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.     

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.     

基于TOPSIS 的区间直觉模糊数排序法

基于传统的逼近理想解排序法(TOPSIS) 思想, 运用区间直觉模糊数的欧氏距离, 给出区间直觉模糊数相对于最大区间直觉模糊数的贴近度公式, 并给出区间直觉模糊数贴近度所具有的优良性质, 这些性质表明贴近度作为排序指标是合理的. 通过与文献中有关区间直觉模糊数排序法的对比分析, 表明基于贴近度的排序方法具有更高的区分能力. 运用新的排序指标提出一种区间直觉模糊多属性决策方法, 并通过实例表明了所提出方法的有效性．

     

Ranking of fuzzy numbers by fuzzy mapping

In this work, the concepts of interval function, the mean value and α lower percentile of a fuzzy number are presented. Also, we defined a large family of fuzzy numbers. Then, we obtained a method to rank them. Herein, the approach proposed is relatively simple in terms of computational efforts and is efficient for ranking fuzzy numbers. Finally, the method is illustrated by numerical examples and compared with other methods.     

Ranking type-2 fuzzy numbers

Type-2 fuzzy sets are a generalization of the ordinary fuzzy sets in which each type-2 fuzzy set is characterized by a fuzzy membership function. In this paper, we consider the problem of ranking a set of type-2 fuzzy numbers. We adopt a statistical viewpoint and interpret each type-2 fuzzy number as an ensemble of ordinary fuzzy numbers. This enables us to define a type-2 fuzzy rank and a type-2 rank uncertainty for each intuitionistic fuzzy number. We show the reasonableness of the results obtained by examining several test cases.     

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 of fuzzy numbers by sign distance

Several different strategies have been proposed for ranking of fuzzy numbers. These include methods based on the coefficient of variation (CV index), distance between fuzzy sets, centroid point and original point, and weighted mean value. Each of these techniques has been shown to produce non-intuitive results in certain cases. In this paper we propose a modification of the distance based approach called the sign distance, which is both efficient to evaluate and able to overcome the shortcomings of the previous techniques. The calculation of the proposed method is far simpler than the other approaches.     

Ranking L-R fuzzy number based on deviation degree

This paper proposed a novel approach to ranking fuzzy numbers based on the left and right deviation degree (L-R deviation degree). In the approach, the maximal and minimal reference sets are defined to measure L-R deviation degree of fuzzy number, and then the transfer coefficient is defined to measure the relative variation of L-R deviation degree of fuzzy number. Furthermore, the ranking index value is obtained based on the L-R deviation degree and relative variation of fuzzy numbers. Additionally, to compare the proposed approach with the existing approaches, five numerical examples are used. The comparative results illustrate that the approach proposed in this paper is simpler and better.     

Ranking fuzzy numbers in the setting of possibility theory

The arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood. Equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals, which is addressed in this paper. A complete set of comparison indices is proposed in the framework of Zadeh's possibility theory. It is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers. Moreover, this approach is related to previous ones, and its possible extension to the ranking of n fuzzy numbers is discussed at length. Finally, it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities.     

基于相似度的区间数排序

为设计一种简便而又科学的区间数排序方法,在综合考虑区间数的特点、提取区间数中的隐形数据、对数据进行处理等基础上,根据各个区间数与目标区间数相似度程度的不同,提出了一种新的对区间数进行排序的方法,并提出了这种排序方法的一些优良性质,如:序关系完全性、传递性、等价替换性、区间数长度相等且区间数的属性为效益型时中心越大越好、区间数中心重合时区间越小越好等,充分表明了这种方法的科学性与可行性。最后,通过一个算例说明这种简单、方便、科学的排序方法的操作过程。     

Ranking of fuzzy numbers by a new metric

In this paper, a new approach for comparison among fuzzy numbers based on new metric distance (D _TM) is proposed. All reasonable properties of ranking function are proved. At first, the distance on the interval numbers based on convex hall of endpoints is proposed. The existing distance measures for interval numbers, (Bardossy and Duckstein in Fuzzy rule-based modeling with applications to geophysical, biological and engineering systems. CRC press, Boca Raton, 1995; Diamond in Info Sci 46:141–157, 1988; Diamond and Korner in Comput Math Appl 33:15–32, 1997; Tran and Duckstein in Fuzzy Set Syst 130:331–341, 2002; Diamond and Tanaka Fuzzy regression analysis. In: Slowinski R (ed) Fuzzy sets in decision analysis, operations research and statistics. Kluwer, Boston, pp 349–387, 1998) do not satisfy the properties of a metric distance, while the proposed distance does. It is extended to fuzzy numbers and its properties are proved in detail. Finally, we compare the proposed definition with some of the known ones.     

Fuzzy risk analysis based on similarity measures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operators

In this paper, we present a new method for fuzzy risk analysis based on a new similarity measure between interval-valued fuzzy numbers and new interval-valued fuzzy number arithmetic operators. First, we present a new similarity measure between interval-valued fuzzy numbers. The proposed similarity measure considers the similarity of the gravities on the X-axis between upper fuzzy numbers, the difference of the spreads between upper fuzzy numbers, the heights of the upper fuzzy numbers, the degree of similarity on the X-axis between interval-valued fuzzy numbers, and the gravities on the Y-axis between interval-valued fuzzy numbers. We also present three properties of the proposed similarity measure between interval-valued fuzzy numbers. Then, we present new interval-valued fuzzy number arithmetic operators. Finally, we apply the proposed similarity measure between interval-valued fuzzy numbers and the proposed interval-valued fuzzy number arithmetic operators to propose a fuzzy risk analysis algorithm to deal with fuzzy risk analysis problems. The proposed method provides a useful way for handling fuzzy risk analysis problems based on interval-valued fuzzy numbers.     

