An improved ranking method for fuzzy numbers with integral values

Ranking fuzzy numbers is a very important decision-making procedure in decision analysis and applications. The last few decades have seen a large number of approaches investigated for ranking fuzzy numbers, yet some of these approaches are non-intuitive and inconsistent. In 1992, Liou and Wang proposed an approach to rank fuzzy number based a convex combination of the right and the left integral values through an index of optimism. Despite its merits, some shortcomings associated with Liou and Wang's approach include: (i) it cannot differentiate normal and non-normal fuzzy numbers, (ii) it cannot rank effectively the fuzzy numbers that have a compensation of areas, (iii) when the left or right integral values of the fuzzy numbers are zero, the index of optimism has no effect in either the left integral value or the right integral value of the fuzzy number, and (iv) it cannot rank consistently the fuzzy numbers and their images.This paper proposes a revised ranking approach to overcome the shortcomings of Liou and Wang's ranking approach. The proposed ranking approach presents the novel left, right, and total integral values of the fuzzy numbers. The median value ranking approach is further applied to differentiate fuzzy numbers that have the compensation of areas. Finally, several comparative examples and an application for market segment evaluation are given herein to demonstrate the usages and advantages of the proposed ranking method for fuzzy numbers.     

Ranking fuzzy numbers based on epsilon-deviation degree

Although numerous research studies in recent years have been proposed for comparing and ranking fuzzy numbers, most of the existing approaches suffer from plenty of shortcomings. In particular, they have produced counter-intuitive ranking orders under certain cases, inconsistent ranking orders of the fuzzy numbers’ images, and lack of discrimination power to rank similar and symmetric fuzzy numbers. This study's goal is to propose a new epsilon-deviation degree approach based on the left and right areas of a fuzzy number and the concept of a centroid point to overcome previous drawbacks. The proposed approach defines an epsilon-transfer coefficient to avoid illogicality when ranking fuzzy numbers with identical centroid points and develops two innovative ranking indices to consistently distinguish similar or symmetric fuzzy numbers by considering the decision maker's attitude. The advantages of the proposed method are illustrated through several numerical examples and comparisons with the existing approaches. The results demonstrate that this approach is effective for ranking generalized fuzzy numbers and overcomes the shortcomings in recent studies.     

Ranking of fuzzy numbers by using value and angle in the epsilon-deviation degree method

In this paper, a modified epsilon-deviation degree method of ranking fuzzy numbers is proposed. The epsilon-deviation degree method and other ranking methods are available in the literature and applied in the field of decision-making. Despite of the merits, some limitations and shortcomings are observed in these methods. Namely, (1) these methods cannot distinguish fuzzy numbers sharing the same support and different cores, (2) these methods cannot distinguish crisp-valued fuzzy numbers with different heights, (3) these methods also cannot make a preference between a crisp-valued fuzzy number and an arbitrary fuzzy number, (4) if the expectation values of the centroid points are the same for the fuzzy numbers to be compared, then these methods give an incorrect ranking, (5) if fuzzy numbers depict compensation of areas, then these methods fail to give a proper ranking, and (6) further inconsistency in ranking the fuzzy numbers and their images is also observed. Hence, a modified epsilon-deviation degree method is developed, based on the concept of the ill-defined magnitude ‘value’ and the angle of the fuzzy set. The proposed method bears all the properties of epsilon-deviation degree method and overcome all the limitations and shortcomings of this method and other existing methods. Various sets of fuzzy numbers are considered for comparative study between the existing ranking methods and the proposed method for validation. Further, the proposed method seems to outperform in all situations. Risk analysis problem under uncertain environment are often studied under fuzzy domain. Hence, a study is done by applying the proposed method to risk analysis in poultry farming.     

