On Pythagorean fuzzy decision making using soft likelihood functions

Multicriteria decision making (MCDM) is to select the optimal candidate which has the best quality from a finite set of alternatives with multiple criteria. One important component of MCDM is to express the evaluation information, and the other one is to aggregate the evaluation results associated with different criteria. For the former, Pythagorean fuzzy set (PFS) is employed to represent uncertain information in this paper, and for the latter, the soft likelihood function developed by Yager is used. To address MCDM issues from a new perspective, the likelihood function of PFS is first proposed in this study and, to improve some of its limitations, the ordered weighted averaging (OWA)-based soft likelihood function is defined, which introduces the attitudinal characteristic to identify decision makers' subjective preferences. In addition, the defined soft likelihood function of PFS is extended by weighted OWA operator considering the importance weight of the argument. Several illustrative cases are provided based on the presented (weighted) OWA-based soft likelihood functions in Pythagorean fuzzy environment for MCDM problem.     

Evidence combination using OWA-based soft likelihood functions

Dempster's combination rule has been widely regarded and applied since it is an effective and rigorous method of synthesizing multisource information with its special information representation (ie, mass function or basic probability assignment). However, it has also been criticized and debated upon regarding some of its unreasonable behaviors and restrictive requirements, such as the counterintuitive results in some cases. To address these issues from different perspectives, in this study, an alternative fusion rule is developed under the framework of Dempster-Shafer evidence theory. A novel evidence combination rule called CR-SLF is proposed based on soft likelihood functions (SLF) considering the ordered weighted average aggregation operator. Some illustrative examples are shown, and the corresponding analyses demonstrate the good performance of CR-SLF to fuse multisource evidence. To extend CR-SLF further, the reliability of multisource evidence is considered from two aspects, subsequently two reliability-based combination rules are presented, including the discount-based rule and the SLF improvement-based rule. The simulation results show that the reliability-based CR-SLF has a better fusion effect than the rule without considering the reliability.     

Generalized intuitionistic fuzzy soft power aggregation operator based on t-norm and their application in multicriteria decision-making

The aim of this paper is to develop some new power aggregation operators for intuitionistic fuzzy (IF) soft numbers. The aggregation operators are named as IF soft power averaging (IFSPA) operator, weighted IFSPA (WIFSPA) operator, ordered WIFSPA operator, IF soft power geometric (IFSPG) operator, and weighted and ordered weighted IFSPG aggregation operators. The salient features of these operators are discussed in detail. Further, these operators are extended to its generalized version and called generalized IFSPA or geometric aggregation operators. Then, we utilized these operators to develop an approach to solve the decision-making problem under IF soft set environment and demonstrated with an illustrative example. A comparative analysis of existing approaches has been done for showing the validity of the proposed work.     

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.     

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。     

Pythagorean Dombi fuzzy aggregation operators with application in multicriteria decision-making

A Pythagorean fuzzy set, an extension of intuitionistic fuzzy sets, is very helpful in representing vague information that occurs in real world scenarios. The Dombi operators with operational parameters, have excellent flexibility. Due to the flexible nature of these Dombi operational parameters, this research paper introduces some new aggregation operators under Pythagorean fuzzy environment, including Pythagorean Dombi fuzzy weighted arithmetic averaging (PDFWAA) operator, Pythagorean Dombi fuzzy weighted geometric averaging (PDFWGA) operator, Pythagorean Dombi fuzzy ordered weighted arithmetic averaging operator and Pythagorean Dombi fuzzy ordered weighted geometric averaging operator. Further, this paper presents several advantageous characteristics, including idempotency, monotonicity, boundedness, reducibility and commutativity of preceding operators. By utilizing PDFWAA and PDFWGA operators, this article describes a multicriteria decision-making (MCDM) technique for solving MCDM problems. Finally, a numerical example related to selection of a leading textile industry is presented to illustrate the applicability of our proposed technique.     

Interval-valued intuitionistic fuzzy cosine similarity measures for multiple attribute decision-making

In this work, considering all information carried by the membership degrees, nonmembership degree, and hesitancy degree in interval-valued intuitionistic fuzzy sets (IVIFSs) as 3-D vector representations, we propose a cosine similarity measure and a weighted cosine similarity measure for IVIFSs based on the extension of the cosine similarity measure (angular coefficient) between intuitionistic fuzzy sets in 2-D vector space. Then, the weighted cosine similarity measure for IVIFSs is applied to multiple attribute decision-making problems under the interval-valued intuitionistic fuzzy environment. Through the similarity measure between the ideal alternative and each alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, two illustrative examples are given to demonstrate the applications and efficiency of the proposed decision-making method.     

