Some new Shapley 2-tuple linguistic Choquet aggregation operators and their applications to multiple attribute group decision making

In this paper, we investigate the multiple attribute group decision making (MAGDM) problems with 2-tuple linguistic information. Firstly, motivated by the ideas of Choquet integral and Shapley index, we propose three 2-tuple linguistic aggregation operators called Shapley 2-tuple linguistic Choquet averaging operator, Shapley 2-tuple linguistic Choquet geometric operator and generalized Shapley 2-tuple linguistic Choquet averaging operator. Then we discuss some properties of these operators, such as idempotency, monotonicity, boundary and commutativity. Secondly, if the information about the weights of decision makers (DMs) and attributes is incompletely known, we build two models to determine the optimal fuzzy measures on DM set and attribute set, respectively. Furthermore, we develop a new method for multiple attribute group decision making under 2-tuple linguistic environment based on the proposed operators. Finally, we apply the developed MAGDM method to select the most desirable emergency alternative and the validity of the developed method is verified by comparing the evaluation results with those obtained from the existing 2-tuple correlated aggregation operators.     

区间直觉二元语言集成算子及其群决策方法

基于下标以零为中心对称的语言评估标度,将区间不确定二元语言集与区间直觉模糊集结合,提出区间直觉二元语言集及变量的概念;讨论区间直觉二元语言变量的运算及可能度;提出区间直觉二元语言加权算术平均算子、区间直觉二元语言有序加权平均算子,并在此基础上,通过可能度矩阵对区间直觉二元语言变量进行排序提出区间直觉二元语言混合加权平均算子;最后基于这些算子构建了一种新的直觉模糊多属性群决策方法,并将其运用于供应商选择过程中。     

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Choquet积分的多属性灰靶群决策方法

针对属性间具有关联性且属性值为区间灰数的多属性群决策问题,提出一种基于Choquet 积分的多属性灰靶群决策方法.根据模糊测度和Choquet积分的性质,定义关于区间灰数的Choquet积分信息集成算子,并证明其相关性质;通过灰靶决策方法,对方案集的综合靶心距进行排序.仿真实例验证了所提出方法的合理性.     

R-WGA 算子的构建及其在群决策分析中的应用

在群决策问题中，决策属性间与专家偏好间均可能存在关联，需定义新的集结算子来计算决策方案的综合评价值．为此，在传统加权几何平均（WGA）算子和模糊测度理论的基础上，构建关联加权几何平均（R—WGA）算子，探讨该算子的性质，给出基于R-WGA算子的群决策分析方法．研究表明，R—WGA算子是WGA算子的推广，与Choquet积分一样，R-WGA算子也可用于求解基于关联的决策问题．     

语言区间直觉模糊Frank算子

针对语言区间直觉模糊信息的集结问题,文中提出Frank集结算子,并构建解决供应商选择问题的群决策方法.首先引入拓展Frank t-模与s-模定义语言区间直觉模糊集的Frank运算法则,提出语言区间直觉模糊Frank加权平均(LIVIFFWA)算子与几何(LIVIFFWG)算子,证明算子的幂等性、封闭性、单调性等基本性质,剖析算子关于参数的退化性.然后,基于LIVIFFWA算子与LIVIFFWG算子构建语言区间直觉模糊多属性群决策方法,用于解决供应商决策问题.最后,通过共享单车回收供应商选择的案例分析验证文中决策方法的可行性和灵活性,讨论参数变化对决策结果的影响,并验证参数具有表征和反馈决策者态度的能力.     

New aggregation operators based on the Choquet integral and 2-tuple linguistic information

Considered in this paper is the group decision making problem with inter-dependent or interactive attributes, where evaluation values of decision makers are in linguistic arguments. By using the Choquet integral, some new aggregation operators are introduced, including the 2-tuple correlated averaging operator, the 2-tuple correlated geometric operator and the generalized 2-tuple correlated averaging operator. The proposed operators can better reflect the correlations among the elements. After investigating properties of these operators, a new multiple attribute decision making method based on the new operators is proposed. Finally, a numerical example is provided to illustrate the feasibility and efficiency of the proposed method.     

