基于勾股模糊集评价的三支决策方法

针对勾股模糊三支决策概率阈值难以确定的问题,文中提出基于优化表示的勾股模糊三支决策概率阈值确定方法.首先从优化的视角研究一对对偶模型,利用KKT条件证明该对偶模型与决策粗糙集模型的等价性.然后,在确定勾股模糊集评价的三支决策概率阈值时引入对偶模型,基于勾股模糊数非线性排序法建立一对非线性规划模型,证明模型最优解的存在性与唯一性.最后,采用优化技术搜索模型最优解,并提出基于勾股模糊集评价的三支决策方法.算例及对比分析表明文中方法能有效克服现有方法难以确定勾股模糊三支决策概率阈值的不足.     

q-Rung orthopair fuzzy sets-based decision-theoretic rough sets for three-way decisions under group decision making

As an extension of Pythagorean fuzzy sets, the q-rung orthopair fuzzy sets (q-ROFSs) can easily solve uncertain information in a broader perspective. Considering the fine property of q-ROFSs, we introduce q-ROFSs into decision-theoretic rough sets (DTRSs) and use it to portray the loss function. According to the Bayesian decision procedure, we further construct a basic model of q-rung orthopair fuzzy decision-theoretic rough sets (q-ROFDTRSs) under the q-rung orthopair fuzzy environment. At the same time, we design the corresponding method for the deduction of three-way decisions by utilizing projection-based distance measures and TOPSIS. Then, we extend q-ROFDTRSs to adapt the group decision-making (GDM) scenario. To fuse different experts’ evaluation results, we propose some new aggregation operators of q-ROFSs by utilizing power average (PA) and power geometric (PG) operators, that is, q-rung orthopair fuzzy power average, q-rung orthopair fuzzy power weighted average (q-ROFPWA), q-rung orthopair fuzzy power geometric, and q-rung orthopair fuzzy power weighted geometric (q-ROFPWG). In addition, with the aid of q-ROFPWA and q-ROFPWG, we investigate three-way decisions with q-ROFDTRSs under the GDM situation. Finally, we give the example of a rural e-commence GDM problem to illustrate the application of our proposed method and verify our results by conducting two comparative experiments.     

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.     

Interval‐valued Pythagorean fuzzy extended Bonferroni mean for dealing with heterogenous relationship among attributes

Owing to the information insufficiency, it might be difficult for decision makers to precisely evaluate their assessments in real decision‐making. As a new extension of the Pythagorean fuzzy sets, the interval‐valued Pythagorean fuzzy sets (IVPFSs) can availably provide enough input space for decision makers to evaluate their assessments with interval numbers. By extending the Bonferroni mean to model the heterogeneous interrelationship among attributes, the extended Bonferroni mean (EBM) was examined. Considering the partition structure of relationship among the attributes, we introduce the EBM into the interval‐valued Pythagorean fuzzy environment and develop two new aggregation operators, namely, interval‐valued Pythagorean fuzzy extended Bonferroni mean and weighted interval‐valued Pythagorean fuzzy extended Bonferroni mean (WIVPFEBM) operators. Meanwhile, some of their special cases and properties are also deeply discussed. Subsequently, by employing the WIVPFEBM operator, we propose an approach for multiple attribute decision making with IVPFSs. Finally, a practical illustration of the E‐commerce project selection problem is investigated by our proposed method, which successfully demonstrates the applicability of our results.     

加型Pythagorean模糊偏好关系的多属性决策方法

考虑Pythagorean模糊偏好关系的多属性决策问题,提出了加性Pythagorean模糊偏好关系的多属性决策方法。基于加性一致性Pythagorean模糊偏好关系提出一种新的Pythagorean模糊权重确定模型。给出了可接受加性一致性Pythagorean模糊偏好关系的定义,并针对不满足可接受加性一致性的Pythagorean模糊偏好关系,提出一种加性一致性调整算法。给出基于Pythagorean模糊偏好关系加性一致性的多属性决策方法,并通过实例分析提出的新方法的可行性和合理性。     

The linear assignment method for multicriteria group decision making based on interval‐valued Pythagorean fuzzy Bonferroni mean

