Intuitionistic uncertain linguistic partitioned Bonferroni means and their application to multiple attribute decision-making

The Bonferroni mean (BM) was originally introduced by Bonferroni and generalised by many other researchers due to its capacity to capture the interrelationship between input arguments. Nevertheless, in many situations, interrelationships do not always exist between all of the attributes. Attributes can be partitioned into several different categories and members of intra-partition are interrelated while no interrelationship exists between attributes of different partitions. In this paper, as complements to the existing generalisations of BM, we investigate the partitioned Bonferroni mean (PBM) under intuitionistic uncertain linguistic environments and develop two linguistic aggregation operators: intuitionistic uncertain linguistic partitioned Bonferroni mean (IULPBM) and its weighted form (WIULPBM). Then, motivated by the ideal of geometric mean and PBM, we further present the partitioned geometric Bonferroni mean (PGBM) and develop two linguistic geometric aggregation operators: intuitionistic uncertain linguistic partitioned geometric Bonferroni mean (IULPGBM) and its weighted form (WIULPGBM). Some properties and special cases of these proposed operators are also investigated and discussed in detail. Based on these operators, an approach for multiple attribute decision-making problems with intuitionistic uncertain linguistic information is developed. Finally, a practical example is presented to illustrate the developed approach and comparison analyses are conducted with other representative methods to verify the effectiveness and feasibility of the developed approach.     

Multiple attribute group decision making methods based on intuitionistic linguistic power generalized aggregation operators

With respect to multiple attribute group decision making (MADM) problems in which attribute values take the form of intuitionistic linguistic numbers, some new group decision making methods are developed. Firstly, some operational laws, expected value, score function and accuracy function of intuitionistic linguistic numbers are introduced. Then, an intuitionistic linguistic power generalized weighted average (ILPGWA) operator and an intuitionistic linguistic power generalized ordered weighted average (ILPGOWA) operator are developed. Furthermore, some desirable properties of the ILPGWA and ILPGOWA operators, such as commutativity, idempotency and monotonicity, etc. are studied. At the same time, some special cases of the generalized parameters in these operators are analyzed. Based on the ILPGWA and ILPGOWA operators, two approaches to multiple attribute group decision making with intuitionistic linguistic information are proposed. Finally, an illustrative example is given to verify the developed approaches and to demonstrate their practicality and effectiveness.     

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.     

q-Rung orthopair uncertain linguistic partitioned Bonferroni mean operators and its application to multiple attribute decision-making method

A q-rung orthopair uncertain linguistic set can be served as an extension of an uncertain linguistic set (ULS) and a q-rung orthopair fuzzy set, which can also be treated as a generalized form of the existing intuitionistic ULS and Pythagorean ULS. The new linguistic set uses the uncertain linguistic variable to express the qualitative evaluation information and allows decision makers to provide their true views freely in a larger membership grade space. In this paper, we investigate the Bonferroni mean under the q-rung orthopair uncertain linguistic environment, then we propose the q-rung orthopair uncertain linguistic Bonferroni mean and its weighted form. Furthermore, considering the specific partition pattern among the attributes, the q-rung orthopair uncertain linguistic partitioned Bonferroni mean and its weighted form are developed. Meanwhile, we discuss several representative cases and attractive properties of our proposed operators in depth. Subsequently, a novel multi-attribute decision-making method is developed based on the above-mentioned aggregation operators. In the end, a comprehensible case is performed to analyze the superiority of the developed method by comparing with other typical studies.     

Correlation Coefficient of Interval-Valued Intuitionistic Uncertain Linguistic Sets and Its Application

With respect to multiattribute group decision-making with interval-valued intuitionistic uncertain linguistic information, correlation and correlation coefficient are defined. To obtain the optimal weight vector, models based on the introduced correlation coefficient are constructed. Furthermore, two aggregation operators called the interval-valued intuitionistic uncertain linguistic Shapley averaging operator and the interval-valued intuitionistic uncertain linguistic Shapley geometric operator are defined, which are used to aggregate the interval-valued intuitionistic uncertain linguistic information. Moreover, an approach to multiattribute group decision-making under interval-valued intuitionistic uncertain linguistic environment is developed. Finally, an illustrative example is selected to verify the practicality and feasibility of the proposed procedure.     