基于Choquet 积分的直觉不确定语言信息集结算子及其应用

基于直觉不确定语言信息,针对属性间不严格相互独立且具有较大关联度的群决策问题,提出了两种基于直觉不确定语言信息的Choquet积分算子.首先,分析了因属性关联使得以往直觉不确定语言信息集结算子失效的现象,对此引入模糊测度,提出了直觉不确定语言的Choquet加权算术平均算子(IULCWA)和直觉不确定语言的Choquet加权几何平均算子(IULCGM);然后,证明了算子的相关性质,研究了属性间相关的、属性值为直觉不确定语言数的多属性群决策方法;最后,通过实例分析说明了以往直觉不确定语言信息集结算子的局限性以及新算子的有效性.     

基于IITFN输入的复杂系统关联MAGDM方法

区间直觉梯形模糊数（Interval-valued intuitionistic trapezoidal fuzzy number,IITFN）是刻画复杂系统不确定性的有效工具. 基于进一步完善的IITFN 运算规则,讨论其局部封闭性. 由此定义IITFN 几何Bonferroni 平均算子,并验证该算子的相关性质. 针对决策者及属性之间均存在关联作用且权重均未知的多属性群决策（Multi-attribute group decision making,MAGDM）问题,提出基于前景混合区间直觉梯形几何 Bonferroni （Prospect hybrid interval-valued intuitionistic trapezoidal fuzzy geometric Bonferroni,PHIITFGB）平均算子 的关联多属性群决策方法. 该方法首先通过依次定义IITFN 的前景效应、前景价值函数和前景价值,获取前景价值矩阵;其次,将前景价值矩阵转化为前景记分函数矩阵,并综合运用基于灰关联深度系数的客观属性权重极大 熵模型和基于2-可加模糊测度与Choquet 积分联合的决策者权重确定模型,获取决策者权重及属性权重;再次,利 用PHIITFGB 算子集结各决策者的方案评估信息,结合决策者权重即可获取相应于各方案的综合前景价值;最后,计算综合前景记分价值函数,基于IITFN 的序关系判别准则确定方案排序. 案例验证决策方法的有效性和可行性.     

Hesitant interval-valued Pythagorean fuzzy VIKOR method

In this article, we investigate multiple attribute decision-making problems with hesitant interval-valued Pythagorean fuzzy information. First, the concepts of hesitant interval-valued Pythagorean fuzzy set are defined, and the operation laws, the score function, and accuracy function have been developed. Then several distance measures for hesitant interval-valued Pythagorean fuzzy values have been presented including the Hamming distance, Euclidean distance, and generalized distance, and so on. Based on the operational laws, a series of aggregation operators have been developed including the hesitant interval-valued Pythagorean fuzzy weighted averaging (HIVPFWA) operator, the hesitant interval-valued Pythagorean fuzzy geometric weighted averaging (HIVPFGWA) operator, the hesitant interval-valued Pythagorean fuzzy ordered weighed averaging (HIVPFOWA) operator, and hesitant interval-valued Pythagorean fuzzy ordered weighed geometric averaging (HIVPFOWGA) operator. By using the generalized mean operator, we also develop the generalized hesitant interval-valued Pythagorean fuzzy weighed averaging (GHIVPFWA) operator, the generalized hesitant interval-valued Pythagorean fuzzy weighed geometric averaging (GHIVPFWGA) operator, the generalized hesitant interval-valued Pythagorean fuzzy ordered weighted averaging (GHIVPFOWA) operator, and generalized hesitant interval-valued Pythagorean fuzzy ordered weighted geometric averaging (GHIVPFOWGA) operator operator. We further develop several hybrid aggregation operators including the hesitant interval-valued Pythagorean fuzzy hybrid averaging (HIVPFHA) operator and the generalized hesitant interval-valued Pythagorean fuzzy hybrid averaging (GHIVPFHA) operator. Based on the distance measures and the aggregation operators, we propose a hesitant interval-valued Pythagorean fuzzy VIKOR method to solve multiple attribute decision problems with multiple periods. Finally, an illustrative example for evaluating the metro project risk is given to demonstrate the feasibility and effectiveness of the proposed method.     

