On consistency measures of linguistic preference relations

Inspired by the concept of deviation measure between two linguistic preference relations, this paper further defines the deviation measure of a linguistic preference relation to the set of consistent linguistic preference relations. Based on this, we present a consistency index of linguistic preference relations and develop a consistency measure method for linguistic preference relations. This method is performed to ensure that the decision maker is being neither random nor illogical in his or her pairwise comparisons using the linguistic label set. Using this consistency measure, we discuss how to deal with inconsistency in linguistic preference relations, and also investigate the consistency properties of collective linguistic preference relations. These results are of vital importance for group decision making with linguistic preference relations.     

On the non-equivalence of weak and strict preference

It is often claimed that the relations of weak preference and strict preference are symmetrical to each other in the sense that weak preference is complete and transitive if and only if strict preference is asymmetric and negatively transitive. The equivalence proof relies on a definitional connection between them, however, that already implies completeness of weak preference. Weakening the connection in order to avoid this leads to a breakdown of the symmetry which gives reason to accept weak preference as the more fundamental relation.     

On the Ferrers property of valued interval orders

We study the relationship between the Ferrers property and the notion of interval order in the context of valued relations. Given a crisp preference structure without incomparability, the strict preference relation satisfies the Ferrers property if and only if the associated weak preference relation does. These conditions characterize a total interval order. For valued relations the Ferrers property can be written in two different and non-equivalent ways. In this work, we compare these properties by finding the kind of completeness they imply. Moreover, we study whether they still characterize a valued total interval orders.     

Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group

Szmidt and Kacprzyk (Lecture Notes in Artificial Intelligence 3070:388–393, 2004a) introduced a similarity measure, which takes into account not only a pure distance between intuitionistic fuzzy sets but also examines if the compared values are more similar or more dissimilar to each other. By analyzing this similarity measure, we find it somewhat inconvenient in some cases, and thus we develop a new similarity measure between intuitionistic fuzzy sets. Then we apply the developed similarity measure for consensus analysis in group decision making based on intuitionistic fuzzy preference relations, and finally further extend it to the interval-valued intuitionistic fuzzy set theory.     

An integrated model-based interactive approach to FMAGDM with incomplete preference information

Incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations are very useful to express decision makers’ incomplete preferences over attributes or alternatives in the process of decision making under fuzzy environments. The aim of this paper is to investigate fuzzy multiple attribute group decision making problems where the attribute values are represented in intuitionistic fuzzy numbers and the information on attribute weights is provided by decision makers by means of one or some of the different preference structures, including weak ranking, strict ranking, difference ranking, multiple ranking, interval numbers, incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations. We transform all individual intuitionistic fuzzy decision matrices into the interval decision matrices and construct their expected decision matrices, and then aggregate all these expected decision matrices into a collective one. We establish an integrated model by unifying the collective decision matrix and all the given different structures of incomplete weight preference information, and develop an integrated model-based approach to interacting with the decision makers so as to adjust all the inconsistent incomplete fuzzy preference relations, inconsistent incomplete linguistic preference relations and inconsistent incomplete multiplicative preference relations into the ones with acceptable consistency. The developed approach can derive the attribute weights and the ranking of the alternatives directly from the integrated model, and thus it has the following prominent characteristics: (1) it does not need to construct the complete fuzzy preference relations, complete linguistic preference relations and complete multiplicative preference relations from the incomplete fuzzy preference relations, incomplete linguistic preference relations and incomplete multiplicative preference relations, respectively; (2) it does not need to unify the different structures of incomplete preferences, and thus can simplify the calculation and avoid distorting the given preference information; and (3) it can sufficiently reflect and adjust the subjective desirability of decision makers in the process of interaction. A practical example is also provided to illustrate the developed approach.     

An approach to group decision making based on interval multiplicative and fuzzy preference relations by using projection

This paper presents a consensus model for group decision making with interval multiplicative and fuzzy preference relations based on two consensus criteria: (1) a consensus measure which indicates the agreement between experts’ preference relations and (2) a measure of proximity to find out how far the individual opinions are from the group opinion. These measures are calculated by using the relative projections of individual preference relations on the collective one, which are obtained by extending the relative projection of vectors. First, the weights of experts are determined by the relative projections of individual preference relations on the initial collective one. Then using the weights of experts, all individual preference relations are aggregated into a collective one. The consensus and proximity measures are calculated by using the relative projections of experts’ preference relations respectively. The consensus measure is used to guide the consensus process until the collective solution is achieved. The proximity measure is used to guide the discussion phase of consensus reaching process. In such a way, an iterative algorithm is designed to guide the experts in the consensus reaching process. Finally the expected value preference relations are defined to transform the interval collective preference relation to a crisp one and the weights of alternatives are obtained from the expected value preference relations. Two numerical examples are given to illustrate the models and approaches.     

