Gradual technical and scale efficiency improvement in DEA

In data envelopment analysis (DEA) an inefficient unit can be projected onto an efficient target that is far away, i.e. reaching the target may demand large reductions in inputs and increases in outputs. When the inputs and outputs modifications planned are large, it may be troublesome to carry them out all at once. In order to help an inefficient unit reach a distant target, a strategy of gradual improvements with successive, intermediate targets has been proposed. This paper extends such approach to the variable returns to scale (VRS) case. In the VRS scenario we distinguish between units that are technical efficient and those that are not. On the one hand, for those units that are not technical efficient the proposed approach determines successive intermediate targets leading to the technical efficiency frontier, i.e. the priority for those units is to attain technical efficiency. On the other hand, for those units that are technical efficient but not scale efficient the proposed approach computes a sequence of targets ending in the global efficiency frontier, i.e. when technical efficiency is guaranteed the goal is then to attain global efficiency. In both cases, the successive targets are obtained by iteratively solving specific DEA models that take into account given bounds on the rates of change in inputs and outputs that the unit can implement in each step.     

Data envelopment analysis with imprecise data

In original data envelopment analysis (DEA) models, inputs and outputs are measured by exact values on a ratio scale. Cooper et al. [Management Science, 45 (1999) 597–607] recently addressed the problem of imprecise data in DEA, in its general form. We develop in this paper an alternative approach for dealing with imprecise data in DEA. Our approach is to transform a non-linear DEA model to a linear programming equivalent, on the basis of the original data set, by applying transformations only on the variables. Upper and lower bounds for the efficiency scores of the units are then defined as natural outcomes of our formulations. It is our specific formulation that enables us to proceed further in discriminating among the efficient units by means of a post-DEA model and the endurance indices. We then proceed still further in formulating another post-DEA model for determining input thresholds that turn an inefficient unit to an efficient one.     

Joint use of DEA and constrained canonical correlation analysis for efficiency valuations involving categorical variables

The research on efficiency valuations has used two distinct approaches. One is the nonparametric approach known as data envelopment analysis (DEA), the other is the parametric approach based on regression analysis or its extension such as constrained canonical correlation analysis (CCCA). Interestingly, a recent study has employed a hybrid approach that cross-fertilizes DEA and CCCA to compensate for the drawbacks of the two methods and capture their positive aspects. This approach first applies DEA to select efficient units and then utilizes CCCA to identify a smooth efficient frontier with the selected efficient units only. We extend it to incorporate a categorical variable that reflects an environmental effect on efficiency performance. The need for considering a categorical variable arises in practice for an equitable efficiency valuation, as illustrated by managerial performance evaluation of the branches of a fast-food company, where the location of branches such as commercial or noncommercial area significantly affects their performance. We demonstrate various possible ways to handle such a categorical variable in the framework of a hybrid approach and characterize each of the methods. Based on this study, we suggest one method that simultaneously utilizes an extension of DEA, referred to as DEA with categorical variable, and CCCA employing a dummy variable, as in multiple regressions with dummy variables. Through an application to the branches of a fast-food company, we show the efficacy of the suggested method in terms of penalizing the advantageous location effect and compensating for the disadvantageous location effect. We also provide some discussions on the limitations underlying the hybrid approach in order to guide a proper use of this approach to the other potential applications.     

Sensitivity and stability analysis on the first and second levels of efficiency score relative to data error

In data envelopment analysis (DEA), efficient decision making units (DMUs) are of primary importance as they define the efficient frontier. The current paper develops a new sensitivity analysis approach for a category DMUs and finds the stability radius for all efficient DMUs. By means of combining some classic DEA models and with the condition that the efficiency scores of efficient DMUs remain unchanged, we are able to determine what perturbations of the data can be tolerated before efficient DMUs become inefficient. Our approach generalizes the conventional sensitivity analysis approach in which the inputs of efficient DMUs increase and their outputs decrease, while the inputs of inefficient DMUs decrease and their outputs increase. We find the maximum quantity of perturbations of data so that all first level efficient DMUs remain at the same level.     