Fuzzy risk analysis based on ranking generalized fuzzy numbers with different left heights and right heights

In this paper, we present a new method for fuzzy risk analysis based on the proposed new fuzzy ranking method for ranking generalized fuzzy numbers with different left heights and right heights. First, we present a fuzzy ranking method for ranking generalized fuzzy numbers with different left heights and right heights. The proposed method considers the areas of the positive side, the areas of the negative side and the centroid values of generalized fuzzy numbers as the factors for calculating the ranking scores of generalized fuzzy numbers with different left heights and right heights. It can overcome the drawbacks of the existing fuzzy ranking methods. Then, we propose a new method for fuzzy risk analysis based on the proposed fuzzy ranking method, where the evaluating values are represented by generalized fuzzy numbers. The proposed fuzzy risk analysis method provides us with a useful way for fuzzy risk analysis based on generalized fuzzy numbers with different left heights and right heights.     

Ranking fuzzy numbers based on the areas on the left and the right sides of fuzzy number

In this paper, a novel method, based on the areas on the left and the right sides of fuzzy numbers is proposed for ranking fuzzy numbers. The merits of the results given here is to overcome certain shortcomings in the recent literature that mostly does not end in the right ordering of fuzzy numbers. The method also has very easy and simple calculations compared to other methods. Moreover, numerical examples are given to compare the proposed method with other existing ones.     

Analyzing the ranking method for L–R fuzzy numbers based on deviation degree

Ranking fuzzy numbers based on their left and right deviation degree (L–R deviation degree) has attracted the attention of many scholars recently, yet most of their ranking methods have two systematic shortcomings that are usually ignored. This paper addresses these shortcomings and proves them through mathematical proofs instead of providing counter-examples. Applying our analyses will help other authors avoid some common errors when building their own ranking index functions. We use Asady’s ranking index function (2010) as an example when we present our arguments and proofs and provide fully detailed analyses of two of the ranking index functions herein. Based on these analyses, an algorithm for detecting inconsistencies in ranking results is proposed, and numerical examples are given to illustrate our arguments.     

A note on ranking generalized fuzzy numbers

Ranking fuzzy numbers plays an important role in decision making under uncertain environment. Recently, Chen and Sanguansat (2011) [Chen, S. M. & Sanguansat, K. (2011). Analyzing fuzzy risk based on a new fuzzy ranking method between generalized fuzzy numbers. Expert Systems with Applications, 38(3), (pp. 2163-2171)] proposed a method for ranking generalized fuzzy numbers. It considers the areas on the positive side, the areas on the negative side and the heights of the generalized fuzzy numbers to evaluate ranking scores of the generalized fuzzy numbers. Chen and Sanguansat’s method (2011) can overcome the drawbacks of some existing methods for ranking generalized fuzzy numbers. However, in the situation when the score is zero, the results of the Chen and Sanguansat’s ranking method (2011) ranking method are unreasonable. The aim of this short note is to give a modification on Chen and Sanguansat’s method (2011) to make the method more reasonable.     

Selecting the best alternatives of fuzzy decision making units with relative distance metric method

Chang et al. (Soft Comput 11:573–584, 2007) proposed a method to rank fuzzy numbers. They employed a relative distance metric method to rank fuzzy numbers; however, there were some problems with the ranking method. In this paper, we want to indicate these problems of Chang’s method, and then propose a revised method, which can avoid these problems for ranking fuzzy numbers. Since the revised method is based on Chang’s method, it is easy to rank fuzzy numbers in a way similar to the original method.     

