Multi-criteria group decision making based on trapezoidal intuitionistic fuzzy information

With respect to multi-criteria group decision making (MCGDM) problems under trapezoidal intuitionistic fuzzy environment, a new MCGDM method is investigated. The proposed method can effectively avoid the failure caused by the use of inconsistent decision information and provides a decision-making idea for the case of “the truth be held in minority”. It consists of three interrelated modules: weight determining mechanism, group consistency analysis, and ranking and selection procedure. For the first module, distance measures, expected values and arithmetic averaging operator for trapezoidal intuitionistic fuzzy numbers are used to determine the weight values of criteria and decision makers. For the second module, a consistency analysis and correction procedure based on trapezoidal intuitionistic fuzzy weighted averaging operator and OWA operator is developed to reduce the influence of conflicting opinions prior to the ranking process. For the third module, a trapezoidal intuitionistic fuzzy TOPSIS is used for ranking and selection. Then a procedure for the proposed MCGDM method is developed. Finally, a numerical example further illustrates the practicality and efficiency of the proposed method.     

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.     

Power geometric operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making

As a special intuitionistic fuzzy set on a real number set, trapezoidal intuitionistic fuzzy numbers (TrIFNs) have the better capability to model ill-known quantities. The purpose of this paper is to develop some power geometric operators of TrIFNs and apply to multi-attribute group decision making (MAGDM) with TrIFNs. First, the lower and upper weighted possibility means of TrIFNs are introduced as well as weighted possibility means. Hereby, a new lexicographic method is developed to rank TrIFNs. The Minkowski distance between TrIFNs is defined. Then, four kinds of power geometric operators of TrIFNs are investigated including the power geometric operator of TrIFNs, power weighted geometric operator of TrIFNs, power ordered weighted geometric operator of TrIFNs and power hybrid geometric operator of TrIFNs. Their desirable properties are discussed. Four methods for MAGDM with TrIFNs are respectively proposed for the four cases whether the weight vectors of attributes and DMs are known or unknown. In these methods, the individual overall attribute values of alternatives are generated by using the power geometric or power weighted geometric operator of TrIFNs. The collective overall attribute values of alternatives are determined through constructing the multi-objective optimization model, which is transformed into the goal programming model to solve. Thus, the ranking order of alternatives is obtained according to the collective overall attribute values of alternatives. Finally, the green supplier selection problem is illustrated to demonstrate the application and validation of the proposed method.     

Study on the ranking problems in multiple attribute decision making based on interval‐valued intuitionistic fuzzy numbers

This paper puts forward a new ranking method for multiple attribute decision‐making problems based on interval‐valued intuitionistic fuzzy set (IIFS) theory. First, the composed ordered weighted arithmetic averaging operator and composed ordered weighted geometric averaging operator are extended to the IIFSs in which they are, respectively, named interval‐valued intuitionistic fuzzy composed ordered weighted arithmetic averaging (IIFCOWA) operator and interval‐valued intuitionistic composed ordered weighted geometric averaging (IIFCOWG) operator. Afterwards, to compare interval‐valued intuitionistic fuzzy numbers, we define the concepts of the maximum, the minimum, and ranking function. Some properties associated with the concepts are investigated. Using the IIFCOWA or IIFCOWG operator, we establish the detailed steps of ranking alternatives (or attributes) in multiple attribute decision making. Finally, an illustrative example is provided to show that the proposed ranking method is feasible in multiple attribute decision making.     

Pythagorean Fuzzy Clustering Analysis: A Hierarchical Clustering Algorithm with the Ratio Index‐Based Ranking Methods

The Pythagorean fuzzy set introduced by R. R. Yager in 2014 is a useful tool to model imprecise and ambiguous information appearing in decision and clustering problems. In this study, we present a general type of distance measure for Pythagorean fuzzy numbers (PFNs) and propose a novel ratio index‐based ranking method of PFNs. The novel ranking method of PFNs has more powerful ability to discriminate the magnitude of PFNs than the existing ranking methods for PFNs, which is further extended to compare the magnitude of interval‐valued Pythagorean fuzzy numbers (IVPFNs). The IVPFN is a new extension of PFN, which is parallel to interval‐valued intuitionistic fuzzy number. We introduce a general type of distance measure for IVPFNs. Afterwards, we study a kind of clustering problems in Pythagorean fuzzy environments in which the evaluation values are expressed by PFNs and/or IVPFNs and develop a novel Pythagorean fuzzy agglomerative hierarchical clustering approach. In the proposed clustering method, we define the concept of the dissimilarity degree between two clusters for each criterion and introduce the clustering procedure in the criteria level. To take all the criteria into account, we also introduce the overall clustering procedure, which is based on the overall dissimilarity degrees for a fixed aggregation operator such as the commonly used weighted arithmetic average operator or the ordered weighted averaging operator. In the overall clustering process, (1) we present a deviation degree‐based method to derive the weights of criteria and further obtain the overall clustering results if the weights of criteria are completely unknown; (2) we employ the ratio index‐based ranking method of IVPFNs to obtain the overall clustering results if the weights of criteria are given in advance and are expressed by IVPFNs. The salient feature of the proposed clustering method is that it not only can address the clustering problems in which the weights of criteria are not given precisely in advance but also can manage simultaneously the PFNs and IVPFNs data.     