Sensitivity analysis of DEA models for simultaneous changes in all the data

In data envelopment analysis (DEA) efficient decision making units (DMUs) are of primary importance as they define the efficient frontier. The current paper develops a new sensitivity analysis approach for the basic DEA models, such as, those proposed by Charnes, Cooper and Rhodes (CCR), Banker, Charnes and Cooper (BCC) and additive models, when variations in the data are simultaneously considered for all DMUs. By means of modified DEA models, in which the specific DMU under examination is excluded from the reference set, we are able to determine what perturbations of the data can be tolerated before efficient DMUs become inefficient. Our approach generalises the usual sensitivity analysis approach developed in which perturbations of the data are only applied to the test DMU while all the remaining DMUs remain fixed. In our framework data are allowed to vary simultaneously for all DMUs across different subsets of inputs and outputs. We study the relations of the infeasibility of modified DEA models employed and the robustness of DEA models. It is revealed that the infeasibility means stability. The empirical applications demonstrate that DEA efficiency classifications are robust with respect to possible data errors, particularly in the convex DEA case.     

Integrated data envelopment analysis: Global vs. local optimum

Chiou et al. (2010) (A joint measurement of efficiency and effectiveness for non-storable commodities: integrated data envelopment analysis approaches. European Journal of Operational Research 201, 477–489) propose an integrated data envelopment analysis model in measuring decision making units (DMUs) that have a two-stage internal network structure with multiple inputs, outputs, and consumptions. They claim that any optimal solutions determined by their DEA model are a global optimum, not a local optimum. We show that such a conclusion is a false statement due to their misuse of Hessian matrix in examining the concavity of the objective function, and their DEA model is actually a non-convex optimization problem. As a result, their DEA model is unusable in practice due to a lack of efficient algorithm for this particular non-convex DEA model. We further show that Chiou et al.’s (2010) model is a special case of a well-known two-stage network DEA model, and it can be transformed into a parametric linear program for which an approximate global optimal solution can be obtained by solving a sequence of linear programs in combination with a simple search algorithm.     

Combining two approaches to efficiency assessment

The advent of data envelopment analysis (DEA) enabled the measurement of efficiency to be extended to the case of multiple outputs. Prior to DEA we had the parametric approach based on multiple regression. We highlight some difficulties associated with these two approaches and present a hybrid which overcomes them whilst maintaining the respective advantages of each. This hybrid models the efficient frontier using an algebraic expression; the resulting smooth representation allows all units to be naturally enveloped and hence slacks to be avoided. (Slacks are potential improvements for inefficient units which are not accounted for in the DEA (radial) score, and so have been problematic for DEA.) The approach identifies the DEA-efficient units and fits a smooth model to them using maximum correlation modelling. This new technique extends the method of multiple regression to the case where there are multiple variables on each side of the model equation (eg outputs and inputs). The resulting expression for the frontier permits managers to estimate the effect on their efficiency score of adjustments in one or more input or output levels.     

The interval efficiency based on the optimistic and pessimistic points of view

Data envelopment analysis (DEA) is a data-oriented approach for evaluating the performances of a set of peer entities called decision-making units (DMUs), whose performance is determined based on multiple measures. The traditional DEA, which is based on the concept of efficiency frontier (output frontier), determines the best efficiency score that can be assigned to each DMU. Based on these scores, DMUs are classified into DEA-efficient (optimistic efficient) or DEA-non-efficient (optimistic non-efficient) units, and the DEA-efficient DMUs determine the efficiency frontier. There is a comparable approach which uses the concept of inefficiency frontier (input frontier) for determining the worst relative efficiency score that can be assigned to each DMU. DMUs on the inefficiency frontier are specified as DEA-inefficient or pessimistic inefficient, and those that do not lie on the inefficient frontier, are declared to be DEA-non-inefficient or pessimistic non-inefficient. In this paper, we argue that both relative efficiencies should be considered simultaneously, and any approach that considers only one of them will be biased. For measuring the overall performance of the DMUs, we propose to integrate both efficiencies in the form of an interval, and we call the proposed DEA models for efficiency measurement the bounded DEA models. In this way, the efficiency interval provides the decision maker with all the possible values of efficiency, which reflect various perspectives. A numerical example is presented to illustrate the application of the proposed DEA models.     

