Generalized Dice measures for multiple attribute decision making under intuitionistic and interval-valued intuitionistic fuzzy environments

This paper proposes Dice measures of intuitionistic fuzzy sets (IFSs) and interval-valued intuitionistic fuzzy sets (IVIFSs) and generalized Dice measures of IFSs and IVIFSs, and then indicates the relations of the generalized Dice measures, Dice measures, and projection measures (asymmetric measures) of IFSs and IVIFSs. Furthermore, we develop the generalized Dice measures-based multiple attribute decision-making methods with intuitionistic and interval-valued intuitionistic fuzzy information. Through the weighted generalized Dice measures between each alternative and the ideal solution (ideal alternative) according to some parameter value selected by decision makers’ preference, all the alternatives can be ranked and the best one can be chosen as well. Finally, an actual example about the selection of manufacturing schemes is provided to demonstrate the applications of the proposed decision-making methods under intuitionistic and interval-valued intuitionistic fuzzy environments, and then, a comparison analysis is conducted between the developed approach and other existing methods to verify the effectiveness and flexibility of the proposed method.

     

Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment

The paper presents the correlation and correlation coefficient of single-valued neutrosophic sets (SVNSs) based on the extension of the correlation of intuitionistic fuzzy sets and demonstrates that the cosine similarity measure is a special case of the correlation coefficient in SVNS. Then a decision-making method is proposed by the use of the weighted correlation coefficient or the weighted cosine similarity measure of SVNSs, in which the evaluation information for alternatives with respect to criteria is carried out by truth-membership degree, indeterminacy-membership degree, and falsity-membership degree under single-valued neutrosophic environment. We utilize the weighted correlation coefficient or the weighted cosine similarity measure between each alternative and the ideal alternative to rank the alternatives and to determine the best one(s). Finally, an illustrative example demonstrates the application of the proposed decision-making method.     

A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making

The ranking of interval-valued intuitionistic fuzzy sets (IVIFSs) is very important for the interval-valued intuitionistic fuzzy decision making. From the probability viewpoint, the possibility degree of comparison between two interval-valued intuitionistic fuzzy numbers (IVIFNs) is defined by using the notion of 2-dimensional random vector, and a new method is then developed to rank IVIFNs. Hereby the ordered weighted average operator and hybrid weighted average operator for IVIFNs are defined based on the Karnik–Mendel algorithms and employed to solve multi-attribute group decision making problems with IVIFNs. The individual overall attribute values of alternatives are obtained by using the weighted average operator for IVIFNs. By using the hybrid weighted average operator for IVIFNs, we can obtain the collective overall attribute values of alternatives, which are used to rank the alternatives. A numerical example is examined to illustrate the effectiveness and flexibility of the proposed method in this paper.     

一种新的区间直觉模糊熵及其应用

提出了区间值直觉模糊集的区间直觉模糊交叉熵,这种交叉熵充分考虑了区间值直觉模糊集的隶属度,非隶属度以及犹豫度。给出一种区间值直觉模糊集的区间直觉模糊熵的公理化体系,并且基于直觉模糊交叉熵公式给出一种区间直觉模糊熵的具体测度公式。利用区间值直觉模糊集的加权相关系数,将提出的熵公式应用于解决属性权重完全未知的区间直觉模糊多属性决策问题。     

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.     

A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems

The objective of this paper is divided into two folds. Firstly, a new generalized improved score function has been presented in the interval-valued intuitionistic fuzzy sets (IVIFSs) environment by incorporating the idea of weighted average of the degree of hesitation between their membership functions. Secondly, an IVIFSs based method for solving the multi-criteria decision making (MCDM) problem has been presented with completely unknown attribute weights. A ranking of the different attributes is based on the proposed generalized improved score functions and the sensitivity analysis on the ranking of the system has been done based on the decision-making parameters. An illustrative examples have been studied to show that the proposed function is more reasonable in the decision-making process than other existing functions.     

Generalized score functions on interval-valued intuitionistic fuzzy sets with preference parameters for different types of decision makers and their application

In this paper, the interval-valued intuitionistic fuzzy sets (IVIFSs) are studied from the viewpoint of the decision makers’ preference. Firstly, two series of principles are proposed to guide the ranking of interval-valued intuitionistic fuzzy numbers (IVIFNs), and two kinds of illustrative generalized score functions on IVIFSs are proposed according to the newly proposed principles. Secondly, two kinds of generalized score functions on IVIFSs are proposed based on decision-makers’ preference. The two generalized score functions are both of two parameters, which represent the decision makers’ attitudinal characters on the classical score values and the classical accuracy values on IVIFNs, respectively. Thirdly, two kinds of generalized score functions on IVIFSs, which are suitable for ranking IVIFNs when there is no information about the importance weights of the classical score values and accuracy values on IVIFNs, are proposed based on integral. Fourthly, three kinds of multi-criteria decision-making (MCDM) methods in interval-valued intuitionistic fuzzy setting are proposed. Finally, an example shows that when a novel generalized score function on IVIFSs is proposed, its suitable application environments should also be pointed out.     

