Computing the Numerical Scale of the Linguistic Term Set for the 2-Tuple Fuzzy Linguistic Representation Model

When using linguistic approaches to solve decision problems, we need the techniques for computing with words (CW). Together with the 2-tuple fuzzy linguistic representation models (i.e., the Herrera and MartÍnez model and the Wang and Hao model), some computational techniques for CW are also developed. In this paper, we define the concept of numerical scale and extend the 2-tuple fuzzy linguistic representation models under the numerical scale. We find that the key of computational techniques based on linguistic 2-tuples is to set suitable numerical scale with the purpose of making transformations between linguistic 2-tuples and numerical values. By defining the concept of the transitive calibration matrix and its consistent index, this paper develops an optimization model to compute the numerical scale of the linguistic term set. The desired properties of the optimization model are also presented. Furthermore, we discuss how to construct the transitive calibration matrix for decision problems using linguistic preference relations and analyze the linkage between the consistent index of the transitive calibration matrix and one of the linguistic preference relations. The results in this paper are pretty helpful to complete the fuzzy 2-tuple representation models for CW.      

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。     

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。     

A Fuzzy Linguistic Methodology to Deal With Unbalanced Linguistic Term Sets

Many real problems dealing with qualitative aspects use linguistic approaches to assess such aspects. In most of these problems, a uniform and symmetrical distribution of the linguistic term sets for linguistic modeling is assumed. However, there exist problems whose assessments need to be represented by means of unbalanced linguistic term sets, i.e., using term sets that are not uniformly and symmetrically distributed. The use of linguistic variables implies processes of computing with words (CW). Different computational approaches can be found in the literature to accomplish those processes. The 2-tuple fuzzy linguistic representation introduces a computational model that allows the possibility of dealing with linguistic terms in a precise way whenever the linguistic term set is uniformly and symmetrically distributed. In this paper, we present a fuzzy linguistic methodology in order to deal with unbalanced linguistic term sets. To do so, we first develop a representation model for unbalanced linguistic information that uses the concept of linguistic hierarchy as representation basis and afterwards an unbalanced linguistic computational model that uses the 2-tuple fuzzy linguistic computational model to accomplish processes of CW with unbalanced term sets in a precise way and without loss of information.     

A 2-tuple fuzzy linguistic representation model for computing withwords

The fuzzy linguistic approach has been applied successfully to many problems. However, there is a limitation of this approach imposed by its information representation model and the computation methods used when fusion processes are performed on linguistic values. This limitation is the loss of information; this loss of information implies a lack of precision in the final results from the fusion of linguistic information. In this paper, we present tools for overcoming this limitation. The linguistic information is expressed by means of 2-tuples, which are composed of a linguistic term and a numeric value assessed in (-0.5, 0.5). This model allows a continuous representation of the linguistic information on its domain, therefore, it can represent any counting of information obtained in a aggregation process. We then develop a computational technique for computing with words without any loss of information. Finally, different classical aggregation operators are extended to deal with the 2-tuple linguistic model     

基于语言值有序对三元组的多类型数据标准化方法

为了标准化处理多类型数据,基于语言值二元组,提出语言值有序对三元组表示模型,研究语言值有序对三元组及其性质,给出语言值有序对三元组之间的相似度.讨论语言值有序对三元组加权平均算子,构造模糊集、直觉模糊集、犹豫模糊集、语言真值直觉模糊集与语言值有序对三元组的标准化转化模型.结合语言值有序对三元组之间的相似度,将模糊集、直觉模糊集、犹豫模糊集、语言真值直觉模糊集的转化模型应用于模式识别中.最后,通过医院智能分诊的实例说明文中方法的合理性和有效性.     

Fusion of imprecise qualitative information

In this paper, we present a new 2-tuple linguistic representation model, i.e. Distribution Function Model (DFM), for combining imprecise qualitative information using fusion rules drawn from Dezert-Smarandache Theory (DSmT) framework. Such new approach allows to preserve the precision and efficiency of the combination of linguistic information in the case of either equidistant or unbalanced label model. Some basic operators on imprecise 2-tuple labels are presented together with their extensions for imprecise 2-tuple labels. We also give simple examples to show how precise and imprecise qualitative information can be combined for reasoning under uncertainty. It is concluded that DSmT can deal efficiently with both precise and imprecise quantitative and qualitative beliefs, which extends the scope of this theory.     