Interval-valued hesitant fuzzy Einstein prioritized aggregation operators and their applications to multi-attribute group decision making

Interval-valued hesitant fuzzy information aggregation plays an important role in interval-valued hesitant fuzzy set theory, which has received more and more attention in recent years. In this paper, we investigate interval-valued hesitant fuzzy multi-attribute group decision-making problems in which there exists a prioritization relationship among the attributes. Firstly, we introduce some Einstein operational laws on interval-valued hesitant fuzzy sets, and discuss some relations of these operations. Then, we develop two interval-valued hesitant fuzzy prioritized aggregation operators with the help of Einstein operations, such as the interval-valued hesitant fuzzy Einstein prioritized weighted average (IVHFEPWA) operator and the interval-valued hesitant fuzzy Einstein prioritized weighted geometric (IVHFEPWG) operator, whose desirable properties are investigated in detail. We further analyze the relationship between these proposed operators and the existing interval-valued hesitant fuzzy prioritized aggregation operators. Moreover, an approach to interval-valued hesitant fuzzy multi-attribute group decision making is given on the basis of the proposed operators. Finally, a numerical example is provided to demonstrate their effectiveness.     

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.     

Some q-rung orthopair fuzzy 2-tuple linguistic Muirhead mean aggregation operators and their applications to multiple-attribute group decision making

Multiple-attribute group decision making (MAGDM) under linguistic environment is an important part of modern decision sciences, and information aggregation operator plays an import role in solving MAGDM problems. In this paper, an approach for solving MAGDM problem with q-rung orthopair fuzzy 2-tuple linguistic information is developed. First, the q-rung orthopair fuzzy 2-tuple linguistic weighted averaging (q-ROFTLWA) operator and the q-rung orthopair fuzzy 2-tuple linguistic weighted geometric (q-ROFTLWG) operator are presented. Furthermore, the q-rung orthopair fuzzy 2-tuple linguistic Muirhead mean (q-ROFTLMM) operator and the q-rung orthopair fuzzy 2-tuple linguistic dual Muirhead mean (q-ROFTLDMM) operator are proposed on the basis of Muirhead mean (MM) operator and dual Muirhead mean (DMM) operator. Then, an approach is developed to deal with MAGDM problem under q-rung orthopair fuzzy 2-tuple linguistic environment based on the proposed operators. Finally, a numerical example for selecting desirable emergency alternative(s) in the process of designing emergency preplan is given to illustrate the application of the developed method and demonstrate its effectiveness.     

Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making

For the real decision making problems, most criteria have inter-dependent or interactive characteristics so that it is not suitable for us to aggregate them by traditional aggregation operators based on additive measures. Thus, to approximate the human subjective decision making process, it would be more suitable to apply fuzzy measures, where it is not necessary to assume additivity and independence among decision making criteria. In this paper, an intuitionistic fuzzy Choquet integral is proposed for multiple criteria decision making, where interactions phenomena among the decision making criteria are considered. First, we introduced two operational laws on intuitionistic fuzzy values. Then, based on these operational laws, intuitionistic fuzzy Choquet integral operator is proposed. Moreover, some of its properties are investigated. It is shown that the intuitionistic fuzzy Choquet integral operator can be represented by some special t-norms and t-conorms, and it is also a generalization of the intuitionistic fuzzy OWA operator and intuitionistic fuzzy weighted averaging operator. Further, the procedure and algorithm of multi-criteria decision making based on intuitionistic fuzzy Choquet integral operator is given under uncertain environment. Finally, a practical example is provided to illustrate the developed approaches.     