基于加权可能性均值的直觉梯形模糊集矩阵博弈求解方法

针对支付值为直觉梯形模糊数（ITFN）的矩阵博弈求解问题,提出了一种基于加权可能性均值的求解方法.定义了ITFN新的运算法则,并引入ITFN的下、上加权可能性均值和加权可能性均值的概念,根据加权可能性均值给出了ITFN新的排序方法;运用新的排序方法,将求解局中人最优策略问题转化为求解双目标线性规划问题.实例分析验证了所提出方法的可行性和有效性.     

An interval arithmetic based fuzzy TOPSIS model

This work suggests a fuzzy TOPSIS model, where ratings of alternatives under criteria and importance weights of criteria are assessed in linguistic values represented by fuzzy numbers. Criteria can be categorized into benefit and cost. Ratings of alternatives versus criteria and the importance weights of criteria are normalized before multiplication. The membership function of each fuzzy weighted rating can be developed by interval arithmetic of fuzzy numbers. A ranking method can then be applied easily to develop positive and negative idea solutions in order to complete the fuzzy TOPSIS model. Finally, a numerical example demonstrates the feasibility of the proposed method.     

Discontinuity of the trapezoidal fuzzy number-valued operators preserving core

We prove that any trapezoidal fuzzy number-valued operator preserving core is discontinuous with respect to any weighted metric on the space of fuzzy numbers. As an application, we obtain the discontinuity of the weighted trapezoidal approximation operator preserving core.     

用区间直觉模糊集方法对属性权重未知的群求解其多属性决策

本文首先提出群区间直觉模糊有序加权几何(groupinterval-valuedintuitionistic fuzzy orderedweighted geometric,GIVIFOWG)算子和群区间直觉模糊有序加权平均(group interval-valued intuitionistic fuzzy ordered weighted averaging,GIVIFOWA)算子.利用GIVIFOWG算子或GIVIFOWA算子聚集群的决策矩阵以获得方案在属性上的综合区间直觉模糊决策矩阵(collectiveinterval-valuedintuitionistic fuzzy decision-matrix,CIVIFDM).然后定义了一个考虑犹豫度的区间直觉模糊熵(interval-valuedintuitionistic fuzzyentropy,IVIFE);通过熵衡量每个属性所含的信息来求解属性权重.最后,提出基于可能度的接近理想解的区间排序法(interval technique for order preference by similarity to an ideal solution,ITOPSIS)和区间得分函数法.在ITOPSIS法中,依据区间距离公式计算候选方案和理想方案的属性加权区间距离,进而采用ITOPSIS准则对各方案进行排序;在区间得分函数法中,算出CIVIFDM中各方案的得分值以及精确值,然后利用区间得分准则对各方案进行排序.实验结果验证了决策方法的有效性和可行性.     

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.     

基于OWA算子的模糊n-cell数排序方法

针对基于模糊n-cell数的多属性排序问题,提出了一种基于有序加权平均算子（OWA算子）的模糊n-cell数排序方法。该方法首先根据样本数据对评估对象的属性构造模糊n-cell数,其次根据均值将属性按照从大到小排列,然后选取合适的权重向量,应用OWA算子进行信息聚合得到综合模糊n-cell数,接着根据各分量均值得到排序结果。最后,将该方法运用到实例中,并与传统的均值方法进行了比较。结果表明该方法不仅灵活有效,可根据具体情况选择不同的OWA权重来消除部分不合理的情况,使结果更有说服力,还弥补了传统均值方法的不足。     

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.     

A simple approach to ranking a group of aggregated fuzzy utilities

When ranking a large quantity of fuzzy numbers, the efficiency, accuracy, and effectiveness of the ranking process is critical. The paper considers the application of "alpha-cut" and "fuzzy arithmetic operations" to the fuzzy weighted average (FWA) method which can be used to rank aggregated fuzzy utilities (or generalized fuzzy numbers). The purpose of this application is to make the method easier to program and the data easier to manipulate, which results in a more practical method for fuzzy decisions.     

