An improved statistical approach for consistency test in AHP

The pair-wise comparison matrix (PCM) is widely used in multi-criteria decision making methods. If the PCM is inconsistent, the resulting priority vector is not reliable. Hence, it is necessary to measure the level of the inconsistency of the PCM. There are two approaches for testing the consistency of the PCM: deterministic approaches and statistical or stochastic approaches. In this paper, an improved statistical approach to test the consistency of the PCM is proposed, which combines hypothesis test and maximum likelihood estimation. The proposed statistical approach is flexible and reliable because it sets a suitable significance level according to different situations. Two numerical examples are introduced to illustrate the proposed statistical approach.     

基于基本回路修正的AHP一致性调整方法研究

为解决AHP一致性问题,提出一种基于基本回路修正的调整方法,能够同时解决数值不一致和逻辑不一致问题,同时保证对原始信息的修改量最小。数值不一致和逻辑不一致均由决策者的不准确判断引起,其中数值不一致可以通过降低一致性比率(CR)值进行改善,而逻辑不一致只有将判断矩阵中所有三阶回路去除才能得到解决。因此,通过对n阶判断矩阵进行基本矩阵分解,得到C³_n个3阶的基本矩阵,其中存在三阶回路的称为基本回路,从而将判断矩阵的一致性修正问题转化为基本回路的一致性修正问题。通过对基本回路的一致性比较,提出了两种确定最不一致元素的方法,即CR和最大法和优化法,并设计了优化模型对最不一致元素进行修正。最后,通过算例分析验证了本文方法的可行性,与已有方法的对比结论证明了本文方法更为有效。     

模糊数互补判断矩阵的乘性一致性检验及改进方法

针对模糊数互补判断矩阵的乘性一致性检验及改进问题进行研究。在文献[11]引入模糊数的Q-算子和模糊数互补判断矩阵的Q-算子矩阵概念的基础上,通过构造具有乘性一致性的特征矩阵及偏差矩阵,建立了衡量乘性一致性程度的指标值并用设定阈值的方法给出了满意乘性一致性的概念,对于不满足满意乘性一致性的情况提出了改进方法。最后通过一个算例说明了此方法的可行性。     

A cosine maximization method for the priority vector derivation in AHP

The derivation of a priority vector from a pair-wise comparison matrix (PCM) is an important issue in the Analytic Hierarchy Process (AHP). The existing methods for the priority vector derivation from PCM include eigenvector method (EV), weighted least squares method (WLS), additive normalization method (AN), logarithmic least squares method (LLS), etc. The derived priority vector should be as similar to each column vector of the PCM as possible if a pair-wise comparison matrix (PCM) is not perfectly consistent. Therefore, a cosine maximization method (CM) based on similarity measure is proposed, which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a PCM. An optimization model for the CM is proposed to derive the reliable priority vector. Using three numerical examples, the CM is compared with the other prioritization methods based on two performance evaluation criteria: Euclidean distance and minimum violation. The results show that the CM is flexible and efficient.     

Enhancing data consistency in decision matrix: Adapting Hadamard model to mitigate judgment contradiction

Cardinal and ordinal inconsistencies are important and popular research topics in the study of decision making with pair-wise comparison matrices (PCMs). Few of the currently-employed tactics are capable of simultaneously dealing with both cardinal and ordinal inconsistency issues in one model, and most are heavily dependent on the method chosen for weight (priorities) derivation or the obtained closest matrix by optimization method that may change many of the original values. In this paper, we propose a Hadamard product induced bias matrix model, which only requires the use of the data in the original matrix to identify and adjust the cardinally inconsistent element(s) in a PCM. Through graph theory and numerical examples, we show that the adapted Hadamard model is effective in identifying and eliminating the ordinal inconsistencies. Also, for the most inconsistent element identified in the matrix, we develop innovative methods to improve the consistency of a PCM. The proposed model is only dependent on the original matrix, is independent of the methods chosen to derive the priority vectors, and preserves most of the original information in matrix A since only the most inconsistent element(s) need(s) to be modified. Our method is much easier to implement than any of the existing models, and the values it recommends for replacement outperform those derived from the literature. It significantly enhances matrix consistency and improves the reliability of PCM decision making.     

An intelligent decomposition of pairwise comparison matrices for large-scale decisions

A Pairwise Comparison Matrix (PCM) has been used to compute for relative priorities of elements and are integral components in widely applied decision making tools: the Analytic Hierarchy Process (AHP) and its generalized form, the Analytic Network Process (ANP). However, PCMs suffer from several issues limiting their applications to large-scale decision problems. These limitations can be attributed to the curse of dimensionality, that is, a large number of pairwise comparisons need to be elicited from a decision maker. This issue results to inconsistent preferences due to the limited cognitive powers of decision makers. To address these limitations, this research proposes a PCM decomposition methodology that reduces the elicited pairwise comparisons. A binary integer program is proposed to intelligently decompose a PCM into several smaller subsets using interdependence scores among elements. Since the subsets are disjoint, the most independent pivot element is identified to connect all subsets to derive the global weights of the elements from the original PCM. As a result, the number of pairwise comparison is reduced and consistency is of the comparisons is improved. The proposed decomposition methodology is applied to both AHP and ANP to demonstrate its advantages.     

