance between two generalized fuzzy numbers and also LR-type fuzzy numbers. The metric properties of the proposed measure are also studied. Some numerical examples have been considered here for applying the proposed fuzzy distance measure and the results are compared.     

A comparative study of ranking fuzzy numbers based on regular weighted function

Fuzzy systems have gained more and more attention from researchers and practitioners of various fields. In such systems, the output represented by a fuzzy set sometimes needs to be transformed into a scalar value, and this task is known as the defuzzification process. Several analytic methods have been proposed for this problem, but in this paper, the researchers suggest a modified approach to the problem of defuzzification, using the bi-symmetric weighted distance between two fuzzy numbers. This defuzzification can be used as a crisp approximation with respect to a fuzzy quantity. By considering this and with benchmark between fuzzy numbers, we can present a method for ranking, which can effectively rank variou s fuzzy numbers and their images and overcome the shortcomings of the previous techniques. After illustrating many numerical examples, following our procedure, the ranking results become valid.     

A general method for calculating functions of fuzzy numbers

This paper discusses the calculation of functions of fuzzy numbers. Our general approach follows that of the fuzzy weighted algorithm (FWA) of Wong and Dong. However, by developing a classification scheme for functions and their arguments, we show that there are many cases for which the FWA, and all other published methods, will give wrong results. We suggest an alternative approach which will work in all cases. For certain restricted classes of functions, we develop methods which require less computation than the FWA.     

A fuzzy ranking method with range reduction techniques

For ranking alternatives based on pairwise comparisons, current analytic hierarchy process (AHP) methods are difficult to use to generate useful information to assist decision makers in specifying their preferences. This study proposes a novel method incorporating fuzzy preferences and range reduction techniques. Modified from the concept of data envelopment analysis (DEA), the proposed approach is not only capable of treating incomplete preference matrices but also provides reasonable ranges to help decision makers to rank decision alternatives confidently.     

A new method for solving fuzzy transportation problems using ranking function

In the literature, several methods are proposed for solving transportation problems in fuzzy environment but in all the proposed methods the parameters are represented by normal fuzzy numbers. [S.H. Chen, Operations on fuzzy numbers with function principal, Tamkang Journal of Management Sciences 6 (1985) 13–25] pointed out that in many cases it is not to possible to restrict the membership function to the normal form and proposed the concept of generalized fuzzy numbers. There are several papers in the literature in which generalized fuzzy numbers are used for solving real life problems but to the best of our knowledge, till now no one has used generalized fuzzy numbers for solving the transportation problems. In this paper, a new method is proposed for solving fuzzy transportation problems by assuming that a decision maker is uncertain about the precise values of the transportation cost, availability and demand of the product. In the proposed method transportation cost, availability and demand of the product are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed method a numerical example is solved and the obtained results are compared with the results of existing methods. Since the proposed method is a direct extension of classical method so the proposed method is very easy to understand and to apply on real life transportation problems for the decision makers.     

The similarity measure of generalized fuzzy numbers based on interval distance

In this paper, we proposed a new interval distance of two fuzzy numbers that satisfy on metric properties. Also, this metric distance satisfies on some of the other properties. Then, we used this metric for similarity measure. Finality, we tested with some examples.     

An original approach to ranking fuzzy numbers by inclusion index and Bitset Encoding

A variety of methods for ranking fuzzy sets has been suggested. Generally, these methods fall under two main categories: a fuzzy-real sets mapping, a dominance relation of one fuzzy set over another. The original approach proposed in this paper belongs to the second category, as the ranking is based on the degree of inclusion in the MIN of two fuzzy numbers. The novelty lies mainly in the intuitive connection between the topological relationship of fuzzy shapes (triangles, trapezoids, etc.) and the measure of inclusion or dominance referred as inclusion index. This connection led to the classification of different topological relationships into classes identified by a binary pattern. This operation is referred to as Bitset Encoding. Consequently, the outcome of a ranking is already decided for most cases by merely identifying its pattern. Ultimately, the method is validated by the axiomatic system of Wang and Kerre and proven to be a reliable, efficient and strong potential alternative to the other prominent methods.     

A new computational method for solving fully fuzzy linear systems of triangular fuzzy numbers

In this paper, a new computational method is proposed to solve fully fuzzy linear systems (FFLS) of triangular fuzzy numbers based on the computation of row reduced echelon form for solving the crisp linear system of equations. The method is illustrated by solving three numerical examples. As compared to the existing methods, the proposed method is easy to understand and to apply for solving FFLS occurring in real life situations and scientific applications. The primary advantage of the proposed method is that, by using it, the consistency of the FFLS can be checked easily and nature of the solutions of an FFLS can also be obtained which may be unique or infinite.     

An interactive fuzzy satisficing method for structured multiobjective linear fractional programs with fuzzy numbers

In this paper, by considering the experts' vague or fuzzy understanding of the nature of the parameters in the problem formulation process, multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated. Using the a-level sets of fuzzy numbers, the corresponding nonfuzzy a-multiobjective linear fractional programming problem is introduced. The fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions including nonlinear ones. Through the introduction of extended Pareto optimality concepts, if the decision maker specifies the degree a and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the minimax problems for which the Dantzig-Wolfe decomposition method and Ritter's partitioning procedure are applicable. Then a linear programming-based interactive fuzzy satisficing method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an extended Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.     

An improved fuzzy time series forecasting method using trapezoidal fuzzy numbers

One of the major drawbacks of the existing fuzzy time series forecasting models is the fact that they only provide a single-point forecasted value just like the output of the traditional time series methods. Hence, they cannot provide a decision analyst more useful information. The aim of this present research is to design an improved fuzzy time series forecasting method in which the forecasted value will be a trapezoidal fuzzy number instead of a single-point value. Furthermore, the proposed method may also increase the forecasting accuracy. Two numerical data sets were used to illustrate the proposed method and compare the forecasting accuracy with three fuzzy time series methods. The results of the comparison indicate that the proposed method can generate forecasting values that are more accurate.     

On ranking fuzzy sets

A mathematical model describing a fuzzy ordering of fuzzy sets is suggested. In this model an induced fuzzy ordering is defined as the inverse image of the natural linear ordering on real numbers under natural fuzzy correspondence between fuzzy sets and real numbers. The transitivity property of the strict fuzzy ordering associated with the induced fuzzy ordering is established.     

Transitive fuzzy orderings of fuzzy numbers

In this paper we introduce a generalization of the Baas-Kwakernaak index by replacing the min operation in their definition by a t-norm. Some properties of the thus defined induced fuzzy ordering are established. In particular, it is shown that restrictions of the induced fuzzy ordering on some special classes of fuzzy numbers are reflexive fuzzy orders.     

New method to posynomial geometric programming of trapezoidal fuzzy numbers

This paper presents a method for solving posynomial geometric programming with fuzzy coefficients. By utilizing comparison of fuzzy numbers with a method, the programming with fuzzy coefficients is reduced to the programming with constant coefficients. Then the programming with fuzzy coefficients can be solved by using a method for posynomial geometric programming. Finally, one comparative example is used to illustrate advantage of the new method.