Ranking function-based solutions of fully fuzzified minimal cost flow problem

The aim of minimal cost flow problem (MCFP) is to find the least transportation cost of a single commodity through a capacitated network. This paper presents a model to deal with one particular group of such problems in which the supply and demand of nodes and the capacity and cost of edges are represented as fuzzy numbers. For easier reference, hereafter, we refer to this group of problems as fully fuzzified MCFP. To represent our model, Hukuhara’s difference and approximated multiplication are used. Thereafter, we sort fuzzy numbers by an order using a ranking function and show that it is a total order, i.e., a reflexive, anti-symmetric, transitive and complete binary relation. Utilizing the proposed ranking function, we transform the fully fuzzified MCFP into three crisp problems solvable in polynomial time. From this standpoint, combinatorial algorithms are provided to solve the above-mentioned problem and find the fuzzy optimal flow. Furthermore, the proposed order is related to the importance weights of the center, the left spread and the right spread of each fuzzy number. Thus, this method is capable of handling the decision maker’s risk taking. By comparing some previous ranking function-based works with our method, the efficiency of the latter is revealed. Finally, an application of our proposed method to petroleum industry is presented.     

A new effect of radius of gyration with neural networks

In 2006, Deng et al. (Comput Math Appl 51: 1127–1136, 2006) proposed a method to evaluate DMU’s. A radius of gyration was employed to rank fuzzy numbers; however, there were some problems with the ranking method. This paper attempts to reveal the issues of Deng’s method and proposes a revised method that can avoid these problems for ranking fuzzy numbers. Since the revised method is based on Deng’s method, it is easy to rank fuzzy numbers in a way similar to the original method.     

Representing complex intuitionistic fuzzy set by quaternion numbers and applications to decision making

Intuitionistic fuzzy sets are useful for modeling uncertain data of realistic problems. In this paper, we generalize and expand the utility of complex intuitionistic fuzzy sets using the space of quaternion numbers. The proposed representation can capture composite features and convey multi-dimensional fuzzy information via the functions of real membership, imaginary membership, real non-membership, and imaginary non-membership. We analyze the order relations and logic operations of the complex intuitionistic fuzzy set theory and introduce new operations based on quaternion numbers. We also present two quaternion distance measures in algebraic and polar forms and analyze their properties. We apply the quaternion representations and measures to decision-making models. The proposed model is experimentally validated in medical diagnosis, which is an emerging application for tackling patient’s symptoms and attributes of diseases.     

Analyzing the ranking method for L–R fuzzy numbers based on deviation degree

Ranking fuzzy numbers based on their left and right deviation degree (L–R deviation degree) has attracted the attention of many scholars recently, yet most of their ranking methods have two systematic shortcomings that are usually ignored. This paper addresses these shortcomings and proves them through mathematical proofs instead of providing counter-examples. Applying our analyses will help other authors avoid some common errors when building their own ranking index functions. We use Asady’s ranking index function (2010) as an example when we present our arguments and proofs and provide fully detailed analyses of two of the ranking index functions herein. Based on these analyses, an algorithm for detecting inconsistencies in ranking results is proposed, and numerical examples are given to illustrate our arguments.     

A novel approach to multi-attribute decision making based on intuitionistic fuzzy sets

In this paper we present a new approach to solve multi-attribute decision making problems in intuitionistic fuzzy environment. This approach is based on a new ranking method of intuitionistic fuzzy sets, in which the evaluated values (in the form of intervals) of the same alternative with different attributes are considered as one unified entity. According to people’s intuition, the ranking method proposed in this paper is mainly grounded on a revised score function and a revised accuracy function of intuitionistic fuzzy sets. Different from the traditional methods, in this new approach, the degree of membership, the degree of nonmembership and the degree of hesitation are considered with various importance in reflecting the true image of the respective alternative. Furthermore, an optimization model is established to estimate the relative degree of importance of each quantity. Finally, two practical examples are provided to illustrate our approach.     

A developed distance method for ranking generalized fuzzy numbers

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 α-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.     

An approximate approach for ranking fuzzy numbers based on left and right dominance

This study presents an approximate approach for ranking fuzzy numbers based on the left and right dominance. The proposed approach only requires a few left and right spreads at some -levels of fuzzy numbers to determine the respective dominance of one fuzzy number over the other. The total dominance is then determined by combining the left and right dominance based on a decision maker's optimistic perspectives. Such a dominance is useful in ranking the fuzzy numbers when membership functions cannot be acquired. The approach proposed herein is relatively simple in terms of computational efforts and is efficient when ranking a large quantity of fuzzy numbers. By using a few left and right spreads, two groups of examples demonstrate the accuracy and applicability of the proposed approach.     

