基于熵和风险态度的二型模糊多属性决策方法

针对属性权重信息完全未知的二型模糊多属性决策问题,提出了一种基于二型模糊熵和决策者风险态度的决策方法。首先,为了准确测度二型模糊集（T2FS）的不确定性,通过引入模糊因子和犹豫因子建立了二型模糊熵的公理化准则,并基于距离测度给出了对应的计算公式。其次,为了减少整体不确定信息对决策结果的影响,结合二型模糊熵构建非线性规划模型来确定属性权重。同时,将决策者的风险态度引入二型模糊信息的得分函数中并给出具体的决策步骤。最后,通过实例分析验证了该决策方法的可行性,并与现有文献对比发现该决策方法更具有灵活性。     

Some results on information measures for complex intuitionistic fuzzy sets

Complex intuitionistic fuzzy sets (CIFSs), modeled by complex-valued membership and nonmembership functions with codomain the unit disc in a complex plane, handle two-dimensional information in a single set. Under this environment, the primary objective of the present study is to introduce some novel formulae of information measures (similarity measures, distance measures, entropies, and inclusion measures) and discuss the transformation relationships among them. To demonstrate the efficiency of the proposed similarity measures, we apply it to pattern recognition problem and a detailed comparative analysis is conducted with some of the existing measures. Further, algorithms based on proposed measures are developed for handing multicriteria decision-making problems and their working is illustrated with the help of an example. Besides this, the practicality of the proposed similarity measure is demonstrated by developing a clustering algorithm under CIFS environment.     

A new type-2 fuzzy set of linguistic variables for the fuzzy analytic hierarchy process

The fuzzy analytic hierarchy process (FAHP) has been used to solve various multi-criteria decision-making problems where trapezoidal type-1 fuzzy sets are utilized in defining decision-makers’ linguistic judgment. Previous theories have suggested that interval type-2 fuzzy sets (IT2 FS) can offer an alternative that can handle vagueness and uncertainty. This paper proposes a new FAHP characterized by IT2 FS for linguistic variables. Differently from the typical FAHP, which directly utilizes trapezoidal type-1 fuzzy numbers, this method introduces IT2 FS to enhance judgment in the fuzzy decision-making environment. This new model includes linguistic variables in IT2 FS and a rank value method for normalizing upper and lower memberships of IT2 FS. The proposed model is illustrated by a numerical example of work safety evaluation. Comparable results are also presented to check the feasibility of the proposed method. It is shown that the ranking order of the proposed method is consistent with the other two methods despite difference in weight priorities.     

On some correlation coefficients in Pythagorean fuzzy environment with applications

A Pythagorean fuzzy set (PFS) is an extension of an intuitionistic FS that can be extended by relaxing the restriction on the grades of satisfaction and dissatisfaction. PFS is a powerful tool for dealing with uncertainty and vagueness. Correlation analysis of PFSs is a hot research topic in Pythagorean fuzzy (PF) theory and has practical applications in many areas, such as decision-making, pattern recognition, medical diagnosis, engineering, and so forth. In this communication, we introduce some novel correlation coefficients in the PF-environment satisfying the condition that the correlation coefficient of two PFSs is one if and only if the two sets are equal. We discuss the properties and applications of the proposed measures in pattern recognition, medical diagnosis, multicriteria decision-making, and clustering analysis. Furthermore, the superiority of our proposed correlation coefficients over some existing ones is also established. We also extend the correlation coefficients to interval-valued PFSs.     

A group decision-making model based on incomplete comparative expressions with hesitant linguistic terms

As a result of uncertainty and complexity for environments of decision-making, it is more suitable for decision makers to use hesitant fuzzy linguistic information. In this paper, a novel group decision making (GDM) model based on fuzzy linear programming is proposed for incomplete comparative expressions with hesitant fuzzy linguistic term set (HFLTSs). We establish an equivalence theorem of additive consistency between 2-tuple fuzzy linguistic preference relation (FLPR) and corresponding fuzzy preference relation. Based on this framework, a fuzzy linear programming is established to address incomplete comparative expressions with HFLTSs. It is more important that the proposed fuzzy linear programming has a double action, finding the highest consistent incomplete 2-tuple FLPR and increasing inconsistent 2-tuple FLPR to the additive consistent 2-tuple FLPR based on given incomplete comparative expressions with HFLTSs. By this means, a novel GDM model is constructed based on importance induced ordered weighted averaging operator. Finally, an investment decision-making in real-world is solved by the proposed model, which shows the result of GDM is effectiveness.     

