A slacks-based measure of super-efficiency in data envelopment analysis: An alternative approach

The current paper proposes a slack-based version of the Super SBM, which is an alternative super-efficiency model for the SBM proposed by Tone. Our two-stage approach provides the same super-efficiency score as that obtained by the Super SBM model when the evaluated DMU is efficient and yields the same efficiency score as that obtained by the SBM model when the evaluated DMU is inefficient. The projection identified by the Super SBM model may not be strongly Pareto efficient; however, the projection identified from our approach is strongly Pareto efficient.     

Ranking efficient units in DEA

Super-efficiency data envelopment analysis (DEA) model can be used in ranking the performance of efficient decision making units (DMUs). Because of the infeasibility problem associated with the super-efficiency DEA model, ranking has been restricted to the model where constant returns to scale and proportional changes in all inputs or all outputs are assumed. In fact, when super-efficiency is used as an efficiency stability measure, infeasibility means the highest super-efficiency. However, if super-efficiency is interpreted as input saving or output surplus achieved by a specific efficient DMU, infeasibility does not necessarily mean the highest super-efficiency. In order to obtain a complete ranking of efficient DMUs when the two assumptions are relaxed, a modified super-efficiency DEA model is proposed to overcome the infeasibility problem and to correctly capture the super-efficiency represented by the input saving or the output surplus. The current paper suggests using both input- and output-oriented super-efficiency models to characterize the super-efficiency when infeasibility occurs. As a result, we can rank the efficient DMUs if infeasibility occurs. The approach is applied to 20 largest Japanese companies and 15 US cities, respectively.     

超效率DEA模型的模糊扩展

在输入、输出数据为模糊数而效率值为准确值的模糊DEA模型中,会出现有两个以上有效单元,从而无法对所有决策单元进行全排序的情形。本文利用模糊数的基于α-截集的比较规则,建立了模糊条件下的超效率DEA模型,有效地解决了模糊决策单元的全排序问题。文末给出了一个实例。     

基于证据推理和前景理论的交叉效率排序方法研究

针对由交叉效率评价策略和交叉效率集结方法的多样性而造成评价结果不一致的问题,提出利用证据推理方法和前景理论,综合各个交叉效率评价策略的评价结果,实现对决策单元的统一评价。首先,分别将选用的交叉效率评价策略以及各个评价策略中的他评效率设置成一级指标和二级指标,依据算数平均和前景理论分别确定一、二级指标的权重;其次,依据他评效率确定二级指标置信度,利用证据推理方法将各个交叉效率评价策略的他评效率综合转换成决策单元被评价为有效的置信度。决策者可通过比较决策单元被识别为有效的置信度的大小来判断决策单元交叉效率的大小,进而实现对决策单元的排序;最后,通过案例验证和说明本文提出方法的有效性和实用性。     

存在保证域的模糊超效率DEA模型

模糊DEA模型是用于解决存在模糊数据的决策单元(DMUs)效率评价问题的,然而现有的模糊DEA模型分辨率低,本文构建了存在保证域的模糊超效率DEA模型,并给出了一种基于截集的求解方法并进行了证明,该模型有效地解决了输入和输出全部或部分为模糊数的决策单元全排序问题。最后给出了一个银行效率评价的实例说明了方法的有效性。     

Super-efficiency based on a modified directional distance function

The problem of infeasibility arises in conventional radial super-efficiency data envelopment analysis (DEA) models under variable returns to scale (VRS). To tackle this issue, a Nerlove–Luenberger (N–L) measure of super-efficiency is developed based on a directional distance function. Although this N–L super-efficiency model does not suffer infeasibility problem as in the conventional radial super-efficiency DEA models, it can produce an infeasible solution in two special situations. The current paper proposes to modify the directional distance function by selecting proper feasible reference bundles so that the resulting N–L measure of super-efficiency is always feasible. As a result, our modified VRS super-efficiency model successfully addresses the infeasibility issues occurring either in conventional VRS models or the N–L super-efficiency model. Numerical examples are used to demonstrate our approach and compare results obtained from various super-efficiency measures.     

