An interactive procedure for linguistic multiple attribute decision making with incomplete weight information

In this paper, we consider the multiple attribute decision making (MADM) problems, in which the information about attribute weights is partly known and the attribute values are expressed in linguistic labels. We first define the concepts of linguistic positive ideal point, linguistic negative ideal point, and satisfactory degree of alternative. Based on these concepts, we then establish some linear programming models, through which the decision maker interacts with the analyst. Furthermore, we establish a practical interactive procedure for solving the MADM problems considered in this paper. The interactive process can be realized by giving and revising the satisfactory degrees of alternatives till an optimum satisfactory solution is achieved. Finally, a practical example is given to illustrate the developed procedure.     

A construction of sound semantic linguistic scales using 4-tuple representation of term semantics

Data semantics plays a fundamental role in computer science, in general, and in computing with words, in particular. The semantics of words arises as a sophisticated problem, since words being actually vague linguistic terms are pieces of information characterized by impreciseness, incompleteness, uncertainty and/or vagueness. The qualitative semantics and the quantitative semantics are two aspects of vague linguistic information, which are closely related. However, the qualitative semantics of linguistic terms, and even the qualitative semantics of the symbolic approaches, seem to be not elaborated on directly in the literature. In this study, we propose an interpretation of the inherent order-based semantics of terms through their qualitative semantics modeled by hedge algebra structures. The quantitative semantics of terms are developed based on the quantification of hedge algebras. With this explicit approach, we propose two concepts of assessment scales to address decision problems: linguistic scales used for representing expert linguistic assessments and semantic linguistic scales based on 4-tuple linguistic representation model, which forms a formalized structure useful for computing with words. An example of a simple multi-criteria decision problem is examined by running a comparative study. We also analyze the main advantages of the proposed approach.     

Some mathematical tools for linguistic probabilities

We propose and develop, in this paper, some concepts and techniques useful for the theory of linguistic probabilisies introduced by L.A. Zadeh. These probabilities are expressed in linguistic rather than numerical terms. The mathematical framework for this study is based upon the possibility theory.We formulate first the problem of optimization under elastic constraints which is not only important for mathematical programming but will be served to justify the extension of possibility measure to linguistic variables. Next, in connection with translation rules in natural languages we study some transformations of fuzzy sets using a relation between random sets and fuzzy sets. Finally, we point out some differences between random variables and fuzzy variables, and present the mathematical notion of possibility, in the setting of set-functions, as a special case of Choquet capacities.     

An architecture of fuzzy neural networks for linguistic processing

Fuzzy sets have been used successfully in order to deal with imprecise data, linguistic terms or not well-defined concepts. Recently, considerable effort has been made in the direction of combining the neural network approach with fuzzy sets. In this paper a fuzzy feed-forward neural network, able to process trapezoidal fuzzy sets, has been investigated. Normalized trapezoidal fuzzy sets have been considered. The fuzzy generalized delta rule with different back-propagation algorithms is discussed. The more interesting and characteristic property of the proposed architecture is the ability of each node to process fuzzy sets or linguistic terms, preserving the simplicity of the back-propagation algorithm. Consequently, the resulting architecture is able to cope with problems in which the input parameters and the desired targets are described by linguistic terms. This methodology has the further interesting characteristic of being able to operate at the linguistic level rather than at the numerical level, that is it can work at a higher data abstraction level. An example in computerized electrocardiography will be illustrated in order to test the proposed approach.     

Multidimensional interval-valued fuzzy reasoning approach based on weighted similarity measure

This paper focuses on presentation of a method to bidirectional interval-valued fuzzy approximate reasoning by employing a weighted similarity measure between the fact and the antecedent (or consequent) portion of production rule in which the vague terms are represented by interval-valued fuzzy concepts rather than plain fuzzy sets. The proposed method is more reasonable and flexible than the one presented in the paper by Chen [Fuzzy Sets and Systems, 91(1997), 339–353] due to the fact that it not only can deal with multidimensional interval-valued fuzzy reasoning scheme, but also consider the different importance degree of linguistic variables in production rule and that of elements in each universe.     

A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts

Group decision making is a type of decision problem in which multiple experts acting collectively, analyze problems, evaluate alternatives, and select a solution from a collection of alternatives. As the natural language is the standard representation of those concepts that humans use for communication, it seems natural that they use words (linguistic terms) instead of numerical values to provide their opinions. However, while linguistic information is readily available, it is not operational and thus it has to be made usable though expressing it in terms of information granules. To do so, Granular Computing, which has emerged as a unified and coherent framework of designing, processing, and interpretation of information granules, can be used. The aim of this paper is to present an information granulation of the linguistic information used in group decision making problems defined in heterogeneous contexts, i.e., where the experts have associated importance degrees reflecting their ability to handle the problem. The granulation of the linguistic terms is formulated as an optimization problem, solved by using the particle swarm optimization, in which a performance index is maximized by a suitable mapping of the linguistic terms on information granules formalized as sets. This performance index is expressed as a weighted aggregation of the individual consistency achieved by each expert.     

