A Probability-Based Approach to Comparison of Fuzzy Numbers and Applications to Target-Oriented Decision Making

In this paper, we introduce a new comparison relation on fuzzy numbers based on their alpha-cut representation and comparison probabilities of interval values. Basically, this comparison process combines a widely accepted interpretation of fuzzy sets together with the uncertain characteristics inherent in the representation of fuzzy numbers. The proposed comparison relation is then applied to the issue of ranking fuzzy numbers using fuzzy targets in terms of target-based evaluations. Some numerical examples are used to illuminate the proposed ranking technique as well as to compare with previous methods. More interestingly, according to the interpretation of the new comparison relation on fuzzy numbers, we provide a fuzzy target-based decision model as a solution to the problem of decision making under uncertainty, with which an interesting link between the decision maker's different attitudes about target and different risk attitudes in terms of utility functions can be established. Moreover, an application of the proposed comparison relation to the fuzzy target-based decision model for the problem of fuzzy decision making with uncertainty is provided. Numerical examples are also given for illustration.     

Fuzzy risk analysis based on fuzzy numbers with different shapes and different deviations

In this paper, we present a new method for fuzzy risk analysis based on fuzzy numbers with different shapes and different deviations. First, we present a new method for ranking trapezoidal fuzzy numbers based on their shapes and deviations. Then, we use some examples to compare the proposed method with the existing methods for ranking fuzzy numbers. Finally, we use the proposed fuzzy ranking method to present a new fuzzy risk analysis algorithm to deal with fuzzy risk analysis problems. The proposed fuzzy risk analysis algorithm is more flexible and simpler than the existing methods due to the fact that it allows the evaluating values to be represented by trapezoidal fuzzy numbers with different shapes and different deviations.     

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.     

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 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 approach for ranking of fuzzy numbers

In order to rank all fuzzy numbers, we modify the method of “a new approach for ranking of trapezoidal fuzzy numbers” by Abbasbandy and Hajjari (2009). Our proposed method is used for ranking symmetric fuzzy numbers. The advantage of this method is illustrated by some comparative examples.     

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 novel meta-heuristic based method for deriving priorities from fuzzy pairwise comparison judgments

This paper proposes a new method to derive the priority vector from fuzzy pairwise comparison matrices. Unlike several known methods, the proposed method derives crisp weights from consistent and inconsistent fuzzy comparison matrices. Therefore, the crisp weights obviate the need of additional aggregation and ranking procedures. To derive the priority vector, a Modified Fuzzy Logarithmic Least Square Model (MFLLSM) is proposed. In order to solve the MFLLSM, a framework based on genetic algorithm is proposed. In the proposed framework, a heuristic algorithm of population initialization, a heuristic algorithm for simulating fuzzy numbers and a heuristic algorithm of fitness evaluation are proposed.The solution of the prioritization problem requires finding priorities such that their ratio approximately satisfies the initial judgments. Computational results reveal the superiority of the proposed method in comparison with five well known methods of literature from the viewpoint of satisfaction of initial judgments by the obtained priority vector. It is shown by ten different examples that the deviation of the priorities ratio from initial judgments in the proposed method is less than five existing methods of literature. In addition, unlike several methods of literature, the proposed method considers fuzzy judgments represented by both triangular and trapezoidal fuzzy numbers. Furthermore, the proposed method for the first time considers judgments represented by triangular shaped fuzzy numbers and trapezoidal shaped fuzzy numbers which are discussed in the paper.     

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.     

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.     

Multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets

In this paper, we present a new method for multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets, where interval-valued intuitionistic fuzzy values are used to represent evaluating values of the decision-maker with respect to alternatives. First, we propose a new method for ranking interval-valued intuitionistic fuzzy values. Based on the proposed fuzzy ranking method of interval-valued intuitionistic fuzzy values, we propose a new method for multicriteria fuzzy decision making. The proposed multicriteria fuzzy decision making method outperforms Ye’s method (2009) due to the fact that the proposed method can overcome the drawback of Ye’s method (2009), where the drawback of Ye’s method is that it can not distinguish the ranking order between alternatives in some situations. The proposed method provides us with a useful way for dealing with multicriteria fuzzy decision making problems based on interval-valued intuitionistic fuzzy sets.     

Fuzzy evaluation of cogeneration alternatives in a petrochemical industry

This paper derives fuzzy net present value (NPV) and pay back year (PBY) models as decision indexes for cogeneration alternatives decision-making. The Mellin transform is employed to establish the means and variances of the fuzzy indexes in order to rank various cogeneration alternatives. It is noted that the mean and variance values depend only on the vertexes of the fuzzy index, and are independent of their height. The current paper verifies the performance of the proposed models by simulating two numerical examples and by considering a cogeneration program in a petrochemical industry. It is shown that the results of the proposed fuzzy economic models are consistent with those of the conventional crisp models, and that the developed concepts are straightforward and easily implemented compared to the fuzzy ranking methods proposed in previous studies. Furthermore, the developed methods serve as readily implemented sensitivity analysis tools for use in the arena of uncertain decision-making.     

Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value

Value and ambiguity are two parameters which were introduced to represent fuzzy numbers. In this paper, we find the nearest trapezoidal approximation and the nearest symmetric trapezoidal approximation to a given fuzzy number, with respect to the average Euclidean distance, preserving the value and ambiguity. To avoid the laborious calculus associated with the Karush–Kuhn–Tucker theorem, the working tool in some recent papers, a less sophisticated method is proposed. Algorithms for computing the approximations, many examples, proofs of continuity and two applications to ranking of fuzzy numbers and estimations of the defect of additivity for approximations are given.     

A new ranking technique for q‐rung orthopair fuzzy values

In intuitionistic fuzzy set and their generalizations such as Pythagorean fuzzy sets and q‐rung orthopair fuzzy sets, ranking is not easy to define. There are several techniques available in literature for ranking values in above mentioned orthopair fuzzy sets. It is interesting to see that almost all the proposed ranking methods produce distinct ranking. Notion of knowledge base is very important to study ranking proposed by different techniques. Aim of this paper is to critically analyze the available ranking techniques for q‐rung orthopair fuzzy values and propose a new graphical ranking method based on hesitancy index and entropy. Several numerical examples are tested with the proposed technique, which shows that the technique is intuitive and convenient for real life applications.     

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.

     

Recommend trustworthy services using interval numbers of four parameters via cloud model for potential users

How to discover the trustworthy services is a challenge for potential users because of the deficiency of usage experiences and the information overload of QoE (quality of experience) evaluations from consumers. Aiming to the limitations of traditional interval numbers in measuring the trustworthiness of service, this paper proposed a novel service recommendation approach using the interval numbers of four parameters (INF) for potential users. In this approach, a trustworthiness cloud model was established to identify the eigenvalue of INF via backward cloud generator, and a new formula of INF possibility degree based on geometrical analysis was presented to ensure the high calculation precision. In order to select the highly valuable QoE evaluations, the similarity of client-side feature between potential user and consumers was calculated, and the multi-attributes trustworthiness values were aggregated into INF by the fuzzy analytic hierarchy process method. On the basis of ranking INF, the sort values of trustworthiness of candidate services were obtained, and the trustworthy services were chosen to recommend to potential user. The experiments based on a realworld dataset showed that it can improve the recommendation accuracy of trustworthy services compared to other approaches, which contributes to solving cold start and information overload problem in service recommendation.     

The implementation of quality function deployment based on linguistic data

Quality function deployment (QFD) is a customer-driven quality management and product development system for achieving higher customer satisfaction. The QFD process involves various inputs in the form of linguistic data, e.g., human perception, judgment, and evaluation on importance or relationship strength. Such data are usually ambiguous and uncertain. An aim of this paper is to examine the implementation of QFD under a fuzzy environment and to develop corresponding procedures to deal with the fuzzy data. It presented a process model using linguistic variables, fuzzy arithmetic, and defuzzification techniques. Based on an example, this paper further examined the sensitivity of the ranking of technical characteristics to the defuzzification strategy and the degree of fuzziness of fuzzy numbers. Results indicated that selection of the defuzzification strategy and membership function are important. This proposed fuzzy approach allows QFD users to avoid subjective and arbitrary quantification of linguistic data. The paper also presents a scheme to represent and interprete the results.     

Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets

In this paper, we present a new method to handle fuzzy multiple attributes group decision-making problems based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. First, we present the arithmetic operations between interval type-2 fuzzy sets. Then, we present a fuzzy ranking method to calculate the ranking values of interval type-2 fuzzy sets. We also make a comparison of the ranking values of the proposed method with the existing methods. Based on the proposed fuzzy ranking method and the proposed arithmetic operations between interval type-2 fuzzy sets, we present a new method to handle fuzzy multiple attributes group decision-making problems. The proposed method provides us with a useful way to handle fuzzy multiple attributes group decision-making problems in a more flexible and more intelligent manner due to the fact that it uses interval type-2 fuzzy sets rather than traditional type-1 fuzzy sets to represent the evaluating values and the weights of attributes.     

Pythagorean fuzzy preference ranking organization method of enrichment evaluations

Recently, a new extension of fuzzy sets, Pythagorean fuzzy sets (PFS), has attracted a lot of attention from scholars in various fields of research. Due to PFS’s powerfulness in modeling the imprecision of human perception in multicriteria decision-making (MCDM) problems, this paper aims to extend the classical preference ranking organization method of enrichment evaluations (PROMETHEE) into the Pythagorean fuzzy environment. The proposed method takes not only the weights related to different criteria but also the preference relations as Pythagorean fuzzy numbers, therefore providing a broader range of choices for the decision-maker to express their preferences. Five properties are put forward to regulate the designing of both intuitionistic and Pythagorean fuzzy PROMETHEE (PF-PROMETHEE) preference functions. Furthermore two illustrative examples are given to demonstrate the detailed procedure of PF-PROMETHEE, and comparisons are made to distinguish the differences among our proposed method, the classical PROMETHEE and intuitionistic PROMETHEE. The results show that PF-PROMETHEE is effective, comprehensive, and applicable to a wide range of MCDM problems.