Evaluating the administrative efficiency of courts

In addition to being held accountable for judicial decision, courts, like other public agencies, can and should be evaluated in terms of their administrative efficiency. This paper illustrates how courts can be evaluated in terms of their relative administrative efficiency, using a new approach—data envelopment analysis (DEA)—first proposed by Charnes et al. [1]. The DEA is based upon the economic notion of Pareto optimality which states that a given decision making unit (DMU) is inefficient if some other DMU, or some combination of other DMUs, can produce at least the same amounts of all outputs with less of some resource input and not more of any other resource. Conversely a DMU is said to be efficient if the above is not possible. Charnes et al. [1] generalized the usual input/output ratio measure of efficiency for a given unit in terms of a fractional linear program with fractional constraints. In the case of courts, the efficiency of any particular court is calculated by forming the ratio of a weighted sum of outputs to a weighted sum of inputs, where the weights for both outputs and inputs are to be selected in a manner that calculates the Pareto-Koopmans efficiency of the court. This paper reviews the DEA method and illustrates its application to a data base for 100 criminal superior courts in North Carolina.     

DEA models for extended two-stage network structures

Liang et al. (2008) [1] developed DEA models based upon game approach to decompose efficiency for two-stage network structures where all outputs of the first stage are the only inputs to the second stage. This paper extends Liang et al. (2008) [1] by assuming that the inputs to the second stage include both the outputs from the first stage and additional inputs to the second stage. Two models are proposed to evaluate the performance of this type general two-stage network structures. One is a non-linear centralized model whose global optimal solutions can be estimated using a heuristic search procedure. The other is a non-cooperative model, in which one of the stages is regarded as the leader and the other is the follower. The newly developed models are applied to a case of regional R&D of China.     

两种改进的超效DEA模型

对DEA有效决策单元不能排序是传统DEA模型的一大缺点。该问题也成为DEA领域研究的热点。AP模型是最常用的区分DEA有效决策单元效率值大小的超效模型,但在实际应用中却出现了无解或解不稳定的情况。国外学者新近提出的MAJ模型、LJK模型则有效解决了这两个问题,但却存在模型区分能力不强的缺点。本文分析了这两种模型缺点产生的原因、予以改进并给出理论证明。标准数据集的实验显示改进效果显著,尤其是改进后的LJK模型表现优异。     

基于最近距离投影的DEA两阶段效率评价方法研究

数据包络分析是进行效率评价最重要的方法之一。传统的数据包络分析理论主要寻找有效前沿面上的最远距离投影，在极大化无效性指数的同时也面临着效率改进的巨大难度和高额成本。对于具有两阶段内部生产结构的决策单元，本文从考虑最小改进难度的视角出发，提出了最近距离投影的两阶段效率评价方法。该方法首先得到所有强有效决策单元的线性组合，且这些组合均占优于被评价的两阶段决策单元。然后建立了两阶段范围调整效率评价模型，在确定具有最近投影距离的占优组合的同时，得到了两阶段评价效率。最后，本文运用我国32家上市银行的年度数据对所提出方法进行了应用验证。     

DEA as a tool for predicting corporate failure and success: A case of bankruptcy assessment

Using an additive super-efficiency data envelopment analysis (DEA) model, this paper develops a new assessment index based on two frontiers for predicting corporate failure and success. The proposed approach is applied to a random sample of 1001 firms, which is composed of 50 large US bankrupt firms randomly selected from Altman's bankruptcy database and 901 healthy matching firms. This sample represents the largest firms that went bankrupt over the period 1991–2004 and represents a full spectrum of industries. Our findings demonstrate that the DEA model is relatively weak in predicting corporate failures compared to healthy firm predictions, and the assessment index improves this weakness by giving the decision maker various options to achieve different precision levels of bankrupt, non-bankrupt, and total predictions.     