The interval‐valued Pythagorean fuzzy sets can easily handle uncertain information more flexibly in the process of decision making. Considering the interrelationship among the input arguments, we extend the Bonferroni mean and the geometric Bonferroni mean to the interval‐valued Pythagorean fuzzy environment and solve its practical application problems. First, we develop the interval‐valued Pythagorean fuzzy Bonferroni mean and the weighted interval‐valued Pythagorean fuzzy Bonferroni mean (WIVPFBM) operators. The properties of these aggregation operators are investigated. Then, we also develop the interval‐valued Pythagorean fuzzy geometric Bonferroni mean and the weighted interval‐valued Pythagorean fuzzy geometric Bonferroni mean (WIVPFGBM) operators and analyze their properties. Third, we utilize the WIVPFBM and WIVPFGBM operators to fuse the information in the interval‐valued Pythagorean fuzzy multicriteria group decision making (IVPFMCGDM) problem, which can obtain much more information in the process of group decision making. With the aid of the linear assignment method, we present its extension and further design a new algorithm for the application of IVPFMCGDM. Finally, an example is given to elaborate our proposed algorithm and validate its excellent performance.     

A ranking function based on principal‐value Pythagorean fuzzy set in multicriteria decision making

Multicriteria decision making (MCDM) has been attracting attention in recent years. There are two essential directions in the research territory, one direction is the research of representation of evaluation information and another is the construction of ranking function. In this paper, we consider some nonstandard fuzzy sets, intuitionistic, and interval‐valued fuzzy sets. Especially, the Pythagorean membership grade and Pythagorean fuzzy set receive much attention. Then, to reflect the importance of principal value, we shall propose the principal‐value Pythagorean fuzzy number (p‐PFN) and principal‐value Pythagorean fuzzy set. Furthermore, a novel ranking function is constructed to select the ideal alternative(s) based on p‐PFNs in MCDM. Finally, an investment strategy decision‐making problem is proposed to reveal the availability and practicability of the function under the new environment.     

The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets

Pythagorean fuzzy sets (PFSs) as a new generalization of fuzzy sets (FSs) can handle uncertain information more flexibly in the process of decision making. In our real life, we also may encounter a hesitant fuzzy environment. In view of the effective tool of hesitant fuzzy sets (HFSs) for expressing the hesitant situation, we introduce HFSs into PFSs and extend the existing research work of PFSs. Concretely speaking, this paper considers that the membership degree and the non-membership degree of PFSs are expressed as hesitant fuzzy elements. First, we propose a new concept of hesitant Pythagorean fuzzy sets (HPFSs) by combining PFSs with HFSs. It provides a new semantic interpretation for our evaluation. Meanwhile, the properties and the operators of HPFSs are studied in detail. For the sake of application, we focus on investigating the normalization method and the distance measures of HPFSs in advance. Then, we explore the application of HPFSs to multi-criteria decision making (MCDM) by employing the technique for order preference by similarity to ideal solution (TOPSIS) method. A new extension of TOPSIS method is further designed in the context of MCDM with HPFSs. Finally, an example of the energy project selection is presented to elaborate on the performance of our approach.     

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.     

Fuzzy rough set on probabilistic approximation space over two universes and its application to emergency decision‐making

Probabilistic approaches to rough sets are still an important issue in rough set theory. Although many studies have been written on this topic, they focus on approximating a crisp concept in the universe of discourse, with less effort on approximating a fuzzy concept in the universe of discourse. This article investigates the rough approximation of a fuzzy concept on a probabilistic approximation space over two universes. We first present the definition of a lower and upper approximation of a fuzzy set with respect to a probabilistic approximation space over two universes by defining the conditional probability of a fuzzy event. That is, we define the rough fuzzy set on a probabilistic approximation space over two universes. We then define the fuzzy probabilistic approximation over two universes by introducing a probability measure to the approximation space over two universes. Then, we establish the fuzzy rough set model on the probabilistic approximation space over two universes. Meanwhile, we study some properties of both rough fuzzy sets and fuzzy rough sets on the probabilistic approximation space over two universes. Also, we compare the proposed model with the existing models to show the superiority of the model given in this paper. Furthermore, we apply the fuzzy rough set on the probabilistic approximation over two universes to emergency decision‐making in unconventional emergency management. We establish an approach to online emergency decision‐making by using the fuzzy rough set model on the probabilistic approximation over two universes. Finally, we apply our approach to a numerical example of emergency decision‐making in order to illustrate the validity of the proposed method.