区间集值信息系统中的粗糙集理论

介绍集值信息系统和区间值信息系统,并提出了同时具有这两种系统特点的区间集值信息系统.依据属性值的语义关系,将区间集值信息系统分为两类：析取（I型）和合取（II型）系统,并对其分别提出了基于优势关系的粗糙集模型,讨论了相关性质.最后用实例分析验证了所提出系统的有效性.     

集中有序集值信息系统

集值信息系统是完备信息系统的一种推广,按照语义可划分为合取集值信息系统和析取集值信息系统。属性偏好关系也有两种：属性递增偏好有序和属性递减偏好有序。提出一种新的属性偏好关系,建立了一种新的优势关系。这种优势关系能够表示一类属性偏好既不是递增有序也不是递减有序,而是趋近于某个标准值的情形,称这样的优势关系为属性集中有序,它可应用于某些集值信息系统。     

Uncertainty measures for fuzzy relations and their applications

Relations and relation matrices are important concepts in set theory and intelligent computation. Some general uncertainty measures for fuzzy relations are proposed by generalizing Shannon's information entropy. Then, the proposed measures are used to calculate the diversity quantity of multiple classifier systems and the granularity of granulated problem spaces, respectively. As a diversity measure, it is shown that the fusion system whose classifiers are of little similarity produces a great uncertainty quantity, which means that much complementary information is achieved with a diverse multiple classifier system. In granular computing, a “coarse–fine” order is introduced for a family of problem spaces with the proposed granularity measures. The problem space that is finely granulated will get a great uncertainty quantity compared with the coarse problem space. Based on the observation, we employ the proposed measure to evaluate the significance of numerical attributes for classification. Each numerical attribute generates a fuzzy similarity relation over the sample space. We compute the condition entropy of a numerical attribute or a set of numerical attribute relative to the decision, where the greater the condition entropy is, the less important the attribute subset is. A forward greedy search algorithm for numerical feature selection is constructed with the proposed measure. Experimental results show that the proposed method presents an efficient and effective solution for numerical feature analysis.     

Multiple-attribute group decision making with different formats of preference information on attributes.

Interval utility values, interval fuzzy preference relations, and interval multiplicative preference relations are three common uncertain-preference formats used by decision-makers to provide their preference information in the process of decision making under fuzziness. This paper is devoted in investigating multiple-attribute group-decision-making problems where the attribute values are not precisely known but the value ranges can be obtained, and the decision-makers provide their preference information over attributes by three different uncertain-preference formats i.e., 1) interval utility values; 2) interval fuzzy preference relations; and 3) interval multiplicative preference relations. We first utilize some functions to normalize the uncertain decision matrix and then transform it into an expected decision matrix. We establish a goal-programming model to integrate the expected decision matrix and all three different uncertain-preference formats from which the attribute weights and the overall attribute values of alternatives can be obtained. Then, we use the derived overall attribute values to get the ranking of the given alternatives and to select the best one(s). The model not only can reflect both the subjective considerations of all decision-makers and the objective information but also can avoid losing and distorting the given objective and subjective decision information in the process of information integration. Furthermore, we establish some models to solve the multiple-attribute group-decision-making problems with three different preference formats: 1) utility values; 2) fuzzy preference relations; and 3) multiplicative preference relations. Finally, we illustrate the applicability and effectiveness of the developed models with two practical examples.     

集值信息系统在变精度相容关系下的决策规则

集值信息系统中对象的属性集值一般不唯一,基于对象属性集值的相似程度在集值信息系统上定义了一种变精度容许关系,并借用极大一致块方法给出了集值信息系统上极大变精度相容类的定义与一些性质,讨论了在这种极大变精度相容类下集值信息系统的决策规则的获取、属性的相对约简及决策规则的优化。     

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.     

基于直觉模糊熵的专家权重确定方法及其验证

针对专家判断信息以直觉模糊集给出的直觉模糊群决策矩阵,提出一种新的客观确定专家权重的方法。与传统的通过专家评价的差异程度来确定专家权重的思路不同,该方法通过定义直觉模糊集的模糊熵计算专家判断信息的模糊程度,进而确定每位专家的权重,并对基于犹豫度、几何距离、相似度量和不确定程度4类模糊熵的定义对专家权重结果的影响进行实验和仿真分析。仿真结果表明,专家的权重不仅取决于不同类模糊熵的定义,还与专家个数和属性个数相关。     

MADM method based on prospect theory and evidential reasoning approach with unknown attribute weights under intuitionistic fuzzy environment

This paper proposes an intuitionistic fuzzy decision method based on prospect theory and the evidential reasoning approach, aiming at analyzing multi-attribute decision making problems in which the criteria values are intuitionistic fuzzy numbers and the information of attributes weights is unknown. Firstly, the measures of entropy and cross entropy are defined for intuitionistic fuzzy sets by taking into consideration the preference of decision maker towards hesitancy degree. Secondly, combined with bounded rationality, the prospect decision matrix is calculated in the light of prospect theory and intuitionistic fuzzy distance. Thirdly, the correlational analyses are conducted between the attribute weights and three indicators which are entropy, cross entropy and prospect value, and optimization models for identifying attribute weights are built under the circumstances that the weights are incomplete and unknown. Finally, in order to avoid the loss of decision making information, the evidential reasoning approach is applied to the calculation of comprehensive prospective values for all alternatives. Following the value calculation, the ranking and the optimal alternative are determined based on the comprehensive prospective values. Illustrating examples demonstrate that the proposed method is reasonable and feasible.     