零模的直觉模糊不确定语言集成算子研究

基于零模与共轭零模算子,探讨了直觉模糊不确定语言变量运算法则,得到了基于零模与共轭零模的直觉模糊不确定语言加权几何算子,并给出了一种使用直觉不确定语言变量的集成算子的多属性群决策方法,最后通过Matlab软件分析了直觉模糊不确定语言加权几何算子的K值与语言术语下标间关系。为多属性群决策提供了有价值的参考,有效地解决了一类具有直觉模糊不确定语言评估信息的多属性群决策问题。     

Some geometric aggregation operators based on intuitionistic fuzzy sets

The weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator are two common aggregation operators in the field of information fusion. But these two aggregation operators are usually used in situations where the given arguments are expressed as crisp numbers or linguistic values. In this paper, we develop some new geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator, which extend the WG and OWG operators to accommodate the environment in which the given arguments are intuitionistic fuzzy sets which are characterized by a membership function and a non-membership function. Some numerical examples are given to illustrate the developed operators. Finally, we give an application of the IFHG operator to multiple attribute decision making based on intuitionistic fuzzy sets.     

Partitioned Bonferroni mean based on two-dimensional uncertain linguistic variables for multiattribute group decision making

The two-dimensional uncertain linguistic variables (2DULVs) add a self-evaluation on the reliability of the assessment results given by decision makers (DMs), so they can better describe some uncertain information, and the partition Bonferroni mean (PBM) operator has the advantages, which assumes that all aggregated arguments are partitioned into several subparts, and members in the same subpart are interrelated and members in different subparts are no interrelationships. However, the traditional PBM can only deal with the crisp numbers and cannot aggregate the 2DULVs. In this paper, we extend the PBM operator to deal with the 2DULVs and propose some PBM operators for 2DULVs. First, we introduce the concepts, properties, operational laws, and comparison methods of 2DULVs, and then we propose the PBM operator for 2DULVs (2DULPBM), the weighted PBM operator for 2DULVs (2DULWPBM), the partitioned geometric Boferroni mean (PGBM) operator for 2DULVs (2DULPGBM), and weighted PGBM operator for 2DULVs (2DULWPGBM). Further, we develop a method to solve multiattribute group decision-making (MAGDM) problems with the 2DULVs. Finally, we give an example to verify that the method based on the proposed operators is effective and influential.     

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。     

Some induced intuitionistic fuzzy aggregation operators applied to multi-attribute group decision making

In this paper, we define various induced intuitionistic fuzzy aggregation operators, including induced intuitionistic fuzzy ordered weighted averaging (OWA) operator, induced intuitionistic fuzzy hybrid averaging (I-IFHA) operator, induced interval-valued intuitionistic fuzzy OWA operator, and induced interval-valued intuitionistic fuzzy hybrid averaging (I-IIFHA) operator. We also establish various properties of these operators. And then, an approach based on I-IFHA operator and intuitionistic fuzzy weighted averaging (WA) operator is developed to solve multi-attribute group decision-making (MAGDM) problems. In such problems, attribute weights and the decision makers' (DMs') weights are real numbers and attribute values provided by the DMs are intuitionistic fuzzy numbers (IFNs), and an approach based on I-IIFHA operator and interval-valued intuitionistic fuzzy WA operator is developed to solve MAGDM problems where the attribute values provided by the DMs are interval-valued IFNs. Furthermore, induced intuitionistic fuzzy hybrid geometric operator and induced interval-valued intuitionistic fuzzy hybrid geometric operator are proposed. Finally, a numerical example is presented to illustrate the developed approaches.     

Induced and uncertain heavy OWA operators

In this paper, we analyse in detail the ordered weighted averaging (OWA) operator and some of the extensions developed about it. We specially focus on the heavy aggregation operators. We suggest some new extensions about the OWA operator such as the induced heavy OWA (IHOWA) operator, the uncertain heavy OWA (UHOWA) operator and the uncertain induced heavy OWA (UIHOWA) operator. For these three new extensions, we consider some of their main properties and a wide range of special cases found in the weighting vector such as the heavy weighted average (HWA) and the uncertain heavy weighted average (UHWA). We further generalize these models by using generalized and quasi-arithmetic means obtaining the generalized heavy weighted average (GHWA), the induced generalized HOWA (IGHOWA) and the uncertain IGHOWA (UIGHOWA) operator. Finally, we develop an application of the new approach in a decision-making problem.     