Application of correlation coefficient to interval-valued intuitionistic fuzzy multiple attribute decision-making with incomplete weight information

With respect to multiple attribute decision-making problems with interval-valued intuitionistic fuzzy information, some operational laws of interval-valued intuitionistic fuzzy numbers, correlation and correlation coefficient of interval-valued intuitionistic fuzzy sets are introduced. An optimization model based on the negative ideal solution and max-min operator, by which the attribute weights can be determined, is established. We utilize the interval-valued intuitionistic fuzzy weighted averaging operator proposed by Xu (Control Decis 22(2):215–219, 2007) to aggregate the interval-valued intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the correlation coefficient. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

Fuzzy multicriteria decision making method based on the improved accuracy function for interval-valued intuitionistic fuzzy sets

A new accuracy function for the theory of interval-valued intuitionistic fuzzy set, which overcomes some difficulties arising in the existing methods for determining rank of interval-valued intuitionistic fuzzy numbers, is proposed by taking into account the hesitancy degree of interval-valued intuitionistic fuzzy sets. By comparing it with several proposed accuracy functions, the necessity and efficiency of our accuracy function are provided by giving related examples. A fuzzy multicriteria decision making method is established to select the best alternative in multicriteria decision making process which is taken as interval-valued intuitionistic fuzzy set of criterion values for alternatives. While aggregating the interval-valued intuitionistic fuzzy information corresponding to each alternative, we utilize the interval-valued intuitionistic fuzzy weighted aggregation operators. Then the accuracy degree of the aggregated interval-valued intuitionistic fuzzy information is computed via the new proposed accuracy function. Thus, we can rank all the alternatives according to the accuracy function and choose the optimal one(s). Finally, an illustrative example is given to demonstrate the practicality and effectiveness of the proposed approach.     

基于直觉模糊集改进算子的多目标决策方法

定义了三角和区间直觉模糊集的一些运算法则,给出了直觉模糊集两个改进算子,即三角模糊数加权算术平均算子（FIFWAA） 和区间直觉模糊数加权几何平均算子（FIFWGA）。在此基础上, 提出用精确函数解决记分函数无法决策的问题,以保证记分函数的严密性与合理性。给出了一种属性权重不完全确定且属性值以三角和区间直觉模糊数给出的多目标决策方法,通过实例分析结果证明了运用直觉模糊集改进算子进行多目标决策方法的有效性和正确性。     

Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications

In this paper we propose an entropy measure for interval-valued intuitionistic fuzzy sets, which generalizes three entropy measures defined independently by Szmidt, Wang and Huang, for intuitionistic fuzzy sets. We also give an approach to construct similarity measures using entropy measures for interval-valued intuitionistic fuzzy sets. In particular, the proposed entropy measure for interval-valued intuitionistic fuzzy sets can yield a similarity measure. Several illustrative examples are given to demonstrate the practicality and effectiveness of the proposed formulas. We apply the similarity measure to solve problems on pattern recognitions, multi-criteria fuzzy decision making and medical diagnosis.     

Intuitionistic fuzzy MST clustering algorithms

In this paper, we investigate graph theory-based clustering techniques for Atanassov’s intuitionistic fuzzy sets (A-IFSs) and interval-valued intuitionistic fuzzy sets (IVIFSs). We start by introducing the concepts of graph, minimum spanning tree (MST), A-IFS, and intuitionistic fuzzy distance, and develop two intuitionistic fuzzy MST clustering algorithms ( and ). Then we extend Algorithm II for clustering IVIFSs, and show the effectiveness of our algorithms through some numerical experiments.     

区间直觉模糊连续熵及其多属性决策方法

提出了区间直觉模糊连续熵,并且研究了一种新的处理区间直觉模糊多属性决策问题的方法。基于连续有序加权平均（COWA）算子,给出了区间直觉模糊连续熵的概念,并且证明了区间直觉模糊连续熵满足区间直觉模糊熵的公理化定义的四个条件。在此基础上,针对属性权重信息完全未知的决策问题,通过衡量每一属性所含的信息量来确定属性权重。依据备选方案与理想方案间的加权相关系数,给出了一种新的区间直觉模糊多属性决策方法。实验结果验证了新的决策方法的可行性和有效性。     

A new preference scale mcdm method based on interval-valued intuitionistic fuzzy sets and the analytic hierarchy process

