Optimal consensus models for group decision making under linguistic preference relations

This paper proposes an optimal consensus model to derive weights for linguistic preference relations (LPRs). Two indexes, an individual‐to‐group consensus index (ICI) and a collective consensus index (CCI), are introduced. An iterative algorithm is presented to describe the consensus reaching process. By changing the weights and modifying a pair of individuals' comparison judgments—which have largest deviation value to the group judgments—the consensus reaching process can terminate, while both ICI and CCI are controlled with predefined thresholds. The algorithm aims to preserve the decision makers’ original information as much as possible. The model and algorithm are then extended to handle the uncertain additive LPRs. Finally, two examples are given to show the effectiveness of the proposed methods.     

纯语言多属性群决策方法研究

研究了属性权重、属性值以及专家权重均以语言形式给出的纯语言多属性群决策问题．定义了语言评估标度的运算法则，给出了一些基于语言评估标度及其运算法则的新算子，提出了一种纯语言多属性群决策方法．该法不但计算简洁便利，而且能充分地利用已有的语言决策信息．最后将该方法应用于解决供应链管理领域中的战略合作伙伴选择问题．     

A new method for probabilistic linguistic multi-attribute group decision making: Application to the selection of financial technologies

“No technology, no finance” has been the consensus in banking industry. Under the background of financial technology (Fintech), how to select an appropriate technology company to cooperate for the banks has become a key. The technology company selection can be regarded as a kind of multi-attribute group decision making (MAGDM) problems. The probabilistic linguistic term set (PLTS) is a useful tool to express decision makers’ (DMs’) evaluations in the technology company selection. This paper develops a new method for MAGDM with PLTSs. Firstly, the possibility degree and range value of PLTSs are defined. Then a possibility degree algorithm is designed for ranking PLTSs. An Euclidean distance measure between PLTSs is presented and extended to probabilistic linguistic matrices. Based on Archimedean t-norm and s-norm, some new operational laws for PLTSs are defined and some desirable properties are discussed. Then, a generalized probabilistic linguistic Hamacher weighted averaging (GPLHWA) operator and a generalized probabilistic linguistic Hamacher ordered weighted averaging (GPLHOWA) operator are developed. Some useful properties for these operators are investigated in detail. Combined with the subjective weights of DMs, the DMs’ weights are determined by the adjusted coefficients. Using the GPLHWA operator, the collective decision matrix is obtained by aggregating all the individual decision matrices. By maximizing the total weighted square possibility degree, a multi-objective program is constructed to derive the attribute weights. The ranking order of alternatives is generated by integrating ELECTRE and TOPSIS. Thereby, a new method is put forward for MAGDM with PLTSs. A Fintech example is analyzed to show the effectiveness of the proposed method. The sensitivity analysis and comparative analyses are conducted to illustrate its advantages.     

区间模糊语言多属性群决策方法及应用研究

基于我国银行业所面临的数据匮乏、信息披露制度不完善等问题,国际上先进的统计量化方法并不能完全适用于对我国操作风险的评价,从定性和定量相结合的角度出发,针对操作风险评价中的多属性群决策问题,提出一种在区间模糊语言信息下基于不确定的拓展有序加权平均算子（UEOWA）和不确定语言混合集结算子（ULHA）的多专家多属性商业银行操作风险评价方法,构建基于区间模糊语言的评价集来描述专家不精确、难量化的意见,梳理评价思路,设计评价步骤,通过实例验证了该方法的可操作性和有效性。     

Partitioned Bonferroni mean based on linguistic 2-tuple for dealing with multi-attribute group decision making