Hesitant 2-tuple linguistic Bonferroni operators and their utilization in group decision making

Hesitant 2-tuple linguistic variable realizes a graded information approach to characterize the uncertainty of human cognition. This study is concerned with the development of new aggregation operators and aims to design a new group decision making approach to address the information fusion involving the interrelationship between aggregated terms and the prioritization relationship among decision makers under hesitant 2-tuple linguistic situation. Firstly, hesitant 2-tuple linguistic Bonferroni mean (H2TLBM) operator and prioritized weighted hesitant 2-tuple linguistic Bonferroni mean (PWH2TLBM) operator are established. Subsequently, some pertinent properties and special forms of the developed operators are studied in detail. To employ the proposed operators to solve group decision making problems, a novel TODIM (an acronym in Portuguese of interactive and multiple attribute decision making) method based on possibility degree is developed under the situation of hesitant 2-tuple linguistic information. The designed decision making method not only inherits the merits of the traditional TODIM approach, but also characterizes the interrelationship of criteria. The detailed process of solving problems is exemplified to highlight the practicality and feasibility of the designed method. Furthermore, comparative analysis with other methods is carried out to further offer insights on the designed decision method.     

Multiple-attribute group decision-making based on power Bonferroni operators of linguistic q-rung orthopair fuzzy numbers

In this paper, a new conception of linguistic q-rung orthopair fuzzy number (Lq-ROFN) is proposed where the membership and nonmembership of the q-rung orthopair fuzzy numbers ( q-ROFNs) are represented as linguistic variables. Compared with linguistic intuitionistic fuzzy numbers and linguistic Pythagorean fuzzy numbers, the Lq-ROFNs can more fully describe the linguistic assessment information by considering the parameter q to adjust the range of fuzzy information. To deal with the multiple-attribute group decision-making (MAGDM) problems with Lq-ROFNs, we proposed the linguistic score and accuracy functions of the Lq-ROFNs. Further, we introduce and prove the operational rules and the related properties characters of Lq-ROFNs. For aggregating the Lq-ROFN assessment information, some aggregation operators are developed, involving the linguistic q-rung orthopair fuzzy power Bonferroni mean (BM) operator, linguistic q-rung orthopair fuzzy weighted power BM operator, linguistic q-rung orthopair fuzzy power geometric BM (GBM) operator, and linguistic q-rung orthopair fuzzy weighted power GBM operator, and then presents their rational properties and particular cases, which cannot only reduce the influences of some unreasonable data caused by the biased decision-makers, but also can take the interrelationship between any two different attributes into account. Finally, we propose a method to handle the MAGDM under the environment of Lq-ROFNs by using the new proposed operators. Further, several examples are given to show the validity and superiority of the proposed method by comparing with other existing MAGDM methods.     

基于折衷型变权向量的直觉语言决策方法

研究一种基于折衷型变权向量的直觉语言决策方法. 首先, 定义折衷型变权向量, 提出与之对应的状态变权向量; 其次, 研究利用马氏效用函数诱导出折衷型变权向量; 再次, 定义直觉语言变量运算法则和大小比较方法, 提出直觉语言信息变权加权平均算子和直觉语言信息变权加权几何平均算子, 进而提出一种初始属性权重确定且属性值以直觉语言形式给出的多属性决策方法; 最后, 通过实例表明了所提出方法的有效性和合理性.

     

Induced generalized intuitionistic fuzzy OWA operator for multi-attribute group decision making

With respect to multi-attribute group decision making (MAGDM) problems in which both the attribute weights and the decision makers (DMs) weights take the form of real numbers, attribute values provided by the DMs take the form of intuitionistic fuzzy numbers, a new group decision making method is developed. Some operational laws, score function and accuracy function of intuitionistic fuzzy numbers are introduced at first. Then a new aggregation operator called induced generalized intuitionistic fuzzy ordered weighted averaging (IG-IFOWA) operator is proposed, which extend the induced generalized ordered weighted averaging (IGOWA) operator introduced by Merigo and Gil-Lafuente [Merigo, J. M., & Gil-Lafuente, A. M. (2009). The induced generalized OWA operator. Information Sciences, 179, 729-741] to accommodate the environment in which the given arguments are intuitionistic fuzzy sets that are characterized by a membership function and a non-membership function. Some desirable properties of the IG-IFOWA operator are studied, such as commutativity, idempotency, monotonicity and boundary. And then, an approach based on the IG-IFOWA and IFWA (intuitionistic fuzzy weighted averaging) operators is developed to solve MAGDM problems with intuitionistic fuzzy information. Finally, a numerical example is used to illustrate the developed approach.     