Common benchmarking and ranking of units with DEA

This paper develops a common framework for benchmarking and ranking units with DEA. In many DEA applications, decision making units (DMUs) experience similar circumstances, so benchmarking analyses in those situations should identify common best practices in their management plans. We propose a DEA-based approach for the benchmarking to be used when there is no need (nor wish) to allow for individual circumstances of the DMUs. This approach identifies a common best practice frontier as the facet of the DEA efficient frontier spanned by the technically efficient DMUs in a common reference group. The common reference group is selected as that which provides the closest targets. A model is developed which allows us to deal not only with the setting of targets but also with the measurement of efficiency, because we can define efficiency scores of the DMUs by using the common set of weights (CSW) it provides. Since these weights are common to all the DMUs, the resulting efficiency scores can be used to derive a ranking of units. We discuss the existence of alternative optimal solutions for the CSW and find the range of possible rankings for each DMU which would result from considering all these alternate optima. These ranking ranges allow us to gain insight into the robustness of the rankings.     

Efficiency measurement for hierarchical network systems

The conventional data envelopment analysis (DEA) models for measuring the relative efficiency of a set of decision making units (DMUs), without considering the operations of the component processes, often produce misleading results, and network models have thus been recommended. This paper discusses the development of a network DEA model for systems with a hierarchical structure. It is shown that the hierarchical structure is equivalent to a parallel structure, with the components being the units at the bottom of the hierarchy. Due to the characteristics of a parallel system, the efficiency of a hierarchical system is thus a weighted average of those of the units at the bottom of the hierarchy. A hypothetic example shows that the proposed model is able to distinguish the order of the efficient DMUs evaluated by the conventional DEA model. Moreover, it provides the efficiencies of the functions of the DMU, which enables managers to identify areas of weakness, and thus better focus efforts to improve overall performance.     

Directional distance based diversification super-efficiency DEA models for mutual funds

Current data envelopment analysis (DEA) models with diversification cannot discriminate the performance of efficient mutual funds. Based on the directional distance function and diversification DEA models, this paper proposes two diversification super-efficiency models for discriminating the performance of efficient mutual funds on financial market. The proposed diversification super-efficiency models as well as the corresponding diversification DEA models are feasible and can deal with negative values in risk measures, transaction costs and return measures. The proposed methods generate bounded super-efficiency scores for all the funds. Under the assumption of discrete return distributions, all the models in the proposed diversification super-efficiency methods can be transformed into linear programming (LP) problems by choosing proper risk and return measures. To demonstrate the validity and practicality of the proposed diversification super-efficiency methods, we apply them to evaluate the performance of mutual funds in the American market. The empirical results show that the proposed diversification super-efficiency models can distinguish efficient funds well and the linear combination of efficient funds might be inefficient. Moreover, the backtesting results show that the proposed diversification super-efficiency models generally have a good practice value for the actual portfolio selection.     

Data envelopment analysis: Prior to choosing a model

In this paper, we address several issues related to the use of data envelopment analysis (DEA). These issues include model orientation, input and output selection/definition, the use of mixed and raw data, and the number of inputs and outputs to use versus the number of decision making units (DMUs). We believe that within the DEA community, researchers, practitioners, and reviewers may have concerns and, in many cases, incorrect views about these issues. Some of the concerns stem from what is perceived as being the purpose of the DEA exercise. While the DEA frontier can rightly be viewed as a production frontier, it must be remembered that ultimately DEA is a method for performance evaluation and benchmarking against best-practice. DEA can be viewed as a tool for multiple-criteria evaluation problems where DMUs are alternatives and each DMU is represented by its performance in multiple criteria which are coined/classified as DEA inputs and outputs. The purpose of this paper is to offer some clarification and direction on these matters.     

Additive slacks-based measure: Computational strategy and extension to network DEA

The current paper visits a set of data envelopment analysis (DEA) models that identify inefficiency by optimizing input and output slacks. These slacks are aggregated either in an additive or ratio form. Only the ratio slacks-based DEA models can be solved as a linear program and generate a DEA score between zero and unity. The additive slacks-based model can be equivalent to the Russell graph measure and converted into a second order cone programming (SOCP) problem whose solving procedure has become a mature technology. As such, the additive slacks-based model can also yield a DEA score between zero and unity. This study shows that the additive slacks-based model can be applied to modelling network DEA where the internal structures of decision making units (DMUs) are of interest. The additive slacks-based network DEA can be solved using SOCP technique and adapted to the preference of the decision maker by choosing the weights for aggregating individual components in the network structures. It is shown that the additive slacks-based approach can yield divisional efficiencies of Pareto optimal equivalences to be selected by the decision maker when compared to the existing ratio slacks-based measure. An example and solving codes are provided in the current study.     