The intuitionistic fuzzy analytic hierarchy process (IF-AHP), in which intuitionistic fuzzy numbers are utilized in defining decision makers’ linguistic judgment, has been used to solve various multi-criteria decision-making problems. Previous theories have suggested that interval-valued intuitionistic fuzzy numbers (IVIFN) with hesitation degree can act as alternative fuzzy numbers that can handle vagueness and uncertainty. This paper proposes a new preference scale in the framework of the interval-valued intuitionistic fuzzy analytic hierarchy process (IVIF-AHP). The comparison matrix judgment is expressed in IVIFN with degree of hesitation. The proposed new preference scale concurrently considers the membership function, the non-membership function and the degree of hesitation of IVIFN. To define the weight entropy of the aggregated matrix of IVIFN, a modified interval-valued intuitionistic fuzzy weighted averaging is proposed, by considering the interval number of the hesitation degree. Three multi-criteria decision-making problems are used to test the proposed method. A comparison of the results is also presented to check the feasibility of the proposed method. It is shown that the ranking order of the proposed method is slightly different from that of the other two methods because of the inclusion of the hesitation degree in defining the preference scale.     

A new fuzzy clustering algorithm for the segmentation of brain tumor

This paper introduces a new method of clustering algorithm based on interval-valued intuitionistic fuzzy sets (IVIFSs) generated from intuitionistic fuzzy sets to analyze tumor in magnetic resonance (MR) images by reducing time complexity and errors. Based on fuzzy clustering, during the segmentation process one can consider numerous cases of uncertainty involving in membership function, distance measure, fuzzifier, and so on. Due to poor illumination of medical images, uncertainty emerges in their gray levels. This paper concentrates on uncertainty in the allotment of values to the membership function of the uncertain pixels. Proposed method initially pre-processes the brain MR images to remove noise, standardize intensity, and extract brain region. Subsequently IVIFSs are constructed to utilize in the clustering algorithm. Results are compared with the segmented images obtained using histogram thresholding, k-means, fuzzy c-means, intuitionistic fuzzy c-means, and interval type-2 fuzzy c-means algorithms and it has been proven that the proposed method is more effective.     

区间直觉二元语言集成算子及其群决策方法

基于下标以零为中心对称的语言评估标度,将区间不确定二元语言集与区间直觉模糊集结合,提出区间直觉二元语言集及变量的概念;讨论区间直觉二元语言变量的运算及可能度;提出区间直觉二元语言加权算术平均算子、区间直觉二元语言有序加权平均算子,并在此基础上,通过可能度矩阵对区间直觉二元语言变量进行排序提出区间直觉二元语言混合加权平均算子;最后基于这些算子构建了一种新的直觉模糊多属性群决策方法,并将其运用于供应商选择过程中。     

基于区间直觉模糊集的多准则决策方法

研究基于区间直觉模糊集的多准则决策方法.首先定义了区间直觉模糊点算子,并讨论了其性质;然后对区间直觉模糊集定义了一系列得分函数,并给出两种基于区间直觉模糊集的多准则决策方法.将该方法应用于区间直觉模糊集多准则决策问题,所得结果推广了有关直觉模糊集的相关结果.     

Multiple attribute group decision-making methods with unknown weights in intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting

To better solve the corresponding multiple attribute group decision-making problem with unknown weights, multiple attribute group decision-making methods with completely unknown weights of decision-makers and incompletely known weights of attributes are proposed in intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting. In the group decision-making method, two weight models are proposed based on the score function to determine the weights of both experts and attributes from the intuitionistic fuzzy decision matrices and the interval-valued intuitionistic fuzzy decision matrices. Then, overall evaluation formulas of weighted scores for each alternative are introduced in the intuitionistic fuzzy setting and the interval-valued intuitionistic fuzzy setting to obtain the ranking order of alternatives and the most desirable one(s). Finally, two numerical examples demonstrate the applicability and benefit of the proposed methods.     

基于云模型的区间直觉模糊数多属性群决策

针对属性值为区间直觉模糊数的多属性群决策问题, 考虑到模糊性和随机性对群决策过程及结果的影响, 本研究将利用云模型理论结合区间直觉模糊数的特征, 运用灰色关联系数法和信息熵理论确定专家和属性权重, 通过信息集结构建综合评价云模型. 不同于传统的区间直觉模糊数的排序方法, 本研究利用云模型的3En规则将区间直觉模糊数进行云转换并通过云相似度确定方案的综合评价值和犹豫度, 然后对决策方案进行比较分析. 研究结果表明: 该方法能够科学有效地进行决策, 进而为决策方提供科学依据.     

直觉区间值模糊集的熵、距离测度和相似测度

熵、距离测度和相似测度是模糊集理论中的三个基本概念.本文中,我们给出了直觉区间值模糊集的熵、距离测度和相似测度的公理化定义并讨论了它们之间的基本关系.有别于以往的交、并运算,我们提出了直觉区间值模糊集的加法、乘法运算,并在此基础上讨论了直觉区间值模糊集的熵、距离测度和相似测度的性质.