基于广义直觉语言算子的多属性群决策方法

针对属性权重未知,属性值为直觉语言数的多属性决策问题,提出了一种基于直觉语言熵和广义直觉语言算子的群决策方法.定义了直觉语言熵,并利用直觉语言熵确定属性权重,提出了三种直觉语言算子:广义直觉语言加权几何平均(GILWGA)算子、广义直觉语言有序加权几何(GILOWG)算子及广义直觉语言混合几何(GILHG)算子.利用GILWGA和GILHG算子集结信息,采用基于直觉语言数的得分函数及精确函数进行方案排序与择优,最后通过一个算例说明了该方法的有效性和合理性.     

Some q-rung orthopair uncertain linguistic aggregation operators and their application to multiple attribute group decision making

q-Rung orthopair fuzzy sets (q-ROFSs), originally presented by Yager, are a powerful fuzzy information representation model, which generalize the classical intuitionistic fuzzy sets and Pythagorean fuzzy sets and provide more freedom and choice for decision makers (DMs) by allowing the sum of the $q\mathrm{t}\mathrm{h}$ power of the membership and the $q\mathrm{t}\mathrm{h}$ power of the nonmembership to be less than or equal to 1. In this paper, a new class of fuzzy sets called q-rung orthopair uncertain linguistic sets (q-ROULSs) based on the q-ROFSs and uncertain linguistic variables (ULVs) is proposed, and this can describe the qualitative assessment of DMs and provide them more freedom in reflecting their belief about allowable membership grades. On the basis of the proposed operational rules and comparison method of q-ROULSs, several q-rung orthopair uncertain linguistic aggregation operators are developed, including the q-rung orthopair uncertain linguistic weighted arithmetic average operator, the q-rung orthopair uncertain linguistic ordered weighted average operator, the q-rung orthopair uncertain linguistic hybrid weighted average operator, the q-rung orthopair uncertain linguistic weighted geometric average operator, the q-rung orthopair uncertain linguistic ordered weighted geometric operator, and the q-rung orthopair uncertain linguistic hybrid weighted geometric operator. Then, some desirable properties and special cases of these new operators are also investigated and studied, in particular, some existing intuitionistic fuzzy aggregation operators and Pythagorean fuzzy aggregation operators are proved to be special cases of these new operators. Furthermore, based on these proposed operators, we develop an approach to solve the multiple attribute group decision making problems, in which the evaluation information is expressed as q-rung orthopair ULVs. Finally, we provide several examples to illustrate the specific decision-making steps and explain the validity and feasibility of two methods by comparing with other methods.     

New q-rung orthopair fuzzy partitioned Bonferroni mean operators and their application in multiple attribute decision making

The q-rung orthopair fuzzy sets are superior to intuitionistic fuzzy sets or Pythagorean fuzzy sets in expressing fuzzy and uncertain information. In this paper, some partitioned Bonferroni means (BMs) for q-rung orthopair fuzzy values have been developed. First, the q-rung orthopair fuzzy partitioned BM (q-ROFPBM) operator and the q-rung orthopair fuzzy partitioned geometric BM (q-ROFPGBM) operator are developed. Some desirable properties and some special cases of the new aggregation operators have been studied. The q-rung orthopair fuzzy weighted partitioned BM (q-ROFWPBM) operator and the q-rung orthopair fuzzy partitioned geometric weighted BM (q-ROFPGWBM) operator are also developed. Then, a new multiple-attribute decision-making method based on the q-ROFWPBM (q-ROFPGWBM) operator is proposed. Finally, a numerical example of investment company selection problem is given to illustrate feasibility and practical advantages of the new method.     

The induced generalized aggregation operators for intuitionistic fuzzy sets and their application in group decision making

In this paper, we present the induced generalized intuitionistic fuzzy ordered weighted averaging (I-GIFOWA) operator. It is a new aggregation operator that generalized the IFOWA operator, including all the characteristics of both the generalized IFOWA and the induced IFOWA operators. It provides a very general formulation that includes as special cases a wide range of aggregation operators for intuitionistic fuzzy information, including all the particular cases of the I-IFOWA operator, GIFOWA operator and the induced intuitionistic fuzzy ordered geometric (I-IFOWG) operator. We also present the induced generalized interval-valued intuitionistic fuzzy ordered weighted averaging (I-GIIFOWA) operator to accommodate the environment in which the given arguments are interval-valued intuitionistic fuzzy sets. Further, we develop procedures to apply them to solve group multiple attribute decision making problems with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information. Finally, we present their application to show the effectiveness of the developed methods.     

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。