Numerical study of melting in an enclosure with discrete protruding heat sources

In order to explore the capability of a solid–liquid phase change material (PCM) for cooling electronic or heat storage applications, melting of a PCM in a vertical rectangular enclosure was studied. Three protruding generating heat sources are attached on one of the vertical walls of the enclosure, and generating heat at a constant and uniform volumetric rate. The horizontal walls are adiabatic. The power generated in heat sources is dissipated in PCM (n-eicosane with the melting temperature, T_m = 36 °C) that filled the rectangular enclosure. The advantage of using PCM is that it is able to absorb high amount of heat generated by heat sources due to its relatively high energy density. To investigate the thermal behaviour and thermal performance of the proposed system, a mathematical model based on the mass, momentum and energy conservation equations was developed. The governing equations are next discretised using a control volume approach in a staggered mesh and a pressure correction equation method is employed for the pressure–velocity coupling. The PCM energy equation is solved using the enthalpy method. The solid regions (wall and heat sources) are treated as fluid regions with infinite viscosity and the thermal coupling between solid and fluid regions is taken into account using the harmonic mean of the thermal conductivity method. The dimensionless independent parameters that govern the thermal behaviour of the system were next identified. After validating the proposed mathematical model against experimental data, a numerical investigation was next conducted in order to examine the thermal behaviour of the system by analyzing the flow structure and the heat transfer during the melting process, for a given values of governing parameters.     

Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm

We propose an algorithm for constrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed Bernstein branch and prune algorithm is based on the Bernstein polynomial approach. We introduce several new features in this proposed algorithm to make the algorithm more efficient. We first present the Bernstein box consistency and Bernstein hull consistency algorithms to prune the search regions. We then give Bernstein contraction algorithm to avoid the computation of Bernstein coefficients after the pruning operation. We also include a new Bernstein cut-off test based on the vertex property of the Bernstein coefficients. The performance of the proposed algorithm is numerically tested on 13 benchmark problems. The results of the tests show the proposed algorithm to be overall considerably superior to existing method in terms of the chosen performance metrics.     

Nonparametric Methods for Checking the Validity of Prior Order Information

A large number of statistical procedures have been proposed in the literature to explicitly utilize available information about the ordering of treatment effects at increasing treatment levels. These procedures are generally more efficient than those ignoring the order information. However, when the assumed order information is incorrect, order restricted procedures are inferior and, strictly speaking, invalid. Just as any statistical model needs to be validated by data, order information to be used in a statistical analysis should also be justified by data first. A common statistical format for checking the validity of order information is to test the null hypothesis of the ordering representing the order information. Parametric tests for ordered null hypotheses have been extensively studied in the literature. These tests are not suitable for data with nonnormal or unknown underlying distributions. The objective of this study is to develop a general distribution-free testing theory for ordered null hypotheses based on rank order statistics and score generating functions. Sufficient and necessary conditions for the consistency of the proposed general tests are rigorously established.     

Statistical Inference for Student Diffusion Process

We consider the problem of parameter estimation for an ergodic diffusion with the symmetric scaled Student invariant distribution, where the spectral representation of the transition density is given in terms of the finite number of polynomial eigenfunctions (Routh–Romanovski polynomials) and absolutely continuous spectrum of the negative infinitesimal generator of observed diffusion. We prove the consistency and asymptotic normality of the proposed estimators and, based on the Stein equation for Student diffusion, consider the statistical test for the Student distributional assumptions.     

Pairwise comparison matrices and the error-free property of the decision maker

Pairwise comparison is a popular assessment method either for deriving criteria-weights or for evaluating alternatives according to a given criterion. In real-world applications consistency of the comparisons rarely happens: intransitivity can occur. The aim of the paper is to discuss the relationship between the consistency of the decision maker—described with the error-free property—and the consistency of the pairwise comparison matrix (PCM). The concept of error-free matrix is used to demonstrate that consistency of the PCM is not a sufficient condition of the error-free property of the decision maker. Informed and uninformed decision makers are defined. In the first stage of an assessment method a consistent or near-consistent matrix should be achieved: detecting, measuring and improving consistency are part of any procedure with both types of decision makers. In the second stage additional information are needed to reveal the decision maker’s real preferences. Interactive questioning procedures are recommended to reach that goal.     

