A DEA based procedure for the selection of subgroups

Data Envelopment Analysis (DEA) is a nonparametric technique originally conceived for efficiency analysis of a set of units. The main characteristic of DEA based procedures is endogenous determination of weighting vectors, i.e., the weighting vectors are determined as variables of the model. Nevertheless, DEA’s applications have vastly exceeded its original target. In this paper, a DEA based model for the selection of a subgroup of alternatives or units is proposed. Considering a set of alternatives, the procedure seeks to determine the group that maximizes overall efficiency. The proposed model is characterized by free selection of weights and allows the inclusion of additional information, such as agent’s preferences in terms of relative importance of the variables under consideration or interactions between alternatives. The solution is achieved by computing a mixed-integer linear programming model. Finally, the proposed model is applied to plan the deployment of filling stations in the province of Seville (Spain).     

Optimizing the rank position of the DMU as secondary goal in DEA cross-evaluation

Cross-efficiency evaluation is an extension of Data Envelopment Analysis (DEA) that permits not only the determination of a ranking of Decision Making Units (DMUs) but also the elimination of unrealistic weighting schemes, thereby rescinding the necessity for the inclusion of individual judgements in the models. The main deficiency of the procedure is the non-uniqueness of the optimal weights, which results in the peer evaluations dependences, for instance, on the software used to determine DMU’s efficiencies. This shortfall justifies the inclusion of secondary goals in order to determine cross-efficiency values. In this paper a new proposal of a secondary goal is studied. The idea is related with that proposed in Wu et al. (2009), in which the objective is the optimization of the rank position of the DMU under evaluation. In the procedure proposed here, an incentive to break level-pegging ties between alternatives is introduced by considering that efficiency scores induce a weak order of alternatives. The model is illustrated with a preference-aggregation application.     

Generating ranking groups in the analytical hierarchy process

The paper introduces a convenient procedure of ranking N alternatives through direct comparisons in AHP. The alternatives are divided into groups in such a way that dominant relationship exists between the groups but not among the alternatives within each group. This method is suitable for situations where the strict ranking in a sequence for all alternatives is not reliable or not necessary. Two procedures are proposed to construct the AHP ranking groups. The proposed grouping procedures can be used in conjunction with the traditional approaches.     

一种组合赋权的两阶段优化算法

针对含有较多决策方案的组合赋权问题,提出了一种两阶段优化算法。第一阶段根据各个属性下的方案排序,采用单边t检验筛选方案,优化组合赋权的样本;第二阶段构建与主客观权重的偏差最小以及各方案之间的离差最大的组合赋权模型;最后运用算例说明方法的有效性。     

A fuzzy simple additive weighting system under group decision-making for facility location selection with objective/subjective attributes

This work presents a new fuzzy multiple attributes decision-making (FMADM) approach, i.e., fuzzy simple additive weighting system (FSAWS), for solving facility location selection problems by using objective/subjective attributes under group decision-making (GDM) conditions. The proposed system integrates fuzzy set theory (FST), the factor rating system (FRS) and simple additive weighting (SAW) to evaluate facility locations alternatives. The FSAWS is applied to deal with both qualitative and quantitative dimensions. The FSAWS process considers the importance of each decision-maker, and the total scores for alternative locations are then derived by homo/heterogeneous group of decision-makers. Finally, a numerical example illustrates the procedure of the proposed FSAWS.     

An efficient method for feature selection in linear regression based on an extended Akaike’s information criterion

A method for feature selection in linear regression based on an extension of Akaike’s information criterion is proposed. The use of classical Akaike’s information criterion (AIC) for feature selection assumes the exhaustive search through all the subsets of features, which has unreasonably high computational and time cost. A new information criterion is proposed that is a continuous extension of AIC. As a result, the feature selection problem is reduced to a smooth optimization problem. An efficient procedure for solving this problem is derived. Experiments show that the proposed method enables one to efficiently select features in linear regression. In the experiments, the proposed procedure is compared with the relevance vector machine, which is a feature selection method based on Bayesian approach. It is shown that both procedures yield similar results. The main distinction of the proposed method is that certain regularization coefficients are identical zeros. This makes it possible to avoid the underfitting effect, which is a characteristic feature of the relevance vector machine. A special case (the so-called nondiagonal regularization) is considered in which both methods are identical.     

Conditional-quantile screening for ultrahigh-dimensional survival data via martingale difference correlation

Using the so-called martingale difference correlation (MDC), we propose a novel censored-conditional-quantile screening approach for ultrahigh-dimensional survival data with heterogeneity (which is often present in such data). By incorporating a weighting scheme, this method is a natural extension of MDC-based conditional quantile screening, as considered in Shao and Zhang (2014), to handle ultrahigh-dimensional survival data. The proposed screening procedure has a sure-screening property under certain technical conditions and an excellent capability of detecting the nonlinear relationship between independent and censored dependent variables. Both simulation results and an analysis of real data demonstrate the effectiveness of the new censored conditional quantile-screening procedure.     