Multiple criteria decision analysis under complex uncertainty: A Pearson-like correlation-based Pythagorean fuzzy compromise approach

The aim of this article is to develop a novel multiple criteria decision analysis (MCDA) method using a Pearson-like correlation-based Pythagorean fuzzy (PF) compromise approach under complex uncertainty based on PF sets and interval-valued Pythagorean fuzzy (IVPF) sets. Because of the complexity and ambiguity involved in real-life decision-making situations, this article utilizes the theory of Pythagorean fuzziness, which is characterized by flexible degrees of membership, nonmembership, and indeterminacy to describe uncertain information more comprehensively. PF and IVPF sets possess exceptional abilities to accurately reflect the uncertainty, fuzziness, and vagueness inherent in the decision information. However, manipulating PF and IVPF information is a complicated and difficult task for most decision makers. In this regard, this article extends the well-known and widely used concept of correlation coefficients to develop simple and effective compromise models for solving MCDA problems in PF and IVPF contexts. This article conducts an extended analysis of Pearson-like correlation coefficients for PF and IVPF sets separately and introduces new concepts of PF and IVPF correlation coefficients to furnish a solid basis for the proposed methodology. Furthermore, this article develops useful concepts of PF and IVPF correlation-based closeness coefficients to simultaneously measure the relative closeness to the positive-ideal PF/IVPF solutions and the relative remoteness from the negative-ideal PF/IVPF solutions. On the basis of the developed concepts, this article proposes a novel Pearson-like correlation-based PF/IVPF compromise approach to address uncertain MCDA problems involving PF/IVPF information and determine the ultimate priority orders among competing alternatives. Finally, this article provides an illustrative application about a financing decision of working capital management to verify the developed approach and demonstrate its feasibility and practicality.     

Interval-valued Pythagorean fuzzy linguistic TODIM based on PCA and its application for emergency decision

Nowadays the emergency decision has become a hot topic in the field of decision-making with interval-valued Pythagorean fuzzy linguistic (IVPFL) information. Moreover, with the increase of attributes and decision makers, the complexity of the operation is a great challenge for making decision. How to overcome multicollinearity is a crucial link in the emergency decision modeling process. In this paper, we treat the attributes and DMs as IVPFL variables and construct the IVPFL principal component analysis (IVPFL-PCA) model to overcome the multicollinearity. Then, a novel TODIM (abbreviation for interactive and multicriterial decision-making in Portuguese) method is proposed to tackle the IVPFL information under several new variables that are independent of each other (i.e., the PCs) and the reasonable weights of PCs obtained based on the IVPFL-PCA model. Finally, a case study on earthquake emergency decision is presented to show the applicability of the proposed approach and some comparisons are presented to illustrate its superiorities.     

Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS

Pythagorean fuzzy sets (PFSs) were proposed by Yager in 2013 to treat imprecise and vague information in daily life more rigorously and efficiently with higher precision than intuitionistic fuzzy sets. In this paper, we construct new distance and similarity measures of PFSs based on the Hausdorff metric. We first develop a method to calculate a distance between PFSs based on the Hasudorff metric, along with proving several properties and theorems. We then consider a generalization of other distance measures, such as the Hamming distance, the Euclidean distance, and their normalized versions. On the basis of the proposed distances for PFSs, we give new similarity measures to compute the similarity degree of PFSs. Some examples related to pattern recognition and linguistic variables are used to validate the proposed distance and similarity measures. Finally, we apply the proposed methods to multicriteria decision-making by constructing a Pythagorean fuzzy Technique for Order Preference by Similarity to an Ideal Solution and then present a practical example to address an important issue related to social sector. Numerical results indicate that the proposed methods are reasonable and applicable and also that they are well suited in pattern recognition, linguistic variables, and multicriteria decision-making with PFSs.     

Emergency decision making with extended axiomatic design approach under picture fuzzy environment

Timely and effective emergency decision making (EDM) is the key to control the spread of disasters and reduce the casualties and property losses caused by emergencies. However, due to limited time and insufficient data, it is difficult for decision makers to provide accurate information about emergency incidents. Moreover, the EDM problems become complicated and unstructured requiring the deployment of advanced mathematical techniques to derive the most acceptable response. In this paper, we propose a new EDM approach by using picture fuzzy sets and axiomatic design technique for determining the optimal rescue plan to reduce the damages of emergencies. The contribution of this paper is to apply the picture fuzzy sets to handle the uncertainty and ambiguity of decision makers' assessments on emergency alternatives, apply the picture fuzzy hybrid averaging operator to aggregate decision makers' opinions into a group evaluation matrix and extend the axiomatic design technique to identify the best emergency solution for EDM. Finally, a real example is provided, and the result is compared with existing methods to demonstrate the feasibility and practicability of our proposed EDM approach.     