基于布尔矩阵的区间数排序方法

讨论了10 个区间数排序的可能度公式, 分析了它们各自的特点. 从可能度的含义和保序性两个角度指出, 基于可能度矩阵的区间数排序方法有时会导出不合理的排序结果. 通过分析可能度矩阵与模糊判断矩阵的关系, 剖 析了导致这种不合理排序结果的原因. 最后, 利用可能度矩阵构造一个布尔矩阵, 基于布尔矩阵给出一个改进的区间 数排序算法, 并从理论上证明了所提出的排序方法的科学性.

     

Ranking fuzzy numbers with an area method using radius of gyration

Ranking fuzzy numbers plays an very important role in linguistic decision making and some other fuzzy application systems. Many methods have been proposed to deal with ranking fuzzy numbers. Chu pointed out some shortcomings of the existing distance method and proposed to rank the fuzzy numbers with the area between the centroid point and original point. However, drawbacks are also found in the area method. For example, it cannot rank fuzzy numbers when some fuzzy numbers have the same centroid point. In this paper, we propose a modified area method to rank fuzzy numbers. The modified method can effectively rank various fuzzy numbers and their images. We also used some comparative examples to illustrate the advantage of the proposed method.     

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.     

Comparison of fuzzy numbers based on preference

Ranking fuzzy numbers plays an important role in a fuzzy decision-making process. However, fuzzy numbers may not be easily ordered into one sequence due to the overlap between them. A new approach is introduced to detect the overlapped fuzzy numbers based on the concept of similarity measure, incorporating the preference of the decision-maker into the fuzzy ranking process. Numerical examples and comparisons with other methods are presented to evaluate the new method. The computational process of the proposed method is straightforward and is practically capable of comparing similar fuzzy numbers. The proposed method is an absolute ranking and no pairwise comparison of fuzzy numbers is necessary. Furthermore, through some examples discussed in this work, it is proved that the proposed method possesses several good characteristics compared to other methods examined in this work.     

A new fuzzy MCDM approach based on centroid of fuzzy numbers

This paper proposes a new fuzzy MCDM (FMCDM) approach based on centroid of fuzzy numbers for ranking of alternatives. The FMCDM approach allows decision makers (DMs) to evaluate alternatives using linguistic terms such as very high, high, slightly high, medium, slightly low, low, very low or none rather than precise numerical values, allows them to express their opinions independently, and also provides an algorithm to aggregate the assessments of alternatives. Three numerical examples are investigated using the FMCDM approach to illustrate its applications. It is shown that the FMCDM approach offers a flexible, practical and effective way of group decision making.     

Ranking numbers of graphs

Given a graph G, a vertex ranking (or simply, ranking) of G is a mapping f from V(G) to the set of all positive integers, such that for any path between two distinct vertices u and v with f(u)=f(v), there is a vertex w in the path with f(w)>f(u). If f is a ranking of G, the ranking number of G under f, denoted γf(G), is defined by , and the ranking number of G, denoted γ(G), is defined by . The vertex ranking problem is to determine the ranking number γ(G) of a given graph G. This problem is a natural model for the manufacturing scheduling problem. We study the ranking numbers of graphs in this paper. We consider the relation between the ranking numbers and the minimal cut sets, and the relation between the ranking numbers and the independent sets. From this, we obtain the ranking numbers of the powers of paths and the powers of cycles, the Cartesian product of P₂ with Pn or Cn, and the caterpilars. And we also find the vertex ranking numbers of the composition of two graphs in this paper.     

Analyzing fuzzy risk based on similarity measures between interval-valued fuzzy numbers

In this paper, we present a new method for handling fuzzy risk analysis problems based on the proposed new similarity measure between interval-valued fuzzy numbers. First, we present a new similarity measure between interval-valued fuzzy numbers. It considers the degrees of closeness between interval-valued fuzzy numbers on the X-axis and the degrees of differences between the shapes of the interval-valued fuzzy numbers on the X-axis and the Y-axis, respectively. We also prove three properties of the proposed similarity measure. Then, we make an experiment to compare the experimental results of the proposed method with the existing similarity measures between interval-valued fuzzy numbers. The proposed method can overcome the drawbacks of the existing methods. Finally, based on the proposed similarity measure between interval-valued fuzzy numbers, we present a new fuzzy risk analysis algorithm for dealing with fuzzy risk analysis problems. Because the proposed method allows the evaluating values of sub-components to be represented by interval-valued fuzzy numbers, it is more flexible than Chen and Chen’s method (2003).     

Ranking fuzzy cognitive map based scenarios with TOPSIS

Scenarios describe events and situations that would occurred in the future real-world. Policy makers use scenario methods as a tool to build landscapes of possible futures at a national level. Based on these future visions, policy and decision-makers are able to explore different courses of action. In recent years, the number of potential scenario methods and applications is increasing. It is because academics and practitioners are increasing their interest about it. In spite of the success of scenario methods’ support, scenario-based decision making still is not a fully structured process.The proposed methodology aims to bring methodological support to scenario-based decision making in scenario analysis. The originality of the proposed approach with respect to other ones is that it aims to use the scenarios’ assessment and ranking as a whole. Traditional approaches consider the future impact of each present entity in isolation. This assumption is a simplification of a more complex reality, in which different entities interact with each other. The model that the authors propose allows decision and policy makers to measure the impact of a entity interactions. To reach this aim, the proposal combine Delphi method, soft computing (fuzzy cognitive maps) and multicriteria (TOPSIS) techniques. In addition, a numerical example is developed for illustrating the proposal.