多目标半定规划的加权中心路径法

对于线性型多目标半定规划问题，引进加权中心路径的概念，并利用单目标半定规划的中心路径法，提出了求解多目标半定规划问题的加权中心路径法，先得型对一个叔向量的有效解，然后在此基础上，提出了通过一次迭代得到对应一定范围内其他任意权向量的有效解的一步修正方法．     

A simple test for the parametric form of the variance function in nonparametric regression

In this paper a new test for the parametric form of the variance function in the common nonparametric regression model is proposed which is applicable under very weak smoothness assumptions. The new test is based on an empirical process formed from pseudo residuals, for which weak convergence to a Gaussian process can be established. In the special case of testing for homoscedasticity the limiting process is essentially a Brownian bridge, such that critical values are easily available. The new procedure has three main advantages. First, in contrast to many other methods proposed in the literature, it does not depend directly on a smoothing parameter. Secondly, it can detect local alternatives converging to the null hypothesis at a rate n ^−1/2. Thirdly, in contrast to most of the currently available tests, it does not require strong smoothness assumptions regarding the regression and variance function. We also present a simulation study and compare the tests with the procedures that are currently available for this problem and require the same minimal assumptions.     

Multi-Objective Interactive Programming

This paper proposes a method for finding optimal, or near optimal, solutions for problems involving m objective functions, where there is an overall criterion which is a weighted sum of the m objective functions, but where the weights are, initially, unknown. The process is an interactive one, beginning with a set within which the actual weighting vector is known to lie, and progressively cutting down the size of the set until an acceptable solution is found. A by-product of the procedure is an iterative method for finding the generators of the polyhedral cones, within which the weighting vector must lie, at each stage.     

基于最大熵原理的线性组合赋权方法

在被评价对象的指标值与理想值之间的广义距离和充分小的情况下,追求不同赋权方法权重组合系数的信息分配最合理.随着广义距离和不断变小,得到一组不同方法赋权后的组合权重,进而得到了评价结果.本文的特色与创新一是本文得到的权重兼顾了信息分配最合理与指标数据距离理想值的广义距离和最小两个目标.二是提出一个单目标模型求解多目标问题Pareto解集的方法,并根据解集对评价对象进行排序, 增加了排序的可靠性,也为多目标模型求解提供了一种新思路.三是改变了组合赋权系数为近似平均的结果.四是解决了多目标线性加权求解时多个目标组合系数不确定问题.     

线性模型中M检验原假设分布的随机加权逼近

在线性模型中M-方法可以用于线性假设检验, 其中M检验、Wald检验和Rao的计分型检验是最常用的检验准则. 但是在计算这些检验的临界值时都涉及到未知参数的估计. 在本文中我们利用随机加权的方法来逼近这些检验的原假设分布. 结果表明在原假设和局部对立假设之下随机加权统计量的渐近分布与原检验统计量在原假设之下的渐近分布相同. 因此我们不需要对冗余参数进行估计,利用随机加权的方法就可以得到这些检验的临界值. 而且在局部对立假设之下可以实现对功效的计算. 当取不同的误差分布和不同的随机权时, 我们对本文的方法进行了蒙特卡洛模拟. 结果表明用随机加权方法来逼近原假设分布是非常精确的.     

A distance-metric methodology for the derivation of weights from a pairwise comparison matrix

A variety of approaches exist for the determination of a weighting scheme from a pairwise comparison matrix describing a scale-relation between objectives or alternatives. The most common context for such an algorithm is that of the analytic hierarchy process (AHP), although uses in other areas of the field of multicriteria decision making (MCDM) can also be found. Typically, the eigenvalue method is the standard method employed in the AHP to determine weights, as in the ExpertChoice software. However, another class of techniques are the distance-metric-based approaches, which are frequently proposed as alternatives to the eigenvalue method. This paper evaluates such distance-metric-based approaches comparing their effectiveness, using the eigenvalue method as a benchmark. A common framework is introduced to establish an efficient frontier for method comparison.     