An integrated fuzzy MCDM approach for supplier evaluation and selection

A fuzzy multi-criteria group decision making approach that makes use of quality function deployment (QFD), fusion of fuzzy information and 2-tuple linguistic representation model is developed for supplier selection. The proposed methodology seeks to establish the relevant supplier assessment criteria while also considering the impacts of inner dependence among them. Two interrelated house of quality matrices are constructed, and fusion of fuzzy information and 2-tuple linguistic representation model are employed to compute the weights of supplier selection criteria and subsequently the ratings of suppliers. The proposed method is apt to manage non-homogeneous information in a decision setting with multiple information sources. The decision framework presented in this paper employs ordered weighted averaging (OWA) operator, and the aggregation process is based on combining information by means of fuzzy sets on a basic linguistic term set. The proposed framework is illustrated through a case study conducted in a private hospital in Istanbul.     

Type‐1 OWA Unbalanced Fuzzy Linguistic Aggregation Methodology: Application to Eurobonds Credit Risk Evaluation

In decision making, a widely used methodology to manage unbalanced fuzzy linguistic information is the linguistic hierarchy (LH), which relies on a linguistic symbolic computational model based on ordinal 2‐tuple linguistic representation. However, the ordinal 2‐tuple linguistic approach does not exploit all advantages of Zadeh's fuzzy linguistic approach to model uncertainty because the membership function shapes are ignored. Furthermore, the LH methodology is an indirect approach that relies on the uniform distribution of symmetric linguistic assessments. These drawbacks are overcome by applying a fuzzy methodology based on the implementation of the type‐1 ordered weighted average (T1OWA) operator. The T1OWA operator is not a symbolic operator and it allows to directly aggregate membership functions, which in practice means that the T1OWA methodology is suitable for both balanced and unbalanced linguistic contexts and with heterogeneous membership functions. Furthermore, the final output of the T1OWA methodology is always fuzzy and defined in the same domain of the original unbalanced fuzzy linguistic labels, which facilitates its interpretation via a visual joint representation. A case study is presented where the T1OWA operator methodology is used to assess the creditworthiness of European bonds based on real credit risk ratings of individual Eurozone member states modeled as unbalanced fuzzy linguistic labels.     

A method based on preference degrees for handling hybrid multiple attribute decision making problems

In this paper, we investigate hybrid multiple attribute decision making problems with various forms of attribute values (real numbers, linguistic labels, interval numbers, intuitionistic fuzzy numbers and interval intuitionistic fuzzy numbers). We propose a method based on preference degrees which may take the forms of fuzzy numbers, intuitionistic fuzzy numbers and interval intuitionistic fuzzy numbers. The method first normalizes various forms of attribute values into preference degrees, and then uses a preference degree-based weighted averaging operator to aggregate the normalized preference degrees. Meanwhile, for convenience of calculation, a new linguistic representation model is presented, whose feasibility is verified by comparing it with the traditional 4-tuple linguistic representation model, and from our model, the mapping relationship between interval intuitionistic fuzzy numbers and linguistic labels can be constructed. Finally, we illustrate the rationality and practicality of the proposed method by an application example.     

A communication model based on the 2-tuple fuzzy linguistic representation for a distributed intelligent agent system on Internet

 Internet users are assisted by means of distributed intelligent agents in the information gathering process to find the fittest information to their needs. In this paper we present a distributed intelligent agent model where the communication of the evaluation of the retrieved information among the agents is carried out by using linguistic operators based on the 2-tuple fuzzy linguistic representation as a way to endow the retrieval process with a higher flexibility, uniformity and precision. The 2-tuple fuzzy linguistic representation model allows to make processes of computing with words without loss of information.     

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.     

A model based on linguistic 2-tuples for dealing with multigranularhierarchical linguistic contexts in multi-expert decision-making

In those problems that deal with multiple sources of linguistic information we can find problems defined in contexts where the linguistic assessments are assessed in linguistic term sets with different granularity of uncertainty and/or semantics (multigranular linguistic contexts). Different approaches have been developed to manage this type of contexts, that unify the multigranular linguistic information in an unique linguistic term set for an easy management of the information. This normalization process can produce a loss of information and hence a lack of precision in the final results. In this paper, we shall present a type of multigranular linguistic contexts we shall call linguistic hierarchies term sets, such that, when we deal with multigranular linguistic information assessed in these structures we can unify the information assessed in them without loss of information. To do so, we shall use the 2-tuple linguistic representation model. Afterwards we shall develop a linguistic decision model dealing with multigranular linguistic contexts and apply it to a multi-expert decision-making problem.     