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.     

一种基于残缺信息的多准则区间直觉模糊决策方法

提出一种权重系数存在残缺信息的多准则区间直觉模糊集的排序算法。该方法通过逻辑集成得到各方案的区间直觉模糊集,计算各种方案的区间直觉模糊数的Hamming距离,并建立非线性规划模型,利用粒子群算法求解所得的优化模型,得出最优准则的权重系数。然后通过比较区间直觉模糊集与优级方案 及次级方案的距离来进行最优排序。最后利用实例对方法的有效性和可行性进行了说明。     

Fuzzy system reliability evaluation using time-dependent intuitionistic fuzzy set

In general, fuzzy sets are used to analyse the system reliability. In this article, the concept of fuzzy set is extended by the idea of intuitionistic fuzzy set (IFS) and a new general procedure is proposed to construct the membership and non-membership functions of the fuzzy reliability using time-dependent IFS. Here, failure rate function of the system is represented by a triangular intuitionistic fuzzy number (IFN). Also, using proposed approach, membership and non-membership functions of fuzzy reliability of series and parallel systems are constructed, where the failure rate of each component is taken as a time-dependent triangular IFN. The major advantage of using IFS over fuzzy sets is that IFS separate the positive and negative evidences for membership of an element in the set. Numerical examples are given to illustrate the proposed approach.     

Experimental analysis of multi-attribute decision-making based on Atanassov intuitionistic fuzzy sets: a discussion of anchor dependency and accuracy functions

This article presents a useful method for relating anchor dependency and accuracy functions to multiple attribute decision-making (MADM) problems in the context of Atanassov intuitionistic fuzzy sets (A-IFSs). Considering anchored judgement with displaced ideals and solution precision with minimal hesitation, several auxiliary optimisation models have proposed to obtain the optimal weights of the attributes and to acquire the corresponding TOPSIS (the technique for order preference by similarity to the ideal solution) index for alternative rankings. Aside from the TOPSIS index, as a decision-maker's personal characteristics and own perception of self may also influence the direction in the axiom of choice, the evaluation of alternatives is conducted based on distances of each alternative from the positive and negative ideal alternatives, respectively. This article originates from Li's [Li, D.-F. (2005), ‘Multiattribute Decision Making Models and Methods Using Intuitionistic Fuzzy Sets’, Journal of Computer and System Sciences, 70, 73–85] work, which is a seminal study of intuitionistic fuzzy decision analysis using deduced auxiliary programming models, and deems it a benchmark method for comparative studies on anchor dependency and accuracy functions. The feasibility and effectiveness of the proposed methods are illustrated by a numerical example. Finally, a comparative analysis is illustrated with computational experiments on averaging accuracy functions, TOPSIS indices, separation measures from positive and negative ideal alternatives, consistency rates of ranking orders, contradiction rates of the top alternative and average Spearman correlation coefficients.     

Multicriteria decision-making using Archimedean aggregation operators in Pythagorean hesitant fuzzy environment

In multicriteria decision-making (MCDM), the existing aggregation operators are mostly based on algebraic t-conorm and t-norm. But, Archimedean t-conorms and t-norms are the generalized forms of t-conorms and t-norms which include algebraic, Einstein, Hamacher, Frank, and other types of t-conorms and t-norms. From that view point, in this paper the concepts of Archimedean t-conorm and t-norm are introduced to aggregate Pythagorean hesitant fuzzy information. Some new operational laws for Pythagorean hesitant fuzzy numbers based on Archimedean t-conorm and t-norm have been proposed. Using those operational laws, Archimedean t-conorm and t-norm-based Pythagorean hesitant fuzzy weighted averaging operator and weighted geometric operator are developed. Some of their desirable properties have also been investigated. Afterwards, these operators are applied to solve MCDM problems in Pythagorean hesitant fuzzy environment. The developed Archimedean aggregation operators are also applicable in Pythagorean fuzzy contexts also. To demonstrate the validity, practicality, and effectiveness of the proposed method, a practical problem is considered, solved, and compared with other existing method.     