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.     

Multiattribute decision making based on interval-valued intuitionistic fuzzy values

In this paper, we present a new multiattribute decision making method based on the proposed interval-valued intuitionistic fuzzy weighted average operator and the proposed fuzzy ranking method for intuitionistic fuzzy values. First, we briefly review the concepts of interval-valued intuitionistic fuzzy sets and the Karnik–Mendel algorithms. Then, we propose the intuitionistic fuzzy weighted average operator and interval-valued intuitionistic fuzzy weighted average operator, based on the traditional weighted average method and the Karnik–Mendel algorithms. Then, we propose a fuzzy ranking method for intuitionistic fuzzy values based on likelihood-based comparison relations between intervals. Finally, we present a new multiattribute decision making method based on the proposed interval-valued intuitionistic fuzzy weighted average operator and the proposed fuzzy ranking method for intuitionistic fuzzy values. The proposed method provides us with a useful way for multiattribute decision making based on interval-valued intuitionistic fuzzy values.     

Fuzzy critical path method based on signed distance ranking offuzzy numbers

We apply a signed distance ranking method for fuzzy numbers to a critical path method for activity-on-edge (AOE) networks. We use signed distance ranking to define ordering simply, which means we can use both positive and negative values to define ordering. The primary result obtained in the paper is the use of signed distance ranking of fuzzy numbers obtaining Properties 3 and 4. We conclude that the fuzzy AOE network is an extension of the crisp AOE network, and thus the fuzzy critical path in a fuzzy AOE network, under some conditions, is the same as the crisp critical path in a crisp AOE network     

RISK ANALYSIS IN DIABETES PREDICTION BASED ON A NEW APPROACH OF RANKING OF GENERALIZED TRAPEZOIDAL FUZZY NUMBERS

A new method of ranking the generalized trapezoidal fuzzy numbers is proposed in this article. Our proposed method is based on mean position, spread along the x-axis, area, and height. Some properties regarding the proposed new method have been derived. It has been compared with existing methods taking different sets of generalized trapezoidal fuzzy numbers and the drawbacks of those existing methods have been overcome by our new method. Then the new proposed method of ranking generalized trapezoidal fuzzy numbers was applied for ordering the risk values to be affected in a diabetes problem.     

Extension of weighted aggregated sum product assessment with interval-valued intuitionistic fuzzy numbers (WASPAS-IVIF)

Different methods are proposed in the framework of multi attribute utility theory for multi criteria decision making. Among the proposed methods, weighted sum and weighted product models (WSM and WPM) are well known and widely accepted. To improve the accuracy of WSM and WPM, the weighted aggregated sum product assessment (WASPAS) method was proposed which used an aggregation operator on WSM and WPM. In this paper, an extended version of WASPAS method is proposed which can be applied in uncertain decision making environment. In the proposed WASPAS-IVIF method, the uncertainty of decision maker(s) in stating their judgments and evaluations regard to criteria importance and/or alternatives performance on criteria are expressed by interval valued intuitionistic fuzzy numbers. Two numerical examples of ranking derelict buildings’ redevelopment decisions and investment alternatives are presented. The results are then compared with the rankings provided by other methods such as TOPSIS-IVIF, COPRAS-IVIF and IFOWA. Combining the strengths of IVIFS in handling uncertainty with the enhanced accuracy of WASPAS makes the proposed method as a desirable method for multi criteria decision making in real world applications.     

区间直觉模糊信息的集成方法及其在决策中的应用

对区间直觉模糊信息的集成方法进行了研究.定义了区间直觉模糊数的一些运算法则,并基于这些运算法则,给出区间直觉模糊数的加权算术和加权几何集成算子.定义了区间直觉模糊数的得分函数和精确函数,进而给出了区间直觉模糊数的一种简单的排序方法.最后提供了一种基于区间直觉模糊信息的决策途径,并进行了实例分析.     

Some new distance measures for type-2 fuzzy sets and distance measure based ranking for group decision making problems

In this paper, we propose some distance measures between type-2 fuzzy sets, and also a new family of utmost distance measures are presented. Several properties of different proposed distance measures have been introduced. Also, we have introduced a new ranking method for the ordering of type-2 fuzzy sets based on the proposed distance measure. The proposed ranking method satisfies the reasonable properties for the ordering of fuzzy quantities. Some properties such as robustness, order relation have been presented. Limitations of existing ranking methods have been studied. Further for practical use, a new method for selecting the best alternative, for group decision making problems is proposed. This method is illustrated with a numerical example.