Linguistic majorities with difference in support

In social choice voting, majorities based on difference of votes and their extension, majorities based on difference in support, implement the crisp preference values (votes) and the intensities of preference provided by voters when comparing pairs of alternatives, respectively. The aim of these rules is declaring which alternative is socially preferred and to that, they require the winner alternative to reach a certain positive difference in its social valuation with respect to the one reached by the loser alternative. This paper introduces a new aggregation rule that extends majorities based on difference of votes from the context of crisp preferences to the framework of linguistic preferences. Under linguistic majorities with difference in support, voters express their intensities of preference between pairs of alternatives using linguistic labels and an alternative defeats another one when a specific support, fixed before the election process, is reached. There exist two main representation methodologies of linguistic preferences: the cardinal one based on the use of fuzzy sets, and the ordinal one based on the use of the 2-tuples. Linguistic majorities with difference in support are formalised in both representation settings, and conditions are given to guarantee that fuzzy linguistic majorities and 2-tuple linguistic majorities are mathematically isomorphic. Finally, linguistic majorities based on difference in support are proved to verify relevant normative properties: anonymity, neutrality, monotonicity, weak Pareto and cancellativeness.     

A Proposal for the Genetic Lateral Tuning of Linguistic Fuzzy Systems and Its Interaction With Rule Selection

Linguistic fuzzy modeling allows us to deal with the modeling of systems by building a linguistic model which is clearly interpretable by human beings. However, since the accuracy and the interpretability of the obtained model are contradictory properties, the necessity of improving the accuracy of the linguistic model arises when complex systems are modeled. To solve this problem, one of the research lines in recent years has led to the objective of giving more accuracy to linguistic fuzzy modeling without losing the interpretability to a high level. In this paper, a new postprocessing approach is proposed to perform an evolutionary lateral tuning of membership functions, with the main aim of obtaining linguistic models with higher levels of accuracy while maintaining good interpretability. To do so, we consider a new rule representation scheme base on the linguistic 2-tuples representation model which allows the lateral variation of the involved labels. Furthermore, the cooperation of the lateral tuning together with fuzzy rule reduction mechanisms is studied in this paper, presenting results on different real applications. The obtained results show the good performance of the proposed approach in high-dimensional problems and its ability to cooperate with methods to remove unnecessary rules.     

Personalized individual semantics in computing with words for supporting linguistic group decision making. An application on consensus reaching

In group decision making (GDM) dealing with Computing with Words (CW) has been highlighted the importance of the statement, words mean different things for different people, because of its influence in the final decision. Different proposals that either grouping such different meanings (uncertainty) to provide one representation for all people or use multi-granular linguistic term sets with the semantics of each granularity, have been developed and applied in the specialized literature. Despite these models are quite useful they do not model individually yet the different meanings of each person when he/she elicits linguistic information. Hence, in this paper a personalized individual semantics (PIS) model is proposed to personalize individual semantics by means of an interval numerical scale and the 2-tuple linguistic model. Specifically, a consistency-driven optimization-based model to obtain and represent the PIS is introduced. A new CW framework based on the 2-tuple linguistic model is then defined, such a CW framework allows us to deal with PIS to facilitate CW keeping the idea that words mean different things to different people. In order to justify the feasibility and validity of the PIS model, it is applied to solve linguistic GDM problems with a consensus reaching process.     

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.     

Measuring consistency of linguistic preference relations: a 2-tuple linguistic approach

By using the 2-tuple fuzzy linguistic representation model and by proposing the concept of the individual consistency evaluations, this paper develops a 2-tuple linguistic index to measure the consistency degree of linguistic preference relations. Comparing with the existing numerical consistency indexes, the proposed linguistic index not only measures the consistency degree of linguistic preference relations via a linguistic way, but also can reflect the individual difference on consistency degree of linguistic preference relations. Further, this paper proposes an algorithm to improving consistency degree in linguistic preference relations, and the proof of the convergence of this algorithm is also given. The results in this paper provide a theoretic basis for the application of consistent linguistic preference relations in decision making problems.     

An interval-valued 2-tuple linguistic group decision-making model based on the Choquet integral operator

The Choquet integral (IL) operator is an effective approach for handling interdependence among decision attributes in complex decision-making problems. However, the fuzzy measures of attributes and attribute sets required by IL are difficult to achieve directly, which limits the application of IL. This paper proposes a new method for determining fuzzy measures of attributes by extending Marichal's concept of entropy for fuzzy measure. To well represent the assessment information, interval-valued 2-tuple linguistic context is utilised to represent information. Then, we propose a Choquet integral operator in an interval-valued 2-tuple linguistic environment, which can effectively handle the correlation between attributes. In addition, we apply these methods to solve multi-attribute group decision-making problems. The feasibility and validity of the proposed operator is demonstrated by comparisons with other models in illustrative example part.