An interval type‐2 fuzzy trust evaluation model in social commerce

Trust is crucial for purchasing decisions in social commerce. However, inexperienced users may not have a direct trust relation to experienced users in practice. Besides, users tend to give their trust degrees to others with linguistic labels rather than crisp values. To evaluate the trust degree for inexperienced users to experienced ones, we propose an interval type‐2 fuzzy trust evaluation model in this paper. Interval type‐2 fuzzy linguistic labels are used to represent trust degree among users. An interval type‐2 fuzzy Algebraic t‐norm is addressed to compute propagative trust degrees. Considering the effect of trust path length, the induced ordered weighted averaging (IOWA) operator is extended to aggregate the interval type‐2 fuzzy trust degrees obtained from multiple paths. In addition, the final interval type‐2 fuzzy trust degree is transferred into the corresponding linguistic label to help users make decisions more naturally. Finally, a case study in social commerce and a related comparison are given to verify the effectiveness of the proposed model.     

基于直觉模糊解析面积的聚类算法

针对现有的直觉模糊集聚类算法对权重的忽视或误用,提出一种基于直觉模糊解析面积的聚类算法.同时给出了直觉模糊集的解析面积和属性权重的计算方法,然后构造了聚类算法的目标函数,并给出聚类算法的详细步骤.算例验证了所提出的算法的合理性与可行性.     

基于Spark框架的FP-Growth大数据频繁项集挖掘算法

针对大数据中的频繁项集挖掘问题,提出一种基于Spark框架的FP-Growth频繁项集并行挖掘算法。首先,根据垂直布局思想将数据按照事务标识符垂直排列,以此解决扫描整个数据集的缺陷。然后,通过FP-Growth算法构建频繁模式树,并生成频繁1-项集。接着,通过扫描垂直数据集来计算项集的支持度,从而识别出非频繁项,并将其从数据集中删除以降低数据尺寸。最后,通过迭代过程来生成频繁 -项集。在标准数据集上的实验结果表明,该算法能够有效挖掘出频繁项集,在执行时间方面具有很大的优越性。     

Finding the numerical compensation in multiple criteria decision-making problems under fuzzy environment

In this paper, we have developed a methodology to derive the level of compensation numerically in multiple criteria decision-making (MCDM) problems under fuzzy environment. The degree of compensation is dependent on the tranquility and anxiety level experienced by the decision-maker while taking the decision. Higher tranquility leads to the higher realisation of the compensation whereas the increased level of anxiety reduces the amount of compensation in the decision process. This work determines the level of tranquility (or anxiety) using the concept of fuzzy sets and its various level sets. The concepts of indexing of fuzzy numbers, the risk barriers and the tranquility level of the decision-maker are used to derive his/her risk prone or risk averse attitude of decision-maker in each criterion. The aggregation of the risk levels in each criterion gives us the amount of compensation in the entire MCDM problem. Inclusion of the compensation leads us to model the MCDM problem as binary integer programming problem (BIP). The solution to BIP gives us the compensatory decision to MCDM. The proposed methodology is illustrated through a numerical example.     

关于二型模糊集合的一些基本问题

采用集合论的方法给出了单位模糊集合和二型模糊集合及其在一点的限制等定义,使得二型模糊集合更易于理解.通过定义嵌入单位模糊集合来描述一般二型模糊集合,并给出离散、半连通二型模糊集合的表达式.根据论域、主隶属度及隶属函数的特性将二型模糊集合分为四种类型:离散、半连通、连通及复合型,并根据连通的特点将连通二型模糊集合分为单连通及多连通两类.利用支集的闭包（Closure of support,CoS）划分法表述主隶属度及区间二型模糊集合.提出了CoS二、三次划分法分别来表述单、复连通二型模糊集合,并使每一个子区域的上下边界及次隶属函数在该子区域上的限制分别具有相同的解析表述式.最后,探讨了二型模糊集合在一点的限制、主隶属度、支集、嵌入单位模糊集合之间的关系.     