Divergence-based cross entropy and uncertainty measures of Atanassov’s intuitionistic fuzzy sets with their application in decision making

The uncertainty measure of Atanassov’s intuitionistic fuzzy sets (AIFSs) is important for information discrimination under intuitionistic fuzzy environment. Although many entropy measures and knowledge measures haven been proposed to depict uncertainty of AIFSs, how to measure the uncertainty of AIFSs is still an open topic. The relation between uncertainty and other measures like entropy measures, fuzziness and intuitionism is not clear. This paper introduces uncertainty measures by using new defined divergence-based cross entropy measure of AIFSs. Axiomatic properties of the developed uncertainty measure are analysis, together with the monotony property of uncertainty degree with respect to fuzziness and intuitionism. To adjust the contribution of fuzzy entropy and intuitionistic entropy on the total uncertainty, the proposed cross entropy and uncertainty measures are parameterized. Numerical examples indicate the effectiveness and agility of the biparametric uncertainty measure in quantifying uncertainty degree. Then we apply the cross entropy and uncertainty measures into an optimal model to determine attribute weights in multi-attribute group decision making (MAGDM) problems. A new method for intuitionistic fuzzy MAGDM problems is proposed to show the efficiency of proposed measures in applications. It is demonstrated by application examples that the proposed measures can get reasonable results coinciding with other existing methods.     

Multiattribute group decision-making based on Pythagorean fuzzy Einstein prioritized aggregation operators

Pythagorean fuzzy set (PFS) is a powerful tool to deal with the imprecision and vagueness. Many aggregation operators have been proposed by many researchers based on PFSs. But the existing methods are under the hypothesis that the decision-makers (DMs) and the attributes are at the same priority level. However, in real group decision-making problems, the attribute and DMs may have different priority level. Therefore, in this paper, we introduce multiattribute group decision-making (MAGDM) based on PFSs where there exists a prioritization relationship over the attributes and DMs. First we develop Pythagorean fuzzy Einstein prioritized weighted average operator and Pythagorean fuzzy Einstein prioritized weighted geometric operator. We study some of its desirable properties such as idempotency, boundary, and monotonicity in detail. Moreover we propose a MAGDM approach based on the developed operators under Pythagorean fuzzy environment. Finally, an illustrative example is provided to illustrate the practicality of the proposed approach.     

信息系统在防治新型冠状病毒的应用

为提高医院防治新型冠状病毒的信息化应急能力,提出基于信息系统的应急处理办法.依托计算机技术、互联网技术、5G技术快速构建线上应急诊疗专区,搭建门急诊分流限流管理平台,开展新冠患者远程会诊.实现慢病、特病、专病患者线上问诊、在线续方、药品配送到家的全流程应用,搭建了机构之间新冠患者远程会诊服务,调整了门诊放号算法,降低了线下就医人群交叉感染率.通过信息系统的应急响应措施,实现医院从线上到线下的疫情应急防治全覆盖,依托信息化手段采用分流、截流、导流快速助力疫情防控,增强了新型冠状病毒的防治效果.     

Adaptive weighted fuzzy rule-based system for the risk level assessment of heart disease

Expert’s knowledge base systems are not effective as a decision-making aid for physicians in providing accurate diagnosis and treatment of heart diseases due to vagueness in information and impreciseness and uncertainty in decision making. For this reason, automatic diagnostic fuzzy systems are very time demanding to improve the diagnostic accuracy. In this paper, we have developed an automatic fuzzy diagnostic system based on genetic algorithm (GA) and a modified dynamic multi-swarm particle swarm optimization (MDMS-PSO) for prognosticating the risk level of heart disease. Our proposed fuzzy diagnostic system (FS) works as follows: i) Preprocess the data sets ii) Effective attributes are selected through statistical methods such as Correlation coefficient, R-Squared and Weighted Least Squared (WLS) method, iii) Weighted fuzzy rules are formed on the basis of selected attributes using GA, iv) MDMS-PSO is employed for the optimization of membership functions (MFs) of FS, v) Build the ensemble FS from the generated fuzzy knowledge base by fusing the different local FSs. Finally, to ascertain the efficiency of the adaptive FS, the applicability of the FS is appraised with quantitative, qualitative and comparative analysis on the publicly available different real-life data sets. From the empirical analysis, we see that this hybrid model can manage the knowledge vagueness and decision-making uncertainty precisely and it has yielded better accuracy on the different publicly available heart disease data sets than other existing methods so that it justifies its adaptability with different data sets.     

Cubic fuzzy Einstein aggregation operators and its application to decision-making

In this paper, we define some Einstein operations on cubic fuzzy set (CFS) and develop three arithmetic averaging operators, which are cubic fuzzy Einstein weighted averaging (CFEWA) operator, cubic fuzzy Einstein ordered weighted averaging (CFEOWA) operator and cubic fuzzy Einstein hybrid weighted averaging (CFEHWA) operator, for aggregating cubic fuzzy data. The CFEHWA operator generalises both the CFEWA and CFEOWA operators. Furthermore, we develop various properties of these operators and derive the relationship between the proposed operators and the exiting aggregation operators. We apply CFEHWA operator to multiple attribute decision-making with cubic fuzzy data. Finally, a numerical example is constructed to demonstrate the established approach.     