Communication and understanding mathematical foundations and practical applications

Since Shannon??s and Weaver??s ??Mathematical theory of Communication?? it is well known that mathematical definitions of information or the degree of information respectively is possible. The great problem for a complete theory of communication is the exact definition of meaning in mathematical terms. I shall demonstrate how such a definition can be achieved in terms of complex systems theory. In particular it is possible to derive exact definitions of the degree of meaning, applied to semantical networks, and of the degree of information that is suited for the analysis of human communication. The degree of information as well as the degree of meaning is dependent on the geometry of the receiving systems, which are modeled as semantical networks. It can be shown that the knowledge about the geometrical structure allows predictions about the degrees of information and of meaning a message has for a certain receiving system, although of course only on the average. In a semantical application of these fundamental concepts it will be shown how a new self organized leaning neural network that we have developed is able to deal with linguistic ambiguities. Additional applications of the communication theory will be demonstrated, in particular an Internet meta search engine based on computing the degrees of information and meaning, and a computer based discourse analysis.     

A consensus-based probabilistic linguistic gained and lost dominance score method

This paper proposes a comprehensive Multiple Criteria Group Decision Making (MCGDM) method with probabilistic linguistic information based on a new consensus measure and a novel outranking method, Gained and Lost Dominance Score (GLDS). Firstly, new operations of the probabilistic linguistic term sets are introduced based on the adjusted rules of probabilistic linguistic term sets and the linguistic scale functions for semantics of linguistic terms. After defining a new consensus measure based on the correlation degree between probabilistic linguistic term sets, we develop a consensus reaching method to improve the consensus degree of a group. To rank alternatives reasonably, we further propose the GLDS method which considers both the “group utility” and the “individual regret” values. The core of the GLDS is to calculate the gained and lost dominance scores that the optimal solution dominates all other alternatives in terms of the net gained dominance flow and the net lost dominance flow. Then, we integrate the GLDS ranking method with the consensus reaching process and develop a consensus-based PL-GLDS method to solve the MCGDM problems with probabilistic linguistic information. Finally, the proposed method is validated by a case study of selecting optimal green enterprises. Some comparative analyses are given to show the efficiency of the proposed method.     

Language-related learning of mathematics: a comparison of kindergarten and primary school as places of learning

The study described in this paper presents research results from two separate projects, as well as two theoretical pillars that form the foundation of these projects. The first of these pillars focuses on the interactional relationship between mathematical and linguistic learning; the second defines an understanding of learning as a process that unfolds through institutions, whereby the most important element in the education of a child is the child him/herself rather than the specific place of learning. That is, for the purposes of this paper, the same linguistic issues arise independently of whether the child is in primary school or kindergarten. The results of both projects are used in the reconstruction of linguistic conditions of the learning of mathematics. These analyses can lead us to conclude that improvement in children’s linguistic competences should be only a secondary goal; the real key to the more effective promotion of children’s education in terms of subject-related learning seems to lie in the improved education and further education of teaching staff. The results should thus form a theoretical basis for the development of unified support and training concepts for children and educators.     

A genetic design of linguistic terms for fuzzy rule based classifiers

The determination of fuzzy information granules including the estimation of their membership functions play a significant role in fuzzy system design as well as in the design of fuzzy rule based classifiers (FRBCSs). However, although linguistic terms are fundamental elements in the process of elucidating expert’s knowledge, the problem of linguistic term design along with their fuzzy-set-based semantics has not been fully addressed, since term-sets of attributes have not been interpreted as a formalized structure. Thus, the essential relationship between linguistic terms, as syntax, and the constructed fuzzy sets, as their quantitative semantics, or in other words, the problem of the natural semantics of terms behind the linguistic literal has not been addressed. In this paper, we introduce the problem of the design of optimal linguistic terms and propose a method of the design of FRBCSs which may incorporate with the design of linguistic terms to ensure that the presence of linguistic literals are supported not only by data but also by their natural semantics. It is shown that this problem plays a primordial role in enhancing the performance and the interpretability of the designed FRBCSs and helps striking a better balance between the generality and the specificity of the desired fuzzy rule bases for fuzzy classification problems. A series of experiments concerning 17 Machine Learning datasets is reported.     

Finite-valued indistinguishability operators

Fuzzy equality relations or indistinguishability operators generalize the concepts of crisp equality and equivalence relations in fuzzy systems where inaccuracy and uncertainty is dealt with. They generate fuzzy granularity and are an essential tool in Computing with Words (CWW). Traditionally, the degree of similarity between two objects is a number between 0 and 1, but in many occasions this assignment cannot be done in such a precise way and the use of indistinguishability operators valued on a finite set of linguistic labels such as small, very much, etc. would be advisable. Recent advances in the study of finite-valued t-norms allow us to combine this kind of linguistic labels and makes the development of a theory of finite-valued indistinguishability operators and their application to real problems possible.     

语言判断矩阵的相容性和一致性研究

语言判断矩阵是决策者常给的一种偏好信息形式.提出了新的语言判断矩阵的导出矩阵、相容性和一致性等概念,在此基础上给出了基于不同粒度的语言判断矩阵的相容性的充要条件,探讨语言判断矩阵的满意一致性以及完全一致性和其导出矩阵之间的关系.     