链式网络DEA模型

数据包络分析(DEA)是评价决策单元(DMU)相对有效性的一种工具,现已得到广泛的应用.传统的DEA不考虑系统内部结构,而是将系统作为一个"黑箱"来度量效率.针对多阶段网络结构提出一个新的网络DEA模型—链式网络DEA模型.研究网络决策单元的网络DEA有效性及各个阶段的弱DEA有效性之间的关系,给出了网络DEA有效的充分必要条件.若网络决策单元不是网络DEA有效的,根据模型可以指出系统在哪些阶段是无效的.     

Employing super-efficiency analysis as an alternative to DEA: An application in outpatient substance abuse treatment

A common technique for conducting efficiency analyses consists of a two-stage procedure that combines data envelopment analysis (DEA) with Tobit regression. As the DEA scores are censored at one, this method has the drawback of masking important information at the upper tail of the distribution of scores. In this paper, we present a DEA-based methodology for a two-stage efficiency analysis where the upper bound constraint of one for the efficiency scores is relaxed. This method, super-efficiency DEA, is contrasted with the two-stage approach that employs traditional, bounded DEA scores. We use data from the National Drug Abuse Treatment Survey to examine how the relative efficiency of the treatment units is affected by the organizational structures, operating characteristics and treatment modalities of a nationally representative sample of outpatient substance abuse treatment units. Our results show that the super-efficiency DEA approach offers advantages over the traditional methodology. It is easy to implement, and, for the same sample size provides more information.     

A conic relaxation model for searching for the global optimum of network data envelopment analysis

Network data envelopment analysis (DEA) models the internal structures of decision-making units (DMUs). Unlike the standard DEA model, multiplier-based network DEA models are often highly non-linear and cannot be converted into linear programs. As such, obtaining a non-linear network DEA's global optimal solution is a challenge because it corresponds to a nonconvex optimization problem. In this paper, we introduce a conic relaxation model that searches for the global optimum to the general multiplier-based network DEA model. We reformulate the general network DEA models and relax the new models into second order cone programming (SOCP) problems. In comparison with linear relaxation models, which is potentially applicable to general network DEA structures, the conic relaxation model guarantees applicability in general network DEA, since McCormick envelopes involved are ensured to be finite. Furthermore, the conic relaxation model avoids unnecessary linear relaxations of some nonlinear constraints. It generates, in a more convenient manner, feasible approximations and tighter upper bounds on the global optimal overall efficiency. Compared with a line-parameter search method that has been applied to solve non-linear network DEA models, the conic relaxation model keeps track of the distances between the optimal overall efficiency and its approximations. As a result, it is able to determine whether a qualified approximation has been achieved or not, with the help of a branch and bound algorithm. Hence, our proposed approach can substantially reduce the computations involved.     

The super-efficiency procedure for outlier identification,not for ranking efficient units

In this paper, we conduct simulation experiments to evaluate the performance of two alternative uses of the super-efficiency procedure in Data Envelopment Analysis (DEA). The first is for outlier identification and the second is for ranking efficient units. We find that the ranking procedure does not perform satisfactorily. In fact, the correlations between the true efficiency and the estimated super-efficiency are negative for the subset of efficient observations, and the conventional DEA model performs as well as the super-efficiency DEA model when all observations are considered. However, when data are contaminated with outliers, the use of the super-efficiency model to identify and remove outliers results in more accurate efficiency estimates than those obtained from the conventional DEA estimation model.     

一类多指标对象评价模型

以C2R模型为基础给出了一类多指标对象评价模型——C2RM模型.我们首先定义了决策单元的优势集,论证了C2RM模型比一类多目标规划模型在优化意义上更进一步——排除了此类多目标规划模型的一些优势集为空的Pareto有效解.我们建立的C2RM模型是线性规划模型,易于判定一个决策单元是否DEA有效.在优势集性质基础上我们进一步定义了C2RM模型下DEA有效决策单元的实际有效率,并给出了所有决策单元的一种排序,最后举例予以说明.     