In this study, a multi-attribute group decision making (MAGDM) problem is investigated, in which decision makers provide their preferences over alternatives by using linguistic 2-tuple. In the process of decision making, we introduce the idea of a specific structure in the attribute set. We assume that attributes are partitioned into several classes and members of intra-partition are interrelated while no interrelationship exists among inter partition. We emphasize the importance of having an aggregation operator, to capture the expressed inter-relationship structure among the attributes, which we will refer to as partition Bonferroni mean (PBM). We also investigate the behavior of the proposed PBM operator. Further to aggregate the given linguistic information to get overall performance value of each alternative in MAGDM, we analyze PBM operator in linguistic 2-tuple environment and develop three new linguistic aggregation operators: 2-tuple linguistic PBM (2TLPBM), weighted 2-tuple linguistic PBM (W2TLPBM) and linguistic weighted 2-tuple linguistic PBM (LW-2TLPBM). Based on the idea that total linguistic deviation between individual decision maker's opinions and group opinion should be minimized, we develop an approach to determine weight of the decision makers. Finally, a practical example is presented to illustrate the proposed method and comparison analysis demonstrates applicability of the proposed method.     

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。     

An approach to multiplicative linguistic group decision making based on possibility degrees

In order to compare two uncertain multiplicative linguistic variables, a possibility degree formula has been developed and its properties are also studied. Then a possibility degree matrix (also considered as a reciprocal preference relation) is constructed to compare a collection of uncertain multiplicative linguistic variables. Two models are further provided to derive the priority vector from the possibility degree matrix based on the additive consistency and multiplicative consistency. Especially, if the parameters are assigned specific values, then the models reduce to the existing ones. A group decision making method has been developed to deal with the situations where the preferences on alternatives are expressed by uncertain multiplicative linguistic variables. In this method, the possibility degree matrix is first constructed, from which the priority of alternatives is obtained using the developed models. At the end, an example is given to illustrate the proposed method.     

模糊语言多属性群决策的客观综合赋权方法

针对一类属性及专家权重完全未知且评价值为直觉不确定语言数的多属性群决策问题,提出一种基于客观综合赋权模型的模糊群决策方法。通过定义直觉不确定语言数的不确定度和距离测度,对单个专家内部以及专家群体之间的评价值进行分析,分别建立基于离差最大化和熵值的属性综合赋权模型以及基于不确定度和偏离度的专家综合赋权模型,提出一种基于客观综合权重的直觉不确定语言多属性群决策方法。通过实例分析表明了该方法的可行性和实用性。     

基于广义优序法的语言型多属性群决策方法

针对具有语言评价信息的多属性群决策问题,提出基于广义优序法的语言型多属性群决策方法。该方法通过对传统优序法进行有效拓展,采用近年来最新发展的二元语义概念,将语言评价信息转化为二元语义形式的广义优序数,并在此基础上利用方案广义优序数的偏差最大化思想求解得到属性权重,最终确定最优方案。该方法对语言信息的处理较为精确,有效地避免了信息的丢失和扭曲。最后,通过对风险投资案例的分析结果表明了所提出方法的简洁性和有效性。     

Double parameters TOPSIS for multi-attribute linguistic group decision making based on the intuitionistic Z-linguistic variables

Multi-attribute linguistic group decision making (MALGDM) is one of the most widespread and important human activities. Therefore, it is extremely significant to study the efficient methods to address linguistic information. Extended by classical fuzzy number, Z-number can describe both restraints and reliability, which has more capacity to represent human knowledge. In this paper, we will propose the intuitionistic Z-linguistic set (IZLS) combined the intuitionistic uncertain linguistic set (IULS) with Z-number. We further introduce the operational rules, new score and accuracy functions of it. Then, the Minkowski distance between two intuitionistic Z-linguistic variables is proposed with the attribute’s weight obtained by the priority relationship among different categories. Finally, we propose an extended TOPSIS method with double parameters which consider both preference of alternatives and optimism of decision makers (DMs). Case study on investment strategy and medical diagnosis are presented to show the applicability of the proposed approach and some comparisons are conducted to illustrate its superiorities.     