Improved Accuracy Function for Interval-Valued Intuitionistic Fuzzy Sets and Its Application to Multi–Attributes Group Decision Making

Accuracy functions proposed by various researchers fail to compare some interval-valued intuitionistic fuzzy sets (IVIFSs) correctly. In the present research paper, we propose an improved accuracy function to compare all comparable IVIFSs correctly. The use of proposed accuracy function is also proposed in a method for multi attribute group decision making (MAGDM) method with partially known attributes’ weight. Finally, the proposed MAGDM method is implemented on a real case study of evaluation teachers’ performance. Sensitivity analysis of this method is also done to show the effectiveness of the proposed accuracy function in MAGDM.     

Soft cooperation systems and games

A cooperative game for a set of agents establishes a fair allocation of the profit obtained for their cooperation. In order to obtain this allocation, a characteristic function is known. It establishes the profit of each coalition of agents if this coalition decides to act alone. Originally players are considered symmetric and then the allocation only depends on the characteristic function; this paper is about cooperative games with an asymmetric set of agents. We introduced cooperative games with a soft set of agents which explains those parameters determining the asymmetry among them in the cooperation. Now the characteristic function is defined not over the coalitions but over the soft coalitions, namely the profit depends not only on the formed coalition but also on the attributes considered for the players in the coalition. The best known of the allocation rules for cooperative games is the Shapley value. We propose a Shapley kind solution for soft games.     

Distance and similarity measures of Pythagorean fuzzy sets and their applications to multiple criteria group decision making

The main feature of Pythagorean fuzzy sets is that it is characterized by five parameters, namely membership degree, nonmembership degree, hesitancy degree, strength of commitment about membership, and direction of commitment. In this paper, we first investigate four existing comparison methods for ranking Pythagorean fuzzy sets and point out by examples that the method proposed by Yager, which considers the influence fully of the five parameters, is more efficient than the other ones. Later, we propose a variety of distance measures for Pythagorean fuzzy sets and Pythagorean fuzzy numbers, which take into account the five parameters of Pythagorean fuzzy sets. Based on the proposed distance measures, we present some similarity measures of Pythagorean fuzzy sets. Furthermore, a multiple criteria Pythagorean fuzzy group decision‐making approach is proposed. Finally, a numerical example is provided to illustrate the validity and applicability of the presented group decision‐making method.     

Some methods for strategic decision‐making problems with immediate probabilities in Pythagorean fuzzy environment

In this article, a new decision‐making model with probabilistic information and using the concept of immediate probabilities has been developed to aggregate the information under the Pythagorean fuzzy set environment. In it, the existing probabilities have been modified by introducing the attitudinal character of the decision maker by using an ordered weighted average operator. Based on it, we have developed some new probabilistic aggregation operator with Pythagorean fuzzy information, namely probabilistic Pythagorean fuzzy weighted average operator, immediate probability Pythagorean fuzzy ordered weighted average operator, probabilistic Pythagorean fuzzy ordered weighted average, probabilistic Pythagorean fuzzy weighted geometric operator, immediate probability Pythagorean fuzzy ordered weighted geometric operator, probabilistic Pythagorean fuzzy ordered weighted geometric, etc. Furthermore, we extended these operators by taking interval‐valued Pythagorean fuzzy information and developed their corresponding aggregation operators. Few properties of these operators have also been investigated. Finally, an illustrative example about the selection of the optimal production strategy has been given to show the utility of the developed method.     

Multicriteria decision making based on intuitionistic fuzzy prioritized arithmetic mean

Atanassov’s intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a nonmembership function, and a hesitancy function, is a generalization of a fuzzy set. Various aggregation operators are defined for AIFSs to deal with multicriteria decision‐making problems in which there exists a prioritization of criteria. However, these existing intuitionistic fuzzy prioritized aggregation operators are not monotone with respect to the total order on Atanassov’s intuitionistic fuzzy values (AIFVs), which is undesirable. We propose an intuitionistic fuzzy prioritized arithmetic mean based on the ?ukasiewicz triangular norm, which is monotone with respect to the total order on AIFVs, and therefore is a true generalization of such operations. We give an example that a consumer selects a car to illustrate the validity and applicability of the proposed method aggregation operator.