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.     

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.     

A revised method for ranking fuzzy numbers using maximizing set and minimizing set

A large number of methods have been proposed for ranking fuzzy numbers in the last few decades. Nevertheless, none of these methods can always guarantee a consistent result for every situation. Some of them are even non-intuitive and not discriminating. Chen proposed a ranking method in 1985 to overcome these limitations and simplify the computational procedure based on the criteria of total utility through maximizing set and minimizing set. However, there were some shortcomings associated with Chen’s ranking method. Therefore, we propose a revised ranking method that can overcome these shortcomings. Instead of considering just a single left and a single right utility in the total utility, the proposed method considers two left and two right utilities. In addition, the proposed method also takes into account the decision maker’s optimistic attitude of fuzzy numbers. Several comparative examples and an application demonstrating the usage, advantages, and applicability of the revised ranking method are presented. It can be concluded that the revised ranking method can effectively resolve the issues with Chen’s ranking method. Moreover, the revised ranking method can be used to differentiate different types of fuzzy numbers.     

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.     

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.     

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.     

A new approach for ranking of L–R type generalized fuzzy numbers

Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. Cheng (Cheng, C. H. (1998). A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets and Systems, 95, 307–317) pointed out that the proof of the statement “Ranking of generalized fuzzy numbers does not depend upon the height of fuzzy numbers” stated by Liou and Wang (Liou, T. S., & Wang, M. J. (1992). Ranking fuzzy numbers with integral value. Fuzzy Sets and Systems, 50, 247–255) is incorrect. In this paper, by giving an alternative proof it is proved that the above statement is correct. Also with the help of several counter examples it is proved that ranking method proposed by Chen and Chen (Chen, S. M., & Chen, J. H. (2009). Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads. Expert Systems with Applications, 36, 6833–6842) is incorrect. The main aim of this paper is to modify the Liou and Wang approach for the ranking of L–R type generalized fuzzy numbers. The main advantage of the proposed approach is that the proposed approach provide the correct ordering of generalized and normal 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 satisfy all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Wang, X., & Kerre, E. E. (2001). Reasonable properties for the ordering of fuzzy quantities (I). Fuzzy Sets and Systems, 118, 375–385).     

Fuzzy decision networks and deconvolution

Three kinds of networks, namely, fuzzy minimal spanning tree, fuzzy PERT, and fuzzy shortest path, are analyzed by the use of a recently developed fuzzy ranking method to handle the various fuzzy quantities. To overcome the problem of double-inclusion of the amounts of uncertainties involved in the fuzzy quantities when extended subtraction is used for problems such as fuzzy PERT, fuzzy deconvolution is used. Since we are only interested in the ranking and aggregation of the results, the existence or nonexistence of the results from deconvolution does not influence the resulting analysis. A technique based on the ranking method is developed to handle negative spreads in the resulting fuzzy quantities. With the use of this fuzzy ranking method, the structure of the network is maintained and conventional algorithms can be applied with appropriate modifications. Emphasis is placed on the use of the subjective decision maker's opinion in the proposed fuzzy network analysis approach. Numerical examples are given to illustrate the approach.     

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.     

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.     

Solving fuzzy multiple objective generalized assignment problems directly via bees algorithm and fuzzy ranking

In this paper, a direct solution approach for solving fuzzy multiple objective generalized assignment problems is proposed. In the problem, the coefficients and right hand side values of the constraints and the objective function coefficients are defined as fuzzy numbers. The addressed problem also has a multiple objective structure where the goals are determined so as to minimize the total cost and the imbalance between the workload of the agents. The direct solution approach utilizes the fuzzy ranking methods to rank the objective function values and to determine the feasibility of the constraints within a metaheuristic search algorithm, known as bees algorithm. Different fuzzy ranking methods, namely signed distance, integral value and area based approach are used in bees algorithm. For the computational study, the effects of these fuzzy ranking methods on the quality of the solutions are also analyzed.     

RM approach for ranking of L–R type generalized fuzzy numbers

Ranking of fuzzy numbers play an important role in decision making, optimization, 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 Syst Appl 36:6833–6842, 2009) is incorrect. The main aim of this paper is to propose a new approach for the ranking of L–R type generalized fuzzy numbers. The proposed ranking approach is based on rank and mode so it is named as RM approach. The main advantage of the proposed approach is that it provides the correct ordering of generalized and normal fuzzy numbers and it 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 Syst 118:375–385, 2001).     

A direct solution approach to fuzzy mathematical programs with fuzzy decision variables

In this paper, a direct solution approach is presented for solving fuzzy mathematical programming problems with fuzzy decision variables. In the proposed approach, a fuzzy ranking procedure for fuzzy numbers and a meta-heuristic algorithm is employed. A basic example is presented in the paper. It has been observed that fuzzy mathematical programs with fuzzy decision variables can be solved effectively by employing direct solution approaches which are based on fuzzy ranking procedures and meta-heuristics.     

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.     

An effective method for solving flow shop scheduling problems with fuzzy processing times

This paper show that fuzzy set theory can be useful in modelling and solving flow shop scheduling problems with uncertain processing times and illustrates a methodology for solving job sequencing problem which the opinions of experts greatly disagree in each processing time. Triangular fuzzy numbers (TFNs) are used to represent the processing times of experts. And the comparison methods based on the dominance property is sued to determine the ranking of the fuzzy numbers. By the dominance criteria, for each job, a major TFN and a minor TFN are selected and a pessimistic sequence with major TFNs and an optimistic sequence with minor TFNs are computer. Branch and bound algorithm for makespan in three-machine flow shop is utilized to illustrate the proposed methodology.     

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 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.     

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 interval type-2 fuzzy PROMETHEE method using a likelihood-based outranking comparison approach

Based on the preference ranking organization method for enrichment evaluations (PROMETHEE), the purpose of this paper is to develop a new multiple criteria decision-making method that uses the approach of likelihood-based outranking comparisons within the environment of interval type-2 fuzzy sets. Uncertain and imprecise assessment of information often occurs in multiple criteria decision analysis (MCDA). The theory of interval type-2 fuzzy sets is useful and convenient for modeling impressions and quantifying the ambiguous nature of subjective judgments. Using the approach of likelihood-based outranking comparisons, this paper presents an interval type-2 fuzzy PROMETHEE method designed to address MCDA problems based on interval type-2 trapezoidal fuzzy (IT2TrF) numbers. This paper introduces the concepts of lower and upper likelihoods for acquiring the likelihood of an IT2TrF binary relationship and defines a likelihood-based outranking index to develop certain likelihood-based preference functions that correspond to several generalized criteria. The concept of comprehensive preference measures is proposed to determine IT2TrF exiting, entering, and net flows in the valued outranking relationships. In addition, this work establishes the concepts of a comprehensive outranking index, a comprehensive outranked index, and a comprehensive dominance index to induce partial and total preorders for the purpose of acquiring partial ranking and complete ranking, respectively, of the alternative actions. The feasibility and applicability of the proposed method are illustrated with two practical applications to the problem of landfill site selection and a car evaluation problem. Finally, a comparison with other relevant methods is conducted to validate the effectiveness of the proposed method.