Acceptable consistency of aggregated comparison matrices in analytic hierarchy process

The analytic hierarchy process is a method for solving multiple criteria decision problems, as well as group decision making. The weighted geometric mean method is appropriate when aggregation of individual judgements is used. This paper presents a new proof which confirms the property that if the comparison matrices of all decision makers are of acceptable consistency, then the weighted geometric mean complex judgement matrix (WGMCJM) also is of acceptable consistency. This property was presented and first proved by Xu (2000), but Lin et al. (2008) rejected the proof. We also discuss under what conditions the WGMCJM is of acceptable consistency when not all comparison matrices of decision makers are of acceptable consistency. For this case we determine the sufficient condition for the WGMCJM to be of acceptable consistency and provide numerical examples. For a special case of two decision makers with 3 × 3 comparison matrices we find out some additional conditions for the WGMCJM to be of acceptable consistency.     

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.     

一种群决策专家客观权重确定的改进方法

针对传统的群决策专家客观权重确定的两类主要方法(即基于判断矩阵一致性程度和基于专家排序向量或判断矩阵元素的聚类分析)的缺陷,提出了一种综合考虑两方面因素的改进方法.首先利用聚类分析方法根据各专家的排序向量得到专家类别间的权重,然后根据单个专家的判断矩阵一致性以及排序向量到类核心的距离确定最终权重.文末给出的算例表明该方法是可行、有效的.     

基于模糊C-OWG算子的模糊DEA模型求解

把Yager和Xu提出的连续区间数据OWG(C-OWG)算子进行拓展,定义了模糊C-OWG(FC-OWG)算子,并研究了它的一些性质.在保持数据一致性的前提下,基于FC-OWG算子,研究了模糊数据包络分析(FDEA)模型向确定型DEA模型的转化问题,并进行了求解.实例表明,新方法计算简单、可行且有效.     

Two sufficient conditions for non-normal Cayley graphs and their applications

A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay(G, S). In this paper, two sufficient conditions for non-normal Cayley graphs are given and by using the conditions, five infinite families of connected non-normal Cayley graphs are constructed. As an application, all connected non-normal Cayley graphs of valency 5 on A5 are determined, which generalizes a result about the normality of Cayley graphs of valency 3 or 4 on A5 determined by Xu and Xu. Further, we classify all non-CI Cayley graphs of valency 5 on A5, while Xu et al. have proved that As is a 4-CI group.     

A new method for similarity measures for pattern recognition

This paper points out three questionable areas in the realm of similarity measures and then provides a new method that will rectify the problem. The purpose of this paper is fourfold. First, we will propose a scenario where the three similarity measures proposed by Hung and Yang (2004) [1] are helpless in aiding a decision maker in deciding pattern recognition problem. Second, we will present our method for solving the dilemma. Third, we will show that our proposed similarity measures satisfy the axioms for well defined similarity measures. Fourth, we will prove that our method could solve pattern recognition problems. Our findings will help researchers handle similarity problems under intuitionistic fuzzy sets environment.     

Representations of nongraded Lie algebras of block type

From his classification of quadratic conformal algebras corresponding to certain Hamiltonian pairs in integrable systems, Xu found a family of simple Lie algebras related to pairs of locally-finite derivations on certain commutative associative algebras. In this paper, we construct a large family of irreducible modules with four parameters for Xu's two-devivation algebras via the corresponding algebras of Weyl type. When the derivations are graded operators, we obtain a large family of uniformly-bounded irreducible weight modules for the Block algebras.     

On dynamics of the system of two coupled nonlinear Schrödinger in

In this paper, we give a new proof of the scattering and blow‐up theory of the two coupled nonlinear Schrödinger system via establishing the corresponding interaction Morawetz estimate and scattering criterion. The method of this paper simplifies the proof in Xu, and the result of the paper improves the result in Xu.     

Adaptive robot under fuzzy control

This paper deals with the problem of ranking n fuzzy subsets of the unit interval. A number of methods suggested in the literature is reviewed and tested on a group of selected examples, where the fuzzy sets can be nonnormal and/or nonconvex.The ranking is obtained from: (i) the index of strict preference defined by Watson, (ii) three indexes proposed by Yager, (iii) the algorithm used by Chang, (iv) three versions of the a-preference index suggested by Adamo, (v) the index defined by Baas and Kwakernaak, (vi) three modified versions used by Baldwin and Guild, (vii) the method proposed by Kerre, (viii) three forms of the index suggested by Jain, (ix) the four grades of dominance studied by Dubois and Prade.In simple cases the results are good for all the methods, with some exceptions. In questionable cases, where the decision must be probably modelled in accordance with the context in which it is imbedded, the best indexes seem to be the dominances suggested by Dubois and Prade. These indexes do not force any particular choice, but clearly describe the situation, hence allowing the decision-maker himself to make his ‘best’ choice.     

An experiment on the consistency of aggregated comparison matrices in AHP

The analytic hierarchy process can be used for group decision making by aggregating individual judgments or individual priorities. The most commonly used aggregation methods are the geometric mean method and the weighted arithmetic mean method. While it is known that the weighted geometric mean comparison matrix is of acceptable consistency if all individual comparison matrices are of acceptable consistency, this paper addresses the following question: Under what conditions would an aggregated geometric mean comparison matrix be of acceptable consistency if some (or all) of the individual comparison matrices are not of acceptable consistency? Using Monte Carlo simulation, results indicate that given a sufficiently large group size, consistency of the aggregate comparison matrix is guaranteed, regardless of the consistency measures of the individual comparison matrices, if the geometric mean is used to aggregate. This result implies that consistency at the aggregate level is a non-issue in group decision making when group size exceeds a threshold value and the geometric mean is used to aggregate individual judgments. This paper determines threshold values for various dimensions of the aggregated comparison matrix.