Continuity, completeness and the definition of weak preferences

This note explores the connections between continuity and completeness under alternative conceptions of preference relations. For non-trivial preorders, it shows that, unlike the standard definitions, the weak preference relation defined in Galaabaatar and Karni (2010) allows for incomplete preferences while maintaining all the continuity properties of complete preference relations. It also makes it possible to distinguish indifference between alternatives from non-comparability of alternatives. If the preference relations are complete, this definition agrees with the customary definitions.     

A Direct Approach to Group Decision Making with Uncertain Additive Linguistic Preference Relations

In this paper, we study the group decision-making problem in which the preference information given by experts takes the form of uncertain additive linguistic preference relations. We define the concept of uncertain additive linguistic preference relation, and introduce a formula based on possibility measure for comparing two uncertain linguistic preference values. We introduce some aggregation operators such as the uncertain linguistic averaging (ULA) operator and uncertain linguistic weighted averaging (ULWA) operator, etc. Based on the ULA and ULWA operators, we develop a direct approach to group decision making with uncertain additive linguistic preference relations without loss of information. Finally, an illustrative numerical example is given to verify the developed approach.     

关于模糊严格偏好关系的传递性

给定一个模糊关系,Ovchinnikov和Roubens引进了非常一般的模糊严格偏好关系定义,本文将详细讨论该偏好关系的弱传递性,一致性,强传递性以及非循环性等传递性有关的性质。     

A consistency and consensus-based method to group decision making with interval linguistic preference relations

Preference relations are a powerful tool to address decision-making problems. In some situations, because of the complexity of decision-making problems and the inherent uncertainty, the decision makers cannot express their preferences by using numerical values. Interval linguistic preference relations, which are more reliable and informative for the decision-makers’ preferences, are a good choice to cope with this issue. Just as with the other types of preference relations, the consistency and consensus analysis is very importance to ensure the reasonable ranking order by using interval linguistic preference relations. Considering this situation, this paper introduces a consistency concept for interval linguistic preference relations. To measure the consistency of interval linguistic preference relations, a consistency measure is defined. Then, a consistency-based programming model is built, by which the consistent linguistic preference relations with respect to each object can be obtained. To cope with the inconsistency case, two models for deriving the adjusted consistent linguistic preference relations are constructed. Then, a consistency-based programming model to estimate the missing values is built. After that, we present a group consensus index and present some of its desirable properties. Furthermore, a group consensus-based model to determine the weights of the decision makers with respect to each object is established. Finally, an approach to group decision making with interval linguistic preference relations is developed, which is based on the consistency and consensus analysis. Meanwhile, the associated numerical examples are offered to illustrate the application of the procedure.     

Aggregation and consensus for preference relations based on fuzzy partial orders

We propose a framework for eliciting and aggregating pairwise preference relations based on the assumption of an underlying fuzzy partial order. We also propose some linear programming optimization methods for ensuring consistency either as part of the aggregation phase or as a pre- or post-processing task. We contend that this framework of pairwise-preference relations, based on the Kemeny distance, can be less sensitive to extreme or biased opinions and is also less complex to elicit from experts. We provide some examples and outline their relevant properties and associated concepts.     

NP-hardness results for the aggregation of linear orders into median orders

Given a collection Π of individual preferences defined on a same finite set of candidates, we consider the problem of aggregating them into a collective preference minimizing the number of disagreements with respect to Π and verifying some structural properties. We study the complexity of this problem when the individual preferences belong to any set containing linear orders and when the collective preference must verify different properties, for instance transitivity. We show that the considered aggregation problems are NP-hard for different types of collective preferences (including linear orders, acyclic relations, complete preorders, interval orders, semiorders, quasi-orders or weak orders), if the number of individual preferences is sufficiently large.     

Derivation of intuitionistic fuzzy weights based on intuitionistic fuzzy preference relations

This paper proposes linear goal programming models for deriving intuitionistic fuzzy weights from intuitionistic fuzzy preference relations. Novel definitions are put forward to define additive consistency and weak transitivity for intuitionistic fuzzy preference relations, followed by a study of their corresponding properties. For any given normalized intuitionistic fuzzy weight vector, a transformation formula is furnished to convert the weights into a consistent intuitionistic fuzzy preference relation. For any intuitionistic fuzzy preference relation, a linear goal programming model is developed to obtain its intuitionistic fuzzy weights by minimizing its deviation from the converted consistent intuitionistic fuzzy preference relation. This approach is then extended to group decision-making situations. Three numerical examples are provided to illustrate the validity and applicability of the proposed models.     

Fuzzy preorders: conditional extensions,extensions and their representations

The crisp literature provides characterizations of the preorders that admit a total preorder extension when some pairwise order comparisons are imposed on the extended relation. It is also known that every preorder is the intersection of a collection of total preorders. In this contribution we generalize both approaches to the fuzzy case. We appeal to a construction for deriving the strict preference and the indifference relations from a weak preference relation, that allows to obtain full characterizations in the conditional extension problem. This improves the performance of the construction via generators studied earlier.     