A consensus reaching model for 2-tuple linguistic multiple attribute group decision making with incomplete weight information

The aim of this paper is to put forward a consensus reaching method for multi-attribute group decision-making (MAGDM) problems with linguistic information, in which the weight information of experts and attributes is unknown. First, some basic concepts and operational laws of 2-tuple linguistic label are introduced. Then, a grey relational analysis method and a maximising deviation method are proposed to calculate the incomplete weight information of experts and attributes respectively. To eliminate the conflict in the group, a weight-updating model is employed to derive the weights of experts based on their contribution to the consensus reaching process. After conflict elimination, the final group preference can be obtained which will give the ranking of the alternatives. The model can effectively avoid information distortion which is occurred regularly in the linguistic information processing. Finally, an illustrative example is given to illustrate the application of the proposed method and comparative analysis with the existing methods are offered to show the advantages of the proposed method.     

Probabilistic linguistic vector-term set and its application in group decision making with multi-granular linguistic information

With the rapid information explosion and sharing, recommender systems (RS) play an auxiliary role in assisting the Internet users to make decision especially in the e-service platform. Normally, the information in this process is related to opinions and preferences, which are usually expressed through a qualitative way such as linguistic evaluation terms (LETs). However, the LETs may come from different sources such as experts, users, etc., which makes the linguistic evaluation scales (LESs) used in this process probably be different due to their different backgrounds and levels of knowledge. The diversity and flexibility of these LESs determine the quality of information, and further affect the effectiveness of a RS. In this paper, we focus on improving the accuracy of the multi-granular linguistic recommender system by supporting customers to find out the most eligible items according their own preferences. We first propose the probabilistic linguistic vector-term sets (PLVTSs) to promote the application of multi-granular linguistic information. Based on the PLVTSs, we then develop a novel algorithm to tackle multi-attribute group decision making (MAGDM) problems with multiple LESs. Furthermore, the effectiveness of the PLVTSs is validated by an illustration of personalized hospital selection-recommender problem. Finally, we point out some possible research directions regrading to the PLVTSs.     

基于诱导型直觉不确定语言集成算子的应急预案评估群决策方法

针对在应急预案评估决策中,决策信息以直觉不确定语言数形式表征的多属性群决策问题,提出了基于诱导型直觉不确定语言集成算子的应急预案评估群决策方法。在直觉不确定语言数的运算法则基础上构建了诱导型直觉不确定语言有序加权平均(I-IULOWA)算子、诱导型直觉不确定语言有序加权几何(I-IULOWG)算子、诱导型直觉不确定语言混合加权(I-IULHA)算子和诱导型直觉不确定语言混合加权几何(I-IULHG)算子,探讨上述算子的若干性质及一些特例;同时,给出了基于距离测度的方法来确定属性权重。最后,通过具体的应急突发事件案例验证该方法的可行性和有效性。     

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.     

Interval programming method for hesitant fuzzy multi-attribute group decision making with incomplete preference over alternatives

The aim of this study is to employ the main structure of LINMAP (LINear programming technique for Multidimensional Analysis of Preference) to propose an interval programming method for solving multi-attribute group decision making (MAGDM) problems in which the ratings of alternatives are taken as hesitant fuzzy elements (HFEs) and all pair-wise comparison judgments over alternatives are represented by interval numbers. The contribution of this study is fivefold: (1) we define the new consistency and inconsistency indices; (2) we construct an interval programming model to determine the hesitant fuzzy positive ideal solution and the optimal weights of attributes, and at the same time present a decision algorithm; (3) we discuss several special cases of the proposed model in detail; (4) we show that compared with intuitionistic fuzzy LINMAP method (Li et al., 2010), the proposed approach reveals more useful information including the interval preference information, and does not need to transform HFEs into intuitionistic fuzzy numbers but directly deals with MAGDM problems and thus obtains better final decision results; and (5) we demonstrate the applicability and implementation process of the proposed approach by using an energy project selection example.     

A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making

The ranking of interval-valued intuitionistic fuzzy sets (IVIFSs) is very important for the interval-valued intuitionistic fuzzy decision making. From the probability viewpoint, the possibility degree of comparison between two interval-valued intuitionistic fuzzy numbers (IVIFNs) is defined by using the notion of 2-dimensional random vector, and a new method is then developed to rank IVIFNs. Hereby the ordered weighted average operator and hybrid weighted average operator for IVIFNs are defined based on the Karnik–Mendel algorithms and employed to solve multi-attribute group decision making problems with IVIFNs. The individual overall attribute values of alternatives are obtained by using the weighted average operator for IVIFNs. By using the hybrid weighted average operator for IVIFNs, we can obtain the collective overall attribute values of alternatives, which are used to rank the alternatives. A numerical example is examined to illustrate the effectiveness and flexibility of the proposed method in this paper.     