Reprioritization of failures in a silane supply system using an intuitionistic fuzzy set ranking technique

Most of the current failure mode, effects, and criticality analysis (FMECA) methods use the risk priority number (RPN) value to evaluate the risk of failure. However, the traditional RPN methodology has been criticized to have several shortcomings. These shortcomings are addressed in this paper. Therefore, an efficient and simplified algorithm to evaluate the risk of failure is needed. This paper proposes a new approach, which utilizes the intuitionistic fuzzy set ranking technique for reprioritization of failures in a system FMECA. The proposed approach has two major advantages: (1) it resolves some of the shortcomings of the traditional RPN method, and (2) it provides an evaluation of the redundancy place, which can assist the designer in making correct decisions to make a safer and more reliable product design. In numerical verification, an FMECA of a silane supply system is presented as a numerical example. After comparing results from the proposed method and two other approaches, this research found that the proposed approach can reduce more duplicate RPN numbers and get a more accurate, reasonable risk ranking.     

Analyzing fuzzy risk based on a new fuzzy ranking method between generalized fuzzy numbers

In this paper, we present a new method for analyzing fuzzy risk based on a new method for ranking generalized fuzzy numbers. First, we present a new method for ranking generalized fuzzy numbers. It considers the areas on the positive side, the areas on the negative side and the heights of the generalized fuzzy numbers to evaluate ranking scores of the generalized fuzzy numbers. The proposed method can overcome the drawbacks of some existing methods for ranking generalized fuzzy numbers. Then, we apply the proposed method for ranking generalized fuzzy numbers to develop a new method for dealing with fuzzy risk analysis problems. The proposed method provides us with a useful way to deal with fuzzy risk analysis problems based on generalized fuzzy numbers.     

GROUNDING EMOTIONS IN ADAPTIVE SYSTEMS: VOLUME II

ABSTRACT

Ranking fuzzy numbers plays a very important role in decision-making problems. Existing centroid-index ranking methods have some drawbacks. In this article, a new centroid-index ranking method of fuzzy numbers is proposed. The proposed method is using the ideal of Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS). Some numerical examples show that the new method can overcome the drawbacks of the existing methods. Finally, a human selection problem is used to illustrate the efficiency of the proposed fuzzy ranking method.     

Conceptual procedure for ranking fuzzy numbers based on adaptive two-dimensions dominance

Many methods for ranking of fuzzy numbers have been proposed. However, these methods just can apply to rank some types of fuzzy numbers (i.e. normal, non-normal, positive, and negative fuzzy numbers), and many ranking cases can just rank by their graphs intuitively. So, it is important to use proper methods in the right condition. In this paper, a conceptual procedure is proposed to describe how to use intuitive ranking and some technical ranking methods properly. We also introduce a new ranking fuzzy numbers approach that can adjust experts confidence and optimistic index of decision maker using two parameters ( and ) to handle the problems and find the best solutions. After illustrate many numerical examples following our conceptual procedure the ranking results are validity.     

基于FVIKOR的三角模糊数型多准则决策方法

针对准则值和准则权重均为三角模糊数的多准则决策问题, 提出一种三角模糊数型VIKOR(FVIKOR) 方法. 首先, 分析FVIKOR方法中直接运用三角模糊数运算规则计算群体效用值、个体遗憾值和妥协解可能违反三角模糊数左中右端点值逐渐增加的基本特性, 提出实施三角模糊数去模糊化的解决策略; 然后, 设计去模糊化参数优化模型, 并给出FVIKOR应用的具体步骤; 最后, 通过具体算例表明了所提出方法的实施过程和有效性.