Allocating a fixed cost based on data envelopment analysis and satisfaction degree

This paper uses the Data Envelopment Analysis (DEA) technique to solve the problem of allocating a fixed cost across a set of comparable decision making units (DMUs) in a fair way. It first investigates the effect of the fixed cost on each DMU and on the collection of DMUs. Next we prove that there exist some cost allocations which can make each DMU and the collection of DMUs efficient. We show that such a cost allocation is unique and equivalent to the proportional sharing method if the fixed cost allocation problem is a one-dimensional case. In a multidimensional case, the fixed cost allocations may not be unique. This paper defines the concept of satisfaction degree, and proposes a maxmin model and a corresponding algorithm to generate a unique fixed cost allocation. Finally, the proposed approach has been applied to a data set from prior literature.     

DEA game cross-efficiency approach to Olympic rankings

A number of studies have used data envelopment analysis (DEA) to evaluate the performance of the countries in Olympic games. While competition exists among the countries in Olympic games/rankings, all these DEA studies do not model competition among peer decision making units (DMUs) or countries. These DEA studies find a set of weights/multipliers that keep the efficiency scores of all DMUs at or below unity. Although cross efficiency goes a further step by providing an efficiency measure in terms of the best multiplier bundle for the unit and all the other DMUs, it is not always unique. This paper presents a new and modified DEA game cross-efficiency model where each DMU is viewed as a competitor via non-cooperative game. For each competing DMU, a multiplier bundle is determined that optimizes the efficiency score for that DMU, with the additional constraint that the resulting score should be at or above that DMU 's estimated best performance. The problem, of course, arises that we will not know this best performance score for the DMU under evaluation until the best performances of all other DMUs are known. To combat this “chicken and egg” phenomenon, an iterative approach leading to the Nash equilibrium is presented. The current paper provides a modified variable returns to scale (VRS) model that yields non-negative cross-efficiency scores. The approach is applied to the last six Summer Olympic Games. Our results may indicate that our game cross-efficiency model implicitly incorporates the relative importance of gold, silver and bronze medals without the need for specifying the exact assurance regions.     

考虑满意度的两阶段DEA固定成本分摊方法

考虑具有相同的投入和产出的两阶段决策单元的固定成本分摊问题。将待分摊的固定成本作为一种新的投入,建立两阶段加性DEA模型。首先证明了各决策单元从自身角度出发可以找到至少一种分摊方案,使其自身整体和阶段效率都达到Pareto有效。然后证明了在公共的权重下,至少存在一种分摊方案可以使得各决策单元整体和阶段效率都达到Pareto有效。并给出了这样的有效成本分摊方案集。为了得到公平的成本分摊方案,定义了各DMU各阶段对分摊方案的满意度,最大化最小的满意度,可以得到最终的成本分摊方案。将问题拓展到两阶段存在共享投入的生产系统中,这些性质定理同样成立。数值算例部分将本文与Yu等（2016）的研究结果进行了对比,验证了本文方法的有效性,指出了本文方法的优越性。本文的方法综合考虑了效率和公平,得到的固定成本分摊方案公平合理。     

基于可变权重的DEA效率评价模型

传统DEA模型将投入、产出权重视为固定变量。为了更合理地对决策单元的效率进行评价,本文研究了如何将有关投入、产出变量的权重的更多信息融入效率评价模型,提出了可变权重的概念,给出了一种基于可变权重的DEA效率评价模型。本文模型是CCR模型的推广。在有关权重信息可得的前提下,本文模型较之CCR模型中效率评价上更为合理。但是,如何准确度量投入、产出权重并将其函数化,是本文方法应用的难点。一个算例分析演示了本文模型。