一种基于前景理论和改进TOPSIS的模糊随机多准则决策方法及其应用

针对准则值为区间灰数直觉模糊数、准则权系数部分已知以及自然状态出现概率为灰数的多准则决策问题,提出一种结合前景理论和改进TOPSIS的决策方法。该方法首先定义了灰色直觉模糊数的前景价值函数和概率权重函数,并利用前景理论构建出前景决策矩阵;接着从两个方面对传统TOPSIS决策方法进行改进:(1)过定义方案间综合差异的概念,采用离差最大化思想,建立平均综合差异最大化规划模型,给出了一种兼顾主客观权重信息确定准则权系数的新方法;(2)用灰关联替换备选方案与正负理想方案的距离,据此刻画了各方案与正负理想方案的贴近度。进而利用改进TOPSIS决策方法中的综合贴近度对方案进行了排序。最后通过实例验证了该方法的有效性。     

A linear optimization problem to derive relative weights using an interval judgement matrix

In this paper, we propose a new Decision Making model, enabling to assess a finite number of alternatives according to a set of bounds on the preference ratios for the pairwise comparisons between alternatives, that is, an “interval judgement matrix”. In the case in which these bounds cannot be achieved by any assessment vector, we analyze the problem of determining of an efficient or Pareto-optimal solution from a multi-objective optimization problem. This multi-objective formulation seeks for assessment vectors that are near to simultaneously fulfil all the bound requirements imposed by the interval judgement matrix. Our new model introduces a linear optimization problem in order to define a consistency index for the interval matrix. By solving this optimization problem it can be associated a weakly efficient assessment vector to the consistency index in those cases in which the bound requirements are infeasible. Otherwise, this assessment vector fulfils all the bound requirements and has geometrical properties that make it appropriate as a representative assessment vector of the set of feasible weights.     

On the use of information criteria in analytic hierarchy process

In this article, we discuss how the model-selection procedures such as Akaike's information criteria (AIC) can be used for selecting the most appropriate one out of several existing statistical models in the literature for the judgment data used in analytic hierarchy process (AHP). Furthermore, once the appropriate model is selected, a procedure is proposed on the basis of AIC for statistical ranking of the alternatives. This ranking procedure does not suffer from the problem of intransitivity and can be based on non-normal distribution. It enables one to obtain the detailed pattern for the ordered priorities of the alternatives in the decision process involving AHP.     

Decision-aid for discrete multiple criteria decision making problems with imprecise data

We describe ways of aiding decision making with a discrete set of alternatives. In many decision situations, it is not possible to obtain explicit preference information from the decision makers. Instead, useful decision-aid can be provided to the decision makers by describing what kind of weighting of the criteria result in certain choices of the alternatives. The suggested treatment is based on the basic ideas of the ELECTRE III method. The modelling of the preferences by pseudo-criteria is especially helpful in case the data, that is, the criterion values are imprecise. Unlike ELECTRE III, no ranking of the alternatives is produced. Based on a minimum-procedure in the exploitation of the outranking relations, we provide information about the weights of the criteria that make a certain alternative the best. We also present an interactive searching procedure in the weighting space. The auxiliary optimization problems to be solved are nondifferentiable. Cases with both single and multiple decision makers are considered.     

Matrix shrinkage of high-dimensional expectation vectors

The shrinkage effect is studied in estimating the expectation vector by weighting of mean vector components in the system of coordinates in which sample covariance matrix is diagonal. The Kolmogorov asymptotic approach is applied, when sample size increases together with the dimension, so that their ratio tends to a constant. Under some weak assumptions on the dependence of variables, the limit expression for the principal part of the quadratic risk function is found in dependence of weighting function. It is proved that the limit risk function does not depend on distributions. The extremum problem is solved, and an approximately unimprovable distribution-free estimator of the expectation vector is proposed.     

Plausibility relations and multiple expected utilities in decision-making under uncertainty

In this article we develop a new method for dealing with decision-making problems under uncertainty, derived from the one proposed by Fourgeaud, Lenclud and Sentis in 1968. We offer an information gathering procedure based on plausibility relations between ‘possible futures’ which leads to the construction of a polyhedron which contains all the admissible probability measures. Thus it is possible to compare alternatives by associating with each one of them a set of expected utilities calculated from these measures.An application concerning the cost effectiveness of the flexible manufacturing systems figures as an illustration. The utilization facilities of the method and its value as an instrument for supporting decision-making are particularly highlighted here.     

Controlled multistage selection procedures for comparison with a standard

Comparison with a standard is a general multiple comparison problem, where each system is required to be compared to a single system, referred to as a “standard”, as well as to other alternative systems. The goal is to determine the best system among a number of systems that are better than the standard, or to select the standard when it is equal to or better than the other alternatives. Kim (2005) proposed an efficient fully sequential procedure for comparison with a standard, that obtains a single observation at each stage from the surviving systems, and is one of the most efficient existing procedures. We develop two provably valid multistage selection procedures that take a number of observations from each system and update the variance estimators at each stage. We also employ appropriate control variate technique for each procedure to further improve the efficiency. Empirical results are provided to demonstrate that the proposed procedures are statistically and computationally more efficient than existing fully sequential procedures.