The fuzzy cross-entropy for intuitionistic hesitant fuzzy sets and their application in multi-criteria decision-making

In this paper, the cross-entropy of intuitionistic hesitant fuzzy sets (IHFSs) is developed by integrating the cross-entropy of intuitionistic fuzzy sets (IFSs) and hesitant fuzzy sets (HFSs). First, several measurement formulae are discussed and their properties are studied. Then, two approaches, which are based on the developed intuitionistic hesitant fuzzy cross-entropy, are proposed for solving multi-criteria decision-making (MCDM) problems within an intuitionistic hesitant fuzzy environment. For both methods, an optimisation model is established in order to determine the weight vector for MCDM problems with incomplete information on criteria weights. Finally, an example is provided in order to illustrate the practicality and effectiveness of the proposed approaches.     

Multicriteria group decision-making method using the distances-based similarity measures between intuitionistic trapezoidal fuzzy numbers

The Hamming and Euclidean distances between intuitionistic trapezoidal fuzzy numbers and the distances-based similarity measures are proposed in this study, then an intuitionistic trapezoidal fuzzy multicriteria group decision-making method is established using the similarity measures and expected weight values, in which linguistic values of intuitionistic trapezoidal fuzzy numbers for linguistic terms are used to assess alternatives with respect to qualitative criteria and criteria weights. We establish simple and exact formulae to solve the multicriteria group decision-making problem based on the similarity measures between the ideal alternative and each alternative, the ranking order of all the alternatives and the best one can be determined by the proposed similarity measures. Finally, an illustrative example demonstrates the implementation process of the technique.     

Choquet integrals of weighted intuitionistic fuzzy information

The Choquet integral is a very useful way of measuring the expected utility of an uncertain event [G. Choquet, Theory of capacities, Annales de l’institut Fourier 5 (1953) 131-295]. In this paper, we use the Choquet integral to propose some intuitionistic fuzzy aggregation operators. The operators not only consider the importance of the elements or their ordered positions, but also can reflect the correlations among the elements or their ordered positions. It is worth pointing out that most of the existing intuitionistic fuzzy aggregation operators are special cases of our operators. Moreover, we propose the interval-valued intuitionistic fuzzy correlated averaging operator and the interval-valued intuitionistic fuzzy correlated geometric operator to aggregate interval-valued intuitionistic fuzzy information, and apply them to a practical decision-making problem involving the prioritization of information technology improvement projects.     

基于加权直觉模糊集合的聚类模型

针对已有基于直觉模糊集的聚类方法的局限性,提出了一种基于加权直觉模糊集合的聚类模型——WIFSCM。在该模型中,提出了特定特征空间下的等价样本和加权直觉模糊集合的概念;并推导出基于等价样本和加权直觉模糊集合的直觉模糊聚类算法的目标函数,利用该目标函数推导出直觉模糊聚类中心迭代算法和隶属度矩阵迭代算法;定义了基于加权直觉模糊集合的密度函数,确定了初始聚类中心,减少了迭代次数。通过灰度图像分割实验,证明了该模型的有效性,同时与普通直觉模糊集FCM聚类算法(IFCM)相比,聚类速度提高近百倍。     

基于直觉模糊Petri网的加权直觉模糊推理

利用直觉模糊集合较好地表现不确定信息的能力和Petri网的并行处理能力,构建了直觉模糊Petri网模型。给出了输入权值、变迁阈值等多种约束条件下的直觉模糊推理算法。该算法将直觉模糊推理过程转化为矩阵的运算过程可充分利用直觉模糊Petri网的并行推理能力,有效地避免同一变迁不必要地重复激发从而节省推理时间。实例分析表明所给出的直觉模糊推理算法较已有算法更加合理并且高效。     

基于改进熵和新得分函数的区间直觉模糊多属性决策

针对决策信息为区间直觉模糊数且属性权重完全未知的多属性决策问题, 提出基于改进的区间直觉模糊熵和新得分函数的决策方法. 首先, 利用改进的区间直觉模糊熵确定属性权重; 然后, 利用区间直觉模糊加权算术平均算子集成信息, 得到各备选方案的综合属性值, 进而指出现有得分函数存在排序失效或排序不符合实际的不足, 同时给出一个新的得分函数, 并以此对方案进行排序; 最后, 通过实例表明了所提出方法的有效性.

     

二元联系数的多准则直觉模糊决策

为了研究信息不完全确定的多准则直觉模糊决策,将直觉模糊数转化为二元联系数,建立了基于二元联系数权系数信息不完全确定的多准则直觉模糊决策综合加权模型,并作不确定性分析.结合具体应用实例,说明了该模型的有效性及合理性.