区间值比例二元组语言集成算子及其决策方法

结合区间不确定性语言集与比例二元组,提出区间值比例二元组的概念;基于得分函数与精确函数,给出了两个区间值比例二元组的比较方法,并介绍了区间值比例二元组与区间数之间的转换函数;在此基础上,提出区间值比例二元组加权几何算子及区间值比例二元组有序加权几何算子;针对不同语言标度的群决策问题,通过统一标度给出了相应的多属性群决策方法,并将其用于投资商投资过程中。     

Some 2-tuple linguistic Pythagorean Heronian mean operators and their application to multiple attribute decision-making

In this paper, we expand the generalised Heronian mean (GHM) operator, generalised weighted Heronian mean (GWHM), geometric Heronian mean (GHM) operator, and weighted geometric Heronian mean (WGHM) operator with 2-tuple linguistic Pythagorean fuzzy numbers (2TLPFNs) to propose generalised 2-tuple linguistic Pythagorean fuzzy Heronian mean (G2TLPFHM) operator, generalised 2-tuple linguistic Pythagorean fuzzy weighted Heronian mean (G2TLPFWHM) operator, 2-tuple linguistic Pythagorean fuzzy geometric Heronian mean (2TLPFGHM) operator, 2-tuple linguistic Pythagorean fuzzy weighted geometric Heronian mean (2TLPFWGHM) operator. Then, the MADM methods are proposed with these operators. In the end, we utilise an applicable example for green supplier selection to prove the proposed methods.     

基于前景理论的犹豫二元语义灰关联群决策法

针对偏好信息为犹豫二元语义形式、专家权重和属性权重均完全未知的多属性群决策问题,基于前景理论和灰色关联分析法的思想,提出一种多属性群决策方法.首先,利用矩阵拉直运算和灰色关联分析法确定专家权重,利用偏差最大化法确定属性权重.其次,给出了两个犹豫二元语义元的比较方法,结合该比较方法确定各决策矩阵的正、负理想方案,并以此作为决策参考点.然后,根据前景理论和灰色关联系数确定犹豫二元语义环境下的前景价值函数,进而确定各方案的收益损失比值,并据此对候选方案进行排序.最后,将所提方法应用于一个投资决策算例,其结果表明了该方法的合理性和有效性.     

区间二元语义Bonferroni集成算子及其决策应用

区间二元语义值是一种常用的不确定环境下决策信息表达形式。考虑到决策信息的交叉影响作用,定义了区间二元语义值的Bonferroni平均算子以及相应的加权形式,在此基础上,给出了组合形式的区间二元语义值的加权Bonferroni平均算子的概念,并研究了算子的幂等性、单调性等数学性质,给出了基于C-I2TLWBA算子区间二元语义值的集成模型和决策应用。实例表明了该模型具有较好的有效性。     

语言比例二元组Bonferroni平均算子的群决策方法

针对语言比例二元组信息集成的问题,提出了语言比例二元组Bonferroni平均算子的群决策方法。基于基本单位区间单调（BUM）函数的定义,提出了三角模糊数的中心有序加权平均算子的概念。由于三角模糊数的中心有序加权平均算子和语言术语的数值表示（NR）之间存在交互关联,提出了用三角模糊数的中心有序加权平均算子代替语言术语的数值表示的方法,并将此方法得到的NR和语言术语的NR进行了对比分析。引入了语言比例二元组Bonferroni平均算子,介绍了语言比例二元组Bonferroni平均算子的群决策方法,并用一个实例来说明其可行性。     

Fuzzy measure identification for criteria coalitions using linguistic information

Modeling interactions between criteria in multiple criteria decision analysis (MCDA) is a complex task. Such complexity arises when there are visible redundancies and synergies among criteria, which traditional MCDA methods cannot deal with. The Choquet integral is a model that has been conceived to deal with these issues, but an appropriate fuzzy measure must be defined. This article shows how to compute a fuzzy measure for criteria coalitions using linguistic information efficiently. Due to the complexity to identify an adequate fuzzy measure when the criteria set cardinality increases, the proposed model reduces the effort to determine the measure of each criteria combination by focusing on relevant interactions. Then, this fuzzy measure is used on Choquet integral to establish the best alternative in a decision-making problem. Finally, a comparison between the arithmetic mean, the OWA operator and the proposed method is presented.