基于ELECTRE方法的模糊多属性群决策

结合模糊集理论知识,将经典的ELECTRE方法拓展到模糊环境下,用来解决模糊多属性群决策问题.决策问题中涉及到的属性评价值与属性权值是以三角模糊数的形式给出.首先集结各决策者的评价矩阵,构造模糊群决策矩阵,在规范化的基础上,分别对其进行模糊和谐性检验与模糊非和谐性检验,最后通过设定否决门槛值得到一致性比较矩阵,最终评价结果也是以三角模糊数的形式给出.该方法操作方便,计算简单.给出的实例验证了此方法的可行性和有效性.     

基于Vague集的群决策方法研究

首先介绍了Vague集的基本概念,将直觉模糊集的一些运算规则重新在Vague集上作了定义,提出用Vague值表示的九级语言术语集.接着指出衡量Vague集(值)相似度要考虑的三个因素,提出了新的度量方法.在同时考虑专家决策结果的一致性和专家权重的基础上,提出了汇总各专家Vague意见的方法.最后以一个案例说明了所提出的方法.     

语言值多属性决策问题研究

指出了目前已有的语言值综合决策模型的不足,建立了一个新的语言值综合决策模型.利用三角形模糊数和扩展原理以及格贴近度和择近原则,证明了所给模型的科学性,同时分析了它的几条良好性质.最后给出一个该模型的应用实例.     

基于犹豫模糊二元语义的多属性决策方法

深入研究了犹豫模糊二元语义多属性决策问题。首先利用幂均算子给出了犹豫模糊二元语义集的均值函数,并基于均匀分布概率准则和二元语义的距离测度提出了犹豫模糊二元语义集两两比较的可能度公式,进一步给出了可能度排序公式的性质。针对属性值为犹豫模糊二元语义集的多属性决策问题,提出了一种基于熵权的多属性决策方法。最后结合实际问题,验证了该方法的有效性和可行性。     

基于语言偏好信息的稳定双边匹配决策方法

针对语言偏好信息下的双边匹配问题,提出一种双边匹配决策方法。首先,将双边主体给出的语言偏好信息转化为三角模糊数;然后,基于去模糊化处理方法将三角模糊数转化为匹配满意度,在此基础上,考虑稳定匹配约束条件,以最大化每方主体的匹配满意度为目标,建立双边匹配多目标优化模型,求解模型,获得双边匹配结果;最后,通过一个算例验证了提出方法的可行性和有效性。     

基于二元语义多属性群决策的投影法

针对解决具有语言评价信息的多属性群决策问题,提出了一种基于二元语义信息处理的群决策方法。该方法采用近年来最新发展的二元语义概念对语言评价信息进行处理和运算,并依据传统投影分析方法的基本思想,通过计算备选方案对正理想方案和负理想方案的投影值,进而计算备选方案对正理想方案相对贴近度,最终确定最优方案。该方法具有对语言信息处理较为精确的特点,避免了以往采用的语言信息处理方法所带来的信息扭臼和损失。最后给出了实例分析。结果表明方法简单,有效和易于计算。     

A new method for multiple criteria group decision making with incomplete weight information under linguistic environment

A new method is proposed to solve multiple criteria group decision making (MCGDM) problems, in which both the criteria values and criteria weights take the form of linguistic information, and the information about linguistic criteria weights is partly known or completely unknown. Firstly, to get reasonable decision result, instead of assigning the same weight to the decision maker (DM) for all criteria, we propose a method to determine the weight of DM with respect to each criterion under linguistic environment by calculating the similarity degree between individual 2-tuple linguistic evaluation value and the mean given by all decision makers (DMs). Secondly, for the situations where the information about the criteria weights is partly known or completely unknown, we establish optimization models to determine the criteria weights by defining 2-tuple linguistic positive ideal solution (TL-PIS), 2-tuple linguistic right negative ideal solution (TL-RNIS) and 2-tuple linguistic left negative ideal solution (TL-LNIS) of the collective 2-tuple linguistic decision matrix. Thirdly, we propose a new method to solve MCGDM problems with partly known or completely unknown linguistic weight information. Finally, an illustrative example is given to demonstrate the calculation process of the proposed method.     

基于区间值直觉模糊不确定语言变量的灰色关联度分析方法

给出了区间值直觉模糊不确定语言环境下的灰色关联度分析方法。首先确定了区间值直觉模糊不确定语言正负理想解, 然后计算每个评价值与正负理想解的灰色关联度, 利用属性的权重向量, 计算方案与正负理想解的灰色关联度, 最后计算出方案的相对关联度, 并根据方案的相对关联度对方案进行排序。如果属性权重部分可知, 则需要根据与正理想解有最大的灰色关联度而与负理想解有最小的关联度的原则建立数学规划确定属性的权重。最后, 为了说明算法的可行性和有效性, 将其应用到房地产开发项目的风险评价上。 实例说明了算法的可行性和有效性。