集值序信息系统的信息熵和知识粒度

针对已有文献中二元优势关系定义过于宽松的不足,在集值序信息系统中结合对象间的不同优势程度,提出δ-优势关系的概念;基于δ-优势关系,将信息熵和知识粒度引入集值序信息系统中进行不确定性的度量。结论表明提出的信息熵和知识粒度可以精确地度量集值序信息系统的不确定性。     

A method based on preference degrees for handling hybrid multiple attribute decision making problems

In this paper, we investigate hybrid multiple attribute decision making problems with various forms of attribute values (real numbers, linguistic labels, interval numbers, intuitionistic fuzzy numbers and interval intuitionistic fuzzy numbers). We propose a method based on preference degrees which may take the forms of fuzzy numbers, intuitionistic fuzzy numbers and interval intuitionistic fuzzy numbers. The method first normalizes various forms of attribute values into preference degrees, and then uses a preference degree-based weighted averaging operator to aggregate the normalized preference degrees. Meanwhile, for convenience of calculation, a new linguistic representation model is presented, whose feasibility is verified by comparing it with the traditional 4-tuple linguistic representation model, and from our model, the mapping relationship between interval intuitionistic fuzzy numbers and linguistic labels can be constructed. Finally, we illustrate the rationality and practicality of the proposed method by an application example.     

直觉模糊信息系统中的信息熵

在直觉模糊信息系统上,直觉模糊信息熵和直觉模糊信息粒度是两种有效地进行不确定性研究的重要工具。在直觉模糊信息系统上,引入直觉模糊粒结构的交、并、差、补等四种运算。提出了基于直觉模糊粒结构的直觉模糊信息熵;并研究了直觉模糊信息熵与直觉模糊信息粒度之间的关系。     

Some information measures for interval-valued intuitionistic fuzzy sets

A new information entropy measure of interval-valued intuitionistic fuzzy set (IvIFS) is proposed by using membership interval and non-membership interval of IvIFS, which complies with the extended form of Deluca-Termini axioms for fuzzy entropy. Then the cross-entropy of IvIFSs is presented and the relationship between the proposed entropy measures and the existing information measures of IvIFSs is discussed. Additionally, some numerical examples are given to illustrate the applications of the proposed entropy and cross-entropy of IvIFSs to pattern recognition and decision-making.     

Hesitant fuzzy information measures and their applications in multi-criteria decision making

Hesitant fuzzy set (HFS) is a powerful decision tool to express uncertain information more flexibly and comprehensively. The aim of this paper is to propose more reasonable information measures for HFSs in comparison with the existing ones. First, a series of distance measures is suggested for hesitant fuzzy element and hesitant fuzzy sets. These measures are directly calculated from hesitant fuzzy elements without judging the decision-makers’ risk preference and adding any values into the hesitant fuzzy element with the smaller number of elements. Then, some similarity and entropy measures are proposed based on the transforming relationship among the information measures. Additionally, based on the proposed information measures, a TOPSIS method for hesitant fuzzy information is provided. Finally, some numerical examples are used in order to illustrate the proposed decision method and a comparative analysis is made to demonstrate that the suggested measures are more objective and feasible in certain cases.     

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.     

Determining objective weights with intuitionistic fuzzy entropy measures: A comparative analysis

In this paper, we propose a new objective weighting method that employs intuitionistic fuzzy (IF) entropy measures to solve multiple-attribute decision-making problems in the context of intuitionistic fuzzy sets. Instead of traditional fuzzy entropy, which uses the probabilistic discrimination of attributes to obtain attribute weights, we utilize the IF entropy to assess objective weights based on the credibility of the input data. We examine various measures for IF entropy with respect to hesitation degree, probability, non-probability, and geometry to calculate the attribute weights. A comparative analysis of different measures to generate attribute rankings is illustrated with both computational experiments as well as analyses of Pearson correlations, Spearman rank correlations, contradiction rates, inversion rates, and consistency rates. The experimental results indicate that ranking the outcomes of attributes not only depends on the type of IF entropy measures but is also affected by the number of attributes and the number of alternatives.     