一种基于二元语义信息处理的多属性群决策方法

为了解决评价信息为语言信息的多属性群决策问题,结合VIKOR方法,提出一种基于二元语义信息处理的多属性群决策方法．该方法使用二元语义信息集结算子获得决策群组的决策信息,通过最大化群效用和最小化个体遗憾来获得决策者满意的折衷方案．该方法计算简单,便于理解,可有效避免信息的丢失和扭曲,并可克服理想解方法不能反映出各方案与正负理想解的接近程度的不足．最后,算例计算结果表明了该方法的有效性．     

An extended compromise ratio method for fuzzy group multi-attribute decision making with SWOT analysis

The technique for order preference by similarity to ideal solution (TOPSIS) is a well-known multi-attribute decision making (MADM) method that is used to identify the most attractive alternative solution among a finite set of alternatives based on the simultaneous minimization of the distance from an ideal solution (IS) and the maximization of the distance from the nadir solution (NS). We propose an alternative compromise ratio method (CRM) using an efficient and powerful distance measure for solving the group MADM problems. In the proposed CRM, similar to TOPSIS, the chosen alternative should be simultaneously as close as possible to the IS and as far away as possible from the NS. The conventional MADM problems require well-defined and precise data; however, the values associated with the parameters in the real-world are often imprecise, vague, uncertain or incomplete. Fuzzy sets provide a powerful tool for dealing with the ambiguous data. We capture the decision makers’ (DMs’) judgments with linguistic variables and represent their importance weights with fuzzy sets. The fuzzy group MADM (FGMADM) method proposed in this study improves the usability of the CRM. We integrate the FGMADM method into a strengths, weaknesses, opportunities and threats (SWOT) analysis framework to show the applicability of the proposed method in a solar panel manufacturing firm in Canada.     

Clustering-based method for large group decision making with hesitant fuzzy linguistic information: Integrating correlation and consensus

Aiming at the large-scale experts and the lower consensus in large group decision making, a novel clustering-based method integrating correlation and consensus of hesitant fuzzy linguistic information is proposed. Firstly, develop a new hesitant degree function for hesitant fuzzy linguistic element considering its scale. Secondly, put forward the correlation measure and consensus measure models combining the hesitant degree. And then present a clustering method integrating the correlation and consensus to divide the large-scale experts into several clusters. The clustering method simultaneously ensures the cohesion of clusters and the gradual increasing of the collective consensus level. After clustering, activate the selection process to update the weights of clusters combining the number of experts in clusters and the consensus level of clusters and use the score function considering the hesitant degree to rank the alternatives. Finally, a case and some comparisons are studied and analyzed to verify the rationality and effectiveness of the method.     

A simulation based multi-attribute group decision making technique with decision constraints

The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a useful technique for solving Multi Attribute Group Decision Making (MAGDM) problems. In MAGDM, the performance scores of the alternatives and the weights of assessment attributes are mostly vague. Therefore, using of deterministic data throughout decision making process may lead to inaccurate results. In order to overcome inherent vagueness and uncertainty, various fuzzy MAGDM techniques were presented in the literature. However, these fuzzy MAGDM techniques are focused on expected and extreme values, which are sometimes insufficient for the precise determination of alternatives’ preference structure. In this paper, in order to eliminate the limitations of deterministic and fuzzy MAGDM methods, we present a probabilistic methodology, which is based on TOPSIS and Monte-Carlo simulation of triangular data. In addition to its straightforward application and thanks to its versatility, simulation enables decision makers to incorporate some decision constraints into decision-making process. Two illustrative examples are also given to show the effectiveness of the proposed methodology. The method is also compared with a fuzzy TOPSIS technique from the literature.