Multiple attribute group decision-making based on order-α divergence and entropy measures under q-rung orthopair fuzzy environment

The q-rung orthopair fuzzy set (^qROPFS), proposed by Yager, is a more effective and proficient tool to represent uncertain or vague information in real-life situations. Divergence and entropy are two important measures, which have been extensively studied in different information environments, including fuzzy, intuitionistic fuzzy, interval-valued fuzzy, and Pythagorean fuzzy. In the present communication, we study the divergence and entropy measures under the q-rung orthopair fuzzy environment. First, the work defines two new order-α divergence measures for ^qROPFSs to quantify the information of discrimination between two ^qROPFSs. We also examine several mathematical properties associated with order-α ^qROPF divergence measures in detail. Second, the paper introduces two new parametric entropy functions called “order-α ^qROPF entropy measures” to measure the degree of fuzziness associated with a ^qROPFS. We show that the proposed order-α divergence and entropy measures include several existing divergence and entropy measures as their particular cases. Further, the paper develops a new decision-making approach to solve multiple attribute group decision-making problems under the ^qROPF environment where the information about the attribute weights is completely unknown or partially known. Finally, an example of selecting the best enterprise resource planning system is provided to illustrate the decision-making steps and effectiveness of the proposed approach.     

The interval-valued hesitant Pythagorean fuzzy set and its applications with extended TOPSIS and Choquet integral-based method

The hesitant Pythagorean fuzzy set is frequently considered as a solution for decision making under uncertainty. Whereas the representation of uncertain information might be not sufficient in the hesitant Pythagorean fuzzy environment, thus the concept of interval-valued hesitant Pythagorean fuzzy sets (IVHPFSs) is proposed. Specifically, we first propose the concept of IVHPFSs and then study the operational rules and distance measures of IVHPFSs in detail. To ease the possible application, we explore two decision-making processes in the setting of IVHPFSs by drawing support from the technique for order preference by similarity to ideal solution and Choquet integral-based method. Finally, the selecting processes of project private partner are also presented to demonstrate the decision-making processes based on IVHPFSs and compared with some similar techniques.     

基于信息测度的勾股模糊决策模型及其应用

相对于直觉模糊集，勾股模糊集能够更为全面和有效地表达描述复杂问题中的不确定和非一致信息，使其受到了广泛研究。对于属性评价值为勾股模糊数并且属性指标权重信息数据完全未知的多属性决策问题，以提出的勾股模糊信息测度为基础，设计了新的多属性决策模型。该模型运用对数函数设计了一种新的勾股模糊数信息熵计算方法；引入了勾股模糊相似度概念，并结合对数行数提出勾股模糊数相似度的衡量方法，随后挖掘出勾股模糊数的信息熵和相似度之间的内在联系；运用提出的勾股模糊熵和相似度计算方法，构建新的多属性决策模型，并进行应用研究。实验结果表明，提出的模型合理有效，同时拓展了模型的使用范围。     

基于改进的直觉模糊核聚类的图像分割方法

针对噪声图像模糊性的本质，提出了基于改进的直觉模糊核聚类的图像分割方法。采用直觉模糊集描述噪声图像包含的不确定性信息，将图像的灰度信息转换到直觉模糊域进行处理；将模糊核聚类拓展为直觉模糊核聚类，在图像的直觉模糊域进行聚类；通过高斯核函数和欧氏距离分别对像素8-邻域的灰度和空间信息进行建模，综合平衡灰度和空间信息对聚类的作用，并将其作为惩罚项加入到直觉模糊核聚类的目标函数中；通过梯度下降法，推导了迭代求解算法；通过典型的合成图像和自然图像分割实例，验证了所提算法的有效性和鲁棒性。     

Intuitionistic fuzzy rough set model based on conflict distance and applications

Due to the complexity and uncertainty of the objective world, as well as the limitation of human ability to understand, it is difficult for one to employ only a single type of uncertainty method to deal with the real-life problem of decision-making, especially problems involving conflicts. On the other hand, by incorporating the advantages of various theories of uncertainty, one is expected to develop a more powerful hybrid method for soft decision making and to solve such problems more effectively. In view of this, in this paper the thought and method of intuitionistic fuzzy set and rough set are used to construct a novel intuitionistic fuzzy rough set model. Corresponding to the fact that the decision-making information system of rough sets is of intuitionistic fuzzy information system, our method defines the conflict distance by using the idea of measuring intuitionistic fuzzy similarity so that it is introduced into the models of rough sets, leading to the development of our intuitionistic fuzzy rough set model. After that, we investigate the properties of the model, introduce a novel tool for conflict analysis based on our hybrid model, and employ this new tool to describe and resolve a real-life conflict problem.