Variables reduction in data envelopment analysis

In real applications of data envelopment analysis (DEA), there are a number of pitfalls that could have a major influence on the efficiency. Some of these pitfalls are avoidable and the others remain problematic. One of the most important pitfalls that the researchers confront is the closeness of the number of operational units and the number of inputs and outputs. In performance measurement using DEA, the closeness of these two numbers could yield a large number of efficient units. In this article, some inputs or outputs will be aggregated and the number of inputs and outputs are reduced iteratively. Numerical examples show that in comparison to the single DEA method, our approach has the fewest efficient units. This means that our approach has a superior ability to discriminate the performance of the DMUs.     

A statistical test for detecting influential observations in DEA

This paper deals with the problem of detecting influential observations in deterministic nonparametric DEA models. The technique we present is intended to classify for a further analysis those sample observations considerably affecting the measured efficiency for the remaining units. Then, the analyst will have to check whether these observations are contaminated by data errors or not. This approach also allows to determine when efficiency changes due to the presence of a given unit in the sample are statistically significant. Thus, ours is a statistical alternative to approach the problem of detecting influential observations in deterministic nonparametric DEA models.     

Centralized DEA models with the possibility of downsizing

While traditional data envelopment analysis (DEA) models assess the relative efficiency of similar, independent decision making units (DMUs) centralized DEA models aim at reallocating inputs and outputs among the units setting new input and output targets for each one. This system point of view is appropriate when the DMUs belong to a common organization that allocates their inputs and appropriates their outputs. This intraorganizational perspective opens up the possibility that greater technical efficiency for the organization as a whole might be achieved by closing down some of the existing DMUs. In this paper, we present three centralized DEA models that take advantage of this possibility. Although these models involve some binary variables, we present efficient solution approaches based on Linear Programming. We also present some numerical results of the proposed models for a small problem from the literature.     

Computation of efficient targets in DEA models with production trade-offs and weight restrictions

Recently new models of data envelopment analysis (DEA) were introduced that incorporate production trade-offs between inputs and outputs or based on them weight restrictions. In this paper, we develop a computational procedure suitable for the practical application of such models. We show that the standard two-stage optimisation procedure used in DEA to test the full efficiency of units and identify their efficient targets may work incorrectly in the new models. The modified procedure consists of three stages: the first evaluates the radial efficiency of the unit, the second identifies its efficient target, and the third its reference set of efficient peers. Each stage requires solving one linear program for each unit.     

关于综合 DEA 模型中的 DEA 有效决策单元集合的几个恒等式

DEA 模型(C~2R 模型,C~2GS~2模型,C~2W 模型和 C~2WH 模型等)是用来评价决策单元之间的相对有效性的.本文将给出关于具有更一般形式的综合 DEA 模型中的 DEA 有效决策单元集合的几个恒等式.这些等式表明在评价相对有效性时,可以将决策单元按照实际情况进行分组,先评价组内决策单元之间的相对有效性,再利用得到的信息进行不同组之间的有效决策单元的效率评价,如此等等.这种分组逐一进行评价的     

Value efficiency analysis of academic research

We propose a systematic approach to analyzing academic research performance at universities and research institutes. The analysis is based on identifying a set of decision-relevant (abstract) criteria. The scales for these criteria are defined by means of concrete indicators, all which are, however, not necessarily quantitative. Qualitative information is quantified using appropriate analytical tools. Once the criteria and indicators have been agreed upon and quantified, data on the research units is collected and a value efficiency analysis is performed. The efficiency of research units is defined in the spirit of data envelopment analysis (DEA), complemented with decision maker's (DM's) (rector in the European university system) preference information. This information is obtained by asking the DM to locate a point on the efficient frontier having the most preferred combination of input and output values. Our approach and the accompanying decision support system enables a university to allocate resources more efficiently than previously to its research units. Using data from the Helsinki school of economics, we illustrate how the approach works.     

Determining a sequence of targets in DEA

Data Envelopment Analysis (DEA) can be used for assessing the relative efficiency of a number of operating units, finding, for each inefficient unit, a target operating point lying on the efficient frontier. Most DEA models project an inefficient unit onto a most distant target, which makes its attainment more difficult. In this paper, we advocate determining a sequence of targets, each one within an appropriate, short distance of the preceding. The proposed Constant Returns to Scale approach has two interesting features: (a) the sequence of targets ends in the efficient frontier and (b) the final, efficient target is generally closer to the original unit than the one-step projection is.