Entropy and Cross‐entropy for Generalized Hesitant Fuzzy Information and Their Use in Multiple Attribute Decision Making

In this paper, we present the entropy, cross‐entropy, and similarity measure for generalized hesitant fuzzy information and discuss their desirable properties. Some measure formulas are developed, and the relationships among them are investigated. We show that the similarity measure and entropy for generalized hesitant fuzzy information can be transformed by each other based on their axiomatic definitions. Then we develop two approaches for solving multiple attribute decision making, in which the attribute values are given in the form of generalized hesitant fuzzy elements (GHFEs). In the first approach, the attribute weight vector is determined by the generalized hesitant fuzzy entropies, and the optimal alternative is obtained by comparing the generalized hesitant fuzzy cross‐entropies between alternatives and positive‐ideal or negative‐ideal solutions; in the second approach, the attribute weight vector is derived from the maximizing deviation method and optimal alternative is obtained by using the technique for order preference by similarly to ideal solution (TOPSIS) method. Finally, an example is provided to illustrate the practicality and effectiveness of the developed approaches.     

On set-valued fuzzy measures

So far, there have been many discussions on Sugeno’s fuzzy measures and fuzzy integrals, but most of them are concentrated on single-valued functions. Motivated by Aumann’s set-valued integral [J. Math. Anal. Appl. 12 (1965) 1], we have introduced the fuzzy integral of set-valued functions [Fuzzy Sets Syst. 56 (1993) 237; Fuzzy Sets Syst. 76 (1995) 365; Fuzzy Sets Syst. 78 (1996) 341; Ph.D. Dissertation, Harbin Institute of Technology, 1998]. It is a well-behaved extension of fuzzy integrals of single-valued functions. It is well known that Arstein’s set-valued measure [Trans. Am. Math. Soc. 165 (1972) 103] is an important branch in set-valued analysis [Set-valued Analysis, Birkhauser, Berlin, 1990] or theory of correspondences [Theory of Correspondences, Wiley, New York, 1984]. Compared to it, the present paper will try to establish the basic idea of set-valued fuzzy measures, which are monotone set-valued set-functions. It is also a natural generalization of (single-valued) fuzzy measures, as well as an extension of set-valued measures in the case of one-dimension. These works include the concept of set-valued fuzzy measures, set-valued fuzzy measures defined by set-valued fuzzy integrals and set-valued pseudo-additive measures. It can be viewed as a continuation of previous work [Fuzzy Sets Syst. 56 (1993) 237; Fuzzy Sets Syst. 76 (1995) 365; Fuzzy Sets Syst. 78 (1996) 341].     

优势-等价关系下序贯三支决策的属性约简

序贯三支决策方法是一种能够表示问题中的多重层次粒度,并将多粒度结合起来解决不确定决策问题的有效途径。优势-等价关系粗糙集则是针对条件属性具有偏好关系的分类问题,提取有序信息,对目标概念进行近似,从而形成决策知识。利用传统的优势关系粗糙集方法进行知识约简和提取的效率低下,而目前大部分序贯三支决策方法则局限在符号值属性的信息系统中,对连续值和有序值不能进行有效处理,造成一定程度的信息丢失。因此,将序贯三支决策的思想应用于优势关系粗糙集模型中,定义了一种新的基于序贯三支决策的属性约简及相应的属性重要度,对具有偏好值属性的信息系统进行更加高效的处理,通过多粒度的表示和关系的研究,加速了知识约简过程。选取了多组UCI数据进行实验,结果表明所提出的基于优势关系的序贯三支决策方法能够在保证约简质量的基础上明显降低时间耗费。     

Set-valued information systems

Set-valued information systems are generalized models of single-valued information systems. Incomplete information systems can be viewed as disjunctively interpreted set-valued information systems. Since some objects in set-valued information systems may have more than one value for an attribute, so we define tolerance relation and use the maximal tolerance classes to classify the universe of discourse. In order to derive optimal decision rules from set-valued decision information systems, we propose the concept of relative reduct of maximal tolerance classes, and define a kind of discernibility function to compute the relative reduct by Boolean reasoning techniques. Finally, we define three kinds of relative reducts for set-valued information systems and used them to evaluate the significance of attributes.     

Set-valued ordered information systems

Set-valued ordered information systems can be classified into two categories: disjunctive and conjunctive systems. Through introducing two new dominance relations to set-valued information systems, we first introduce the conjunctive/disjunctive set-valued ordered information systems, and develop an approach to queuing problems for objects in presence of multiple attributes and criteria. Then, we present a dominance-based rough set approach for these two types of set-valued ordered information systems, which is mainly based on substitution of the indiscernibility relation by a dominance relation. Through the lower/upper approximation of a decision, some certain/possible decision rules from a so-called set-valued ordered decision table can be extracted. Finally, we present attribute reduction (also called criteria reduction in ordered information systems) approaches to these two types of ordered information systems and ordered decision tables, which can be used to simplify a set-valued ordered information system and find decision rules directly from a set-valued ordered decision table. These criteria reduction approaches can eliminate those criteria that are not essential from the viewpoint of the ordering of objects or decision rules.