基于偏序集的模糊互补矩阵次序一致性检验与调整

针对模糊互补矩阵次序一致性检验和调整存在争议和繁杂问题,提出了模糊互补矩阵次序一致性检验与调整的偏序集表示方法。在定义了偏序集、模糊互补矩阵、截集矩阵等相关概念基础上,求出模糊互补矩阵B的0.5水平截集矩阵,证明了模糊互补判断矩阵次序一致性和0.5水平截集矩阵为偏序关系矩阵的等价性;模糊互补判断矩阵完全次序一致性的充要条件是任意截集均满足传递性;任意截集满足传递性和偏序关系矩阵的等价性。结果表明,利用0.5水平截集矩阵转换为矩阵来检验模糊互补矩阵的次序一致性;通过调整每个截集矩阵满足传递性并赋值,能够达到模糊互补矩阵完全次序一致性。最后,通过两个算例验证该检验和调整方法的合理性和可行性。     

模糊判断矩阵一致性的调整方法

给出了模糊判断矩阵一致性调整的新方法 .该方法是在模糊判断矩阵一致性定义及判定方法的基础上 ,通过构造和分析模糊判断矩阵的调和矩阵 ,进一步给出了将模糊判断矩阵改进为满意一致性矩阵的计算步骤 .最后给出了一个算例 .     

An action learning approach for assessing the consistency of pairwise comparison data

Pairwise comparison data are used in various contexts including the generation of weight vectors for multiple criteria decision making problems. If this data is not sufficiently consistent, then the resulting weight vector cannot be considered to be a reliable reflection of the evaluator’s opinion. Hence, it is necessary to measure its level of inconsistency. Different approaches have been proposed to measuring the level of inconsistency, but they are often based on ‘rules of thumb” and/or randomly generated matrices, and are not interpretable. In this paper we present an action learning approach for assessing the consistency of the input pairwise comparison data that offer interpretable consistency measures.     

Tests of fit for the Rayleigh distribution based on the empirical Laplace transform

In this paper a class of goodness-of-fit tests for the Rayleigh distribution is proposed. The tests are based on a weighted integral involving the empirical Laplace transform. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight function tends to infinity the test statistics approach limit values. In a particular case the resulting limit statistic is related to the first nonzero component of Neyman’s smooth test for this distribution. The new tests are compared with other omnibus tests for the Rayleigh distribution.     

A statistical model for global optimization by means of select and clone

A statistical model tor giobai optimization is constructed generalizing some properties ofthe Wiener process to the multidimensional case. A new approach, which is similar to the Branch and Bound approach, is proposed to the construction of algorithms based on statistical models. A two dimensional version of the algorithm is implemented, and test results are presented     

A Bayesian finite element model updating with combined normal and lognormal probability distributions using modal measurements

The present work is associated with Bayesian finite element (FE) model updating using modal measurements based on maximizing the posterior probability instead of any sampling based approach. Such Bayesian updating framework usually employs normal distribution in updating of parameters, although normal distribution has usual statistical issues while using non-negative parameters. These issues are proposed to be dealt with incorporating lognormal distribution for non-negative parameters. Detailed formulations are carried out for model updating, uncertainty-estimation and probabilistic detection of changes/damages of structural parameters using combined normal-lognormal probability distribution in this Bayesian framework. Normal and lognormal distributions are considered for eigen-system equation and structural (mass and stiffness) parameters respectively, while these two distributions are jointly considered for likelihood function. Important advantages in FE model updating (e.g. utilization of incomplete measured modal data, non-requirement of mode-matching) are also retained in this combined normal-lognormal distribution based proposed FE model updating approach. For demonstrating the efficiency of this proposed approach, a two dimensional truss structure is considered with multiple damage cases. Satisfactory performances are observed in model updating and subsequent probabilistic estimations, however level of performances are found to be weakened with increasing levels in damage scenario (as usual). Moreover, performances of this proposed FE model updating approach are compared with the typical normal distribution based updating approach for those damage cases demonstrating quite similar level of performances. The proposed approach also demonstrates better computational efficiency (achieving higher accuracy in lesser computation time) in comparison with two prominent Markov Chain Monte Carlo (MCMC) techniques (viz. Metropolis-Hastings algorithm and Gibbs sampling).     

A heuristic approach for deriving the priority vector in AHP

This paper proposes an approach for deriving the priority vector from an inconsistent pair-wise comparison matrix through the nearest consistent matrix and experts judgments, which enables balancing the consistency and experts judgments. The developed algorithm for achieving a nearest consistent matrix is based on a logarithmic transformation of the pair-wise comparison matrix, and follows an iterative feedback process that identifies an acceptable level of consistency while complying with experts preferences. Three numerical examples are examined to illustrate applications and advantages of the developed approach.     

Copula函数的加权平均距离检验方法

针对无法从定性的角度选择最优Copula函数,本文从定量的角度给出Copula函数的一种加权平均距离检验方法,并给出该检验方法的拒绝域与检验p值,最后证明了此检验方法具有相合性,并给出检验统计量的渐近分布,从而说明此方法可以用于最优Copula函数的选择。