Fuzzy ordered distance measures

Xu and Chen (J Syst Sci Syst Eng 17:432–445, 2008) introduced a type of ordered weighted distance measures called ordered weighted distance (OWD) measures whose fundamental aspect is the reordering step, which can be used in many actual fields, including group decision making, medical diagnosis, data mining, and pattern recognition, etc. The OWD measures are very suitable to deal with situations where the input data are expressed in exact numerical values. In this paper, we consider situations with linguistic, interval or fuzzy preference information, and develop some fuzzy ordered distance measures, such as linguistic ordered weighted distance measure, uncertain ordered weighted distance measure, linguistic hybrid weighted distance measure, and uncertain hybrid weighted distance measure, etc. After that, based on hybrid weighted distance measures, we establish a consensus reaching process of group decision making with linguistic, interval, triangular or trapezoidal fuzzy preference information.     

Ordinal regression revisited: Multiple criteria ranking using a set of additive value functions

We present a new method, called UTA^GMS, for multiple criteria ranking of alternatives from set A using a set of additive value functions which result from an ordinal regression. The preference information provided by the decision maker is a set of pairwise comparisons on a subset of alternatives A^R ⊆ A, called reference alternatives. The preference model built via ordinal regression is the set of all additive value functions compatible with the preference information. Using this model, one can define two relations in the set A: the necessary weak preference relation which holds for any two alternatives a, b from set A if and only if for all compatible value functions a is preferred to b, and the possible weak preference relation which holds for this pair if and only if for at least one compatible value function a is preferred to b. These relations establish a necessary and a possible ranking of alternatives from A, being, respectively, a partial preorder and a strongly complete relation. The UTA^GMS method is intended to be used interactively, with an increasing subset A^R and a progressive statement of pairwise comparisons. When no preference information is provided, the necessary weak preference relation is a weak dominance relation, and the possible weak preference relation is a complete relation. Every new pairwise comparison of reference alternatives, for which the dominance relation does not hold, is enriching the necessary relation and it is impoverishing the possible relation, so that they converge with the growth of the preference information. Distinguishing necessary and possible consequences of preference information on the complete set of actions, UTA^GMS answers questions of robustness analysis. Moreover, the method can support the decision maker when his/her preference statements cannot be represented in terms of an additive value function. The method is illustrated by an example solved using the UTA^GMS software. Some extensions of the method are also presented.     

直觉模糊集之间的距离测度

介绍了直觉模糊集之间距离测度公理化的定义,引入了一些新的距离测度并给出了这些距离测度性质的证明.讨论了相似测度与距离测度之间的关系.最后,指出直觉模糊集距离测度在模式识别中的应用.     

An Approximation for the Zakai Equation

In this article we consider a polygonal approximation to the unnormalized conditional measure of a filtering problem, which is the solution of the Zakai stochastic differential equation on measure space. An estimate of the convergence rate based on a distance which is equivalent to the weak convergence topology is derived. We also study the density of the unnormalized conditional measure, which is the solution of the Zakai stochastic partial differential equation. An estimate of the convergence rate is also given in this case. 60F25, 60H10.} Accepted 23 April 2001. Online publication 14 August 2001.     

不完全信息群体多属性决策的过滤函数法

对于不完全信息群体多属性决策问题，本文通过构造适当的过滤函数，充分利用高质量的信息和过滤掉低质量的信息，使效用信息的集结过程充分体现群体的意愿．在引进一致优、严格优和弱优等偏好指标的基础上，利用它们建立起方案集上合理的全序关系，从而构造了一个交互式的群体多属性决策方法．     

Aggregation of ordinal data using ordered weighted averaging operator weights

In multi-criteria decision-making problems, ordinal data themselves provide a convenient instrument for articulating preferences but they impose some difficulty on the aggregation process since ambiguity prevails in the preference structure inherent in the ordinal data. One of the key concerns in the aggregation of ordinal data is to differentiate among the rank positions by reflecting decision-maker??s preferences. Since individual attitude is fairly different, it is presumable that each ranking position has different importance. In other words, the quantification schemes among the rank positions could vary depending on the individual preference structure. We find that, among others, the ordered weighted averaging (OWA) operator can help to take this concept into effect on several reasons. First, the OWA operator provides a means to take into account a discriminating factor by introducing the measure of attitudinal character. Second, it can produce appropriate ranking weights corresponding to each rank position by solving a mathematical program subject to the constraint of attitudinal character. To better understand the attitudinal character playing a role as a discriminating factor, we develop centered ranking weights from ordinal weak relations among the ranking positions and then investigate their properties to relate them with the OWA operator weights having the maximum entropy. Finally, we present a method for generating the OWA operator weights via rank-based weighting functions.