A modified super-efficiency DEA model for infeasibility

The super-efficiency data envelopment analysis (DEA) model is obtained when a decision making unit (DMU) under evaluation is excluded from the reference set. This model provides for a measure of stability of the “efficient” status for frontier DMUs. Under the assumption of variable returns to scale (VRS), the super efficiency model can be infeasible for some efficient DMUs, specifically those at the extremities of the frontier. The current study develops an approach to overcome infeasibility issues. It is shown that when the model is feasible, our approach yields super-efficiency scores that are equivalent to those arising from the original model. For efficient DMUs that are infeasible under the super-efficiency model, our approach yields optimal solutions and scores that characterize the extent of super-efficiency in both inputs and outputs. The newly developed approach is illustrated with two real world data sets.     

Measuring super-efficiency in DEA in the presence of infeasibility

Super-efficiency data envelopment analysis (DEA) model is obtained when a decision making unit (DMU) under evaluation is excluded from the reference set. Because of the possible infeasibility of super-efficiency DEA model, the use of super-efficiency DEA model has been restricted to the situations where constant returns to scale (CRS) are assumed. It is shown that one of the input-oriented and output-oriented super-efficiency DEA models must be feasible for a any efficient DMU under evaluation if the variable returns to scale (VRS) frontier consists of increasing, constant, and decreasing returns to scale DMUs. We use both input- and output-oriented super-efficiency models to fully characterize the super-efficiency. When super-efficiency is used as an efficiency stability measure, infeasibility means the highest super-efficiency (stability). If super-efficiency is interpreted as input saving or output surplus achieved by a specific efficient DMU, infeasibility does not necessary mean the highest super-efficiency.     

A directional distance based super-efficiency DEA model handling negative data

This paper develops a new radial super-efficiency data envelopment analysis (DEA) model, which allows input–output variables to take both negative and positive values. Compared with existing DEA models capable of dealing with negative data, the proposed model can rank the efficient DMUs and is feasible no matter whether the input–output data are non-negative or not. It successfully addresses the infeasibility issue of both the conventional radial super-efficiency DEA model and the Nerlove–Luenberger super-efficiency DEA model under the assumption of variable returns to scale. Moreover, it can project each DMU onto the super-efficiency frontier along a suitable direction and never leads to worse target inputs or outputs than the original ones for inefficient DMUs. Additional advantages of the proposed model include monotonicity, units invariance and output translation invariance. Two numerical examples demonstrate the practicality and superiority of the new model.     

Super-efficiency infeasibility and zero data in DEA

Lee et al. (2011) and Chen and Liang (2011) develop a data envelopment analysis (DEA) model to address the infeasibility issue in super-efficiency models. In this paper, we point out that their model is feasible when input data are positive but can be infeasible when some of input is zero. Their model is modified so that the new super-efficiency DEA model is always feasible when data are non-negative. Note that zero data can make the super-efficiency model under constant returns to scale (CRS) infeasible. Our discussion is based upon variable returns to scale (VRS) and can be applied to CRS super-efficiency models.     

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.     

Super-efficiency and DEA sensitivity analysis

This paper discusses and reviews the use of super-efficiency approach in data envelopment analysis (DEA) sensitivity analyses. It is shown that super-efficiency score can be decomposed into two data perturbation components of a particular test frontier decision making unit (DMU) and the remaining DMUs. As a result, DEA sensitivity analysis can be done in (1) a general situation where data for a test DMU and data for the remaining DMUs are allowed to vary simultaneously and unequally and (2) the worst-case scenario where the efficiency of the test DMU is deteriorating while the efficiencies of the other DMUs are improving. The sensitivity analysis approach developed in this paper can be applied to DMUs on the entire frontier and to all basic DEA models. Necessary and sufficient conditions for preserving a DMU’s efficiency classification are developed when various data changes are applied to all DMUs. Possible infeasibility of super-efficiency DEA models is only associated with extreme-efficient DMUs and indicates efficiency stability to data perturbations in all DMUs.     

A comment on “Measuring super-efficiency in DEA in the presence of infeasibility”

In a recent paper by Chen [Chen, Y., 2005. Measuring super-efficiency in DEA in the presence of infeasibility. European Journal of Operational Research 161 (1), 447–468], he deals with the infeasibility of super-efficiency DEA models in variable returns to scale (VRS) technology. He provides a necessary and sufficient condition for simultaneous infeasibility of input- and output-oriented super-efficiency DEA models in VRS case, then he claims that both of these models are infeasible only for a rare situation. In this paper, we present some counterexamples and comments to the contention by Chen.     

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.     

Deriving the DEA frontier for two-stage processes

Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs). Recently DEA has been extended to examine the efficiency of two-stage processes, where all the outputs from the first stage are intermediate measures that make up the inputs to the second stage. The resulting two-stage DEA model provides not only an overall efficiency score for the entire process, but as well yields an efficiency score for each of the individual stages. Due to the existence of intermediate measures, the usual procedure of adjusting the inputs or outputs by the efficiency scores, as in the standard DEA approach, does not necessarily yield a frontier projection. The current paper develops an approach for determining the frontier points for inefficient DMUs within the framework of two-stage DEA.     

Super efficiencies or super inefficiencies? Insights from a joint computation model for slacks-based measures in DEA

The slacks-based measure (SBM) can incorporate input and output slacks that would otherwise be neglected in the classical DEA model. In parallel, the super-efficiency model for SBM (S-SBM) has been developed for the purpose of ranking SBM efficient decision-making units (DMUs). When implementing SBM in conjunction with S-SBM, however, several issues can arise. First, unlike the standard super-efficiency model, S-SBM can only solve for super-efficiency scores but not SBM scores. Second, the S-SBM model may result in weakly efficient reference points. Third, the S-SBM and SBM scores for certain DMUs may be discontinuous with a perturbation to their inputs and outputs, making it hard to interpret and justify the scores in applications and the efficiency scores may be sensitive to small changes/errors in data. Due to this discontinuity, the S-SBM model may overestimate the super-efficiency score. This paper extends the existing SBM approaches and develops a joint model (J-SBM) that addresses the above issues; namely, the J-SBM model can (1) simultaneously compute SBM scores for inefficient DMUs and super-efficiency for efficient DMUs, (2) guarantee the reference points generated by the joint model are Pareto-efficient, and (3) the J-SBM scores of a firm are continuous in the input and output space. Interestingly, the radial DEA efficiency and super-efficiency scores for a DMU are continuous in the input–output space. The J-SBM model combines the merits of the radial and SBM models (i.e., continuity and Pareto-efficiency).     

对DEA中有效单元排序的一种新方法

文章对数据包络分析(DEA)的强有效性问题提出了一种新的研究方法.利用有效值和负有效值来构造复合输入和输出这种方法可以实现有效决策单元的完全排序.文章还给出了新方法中模型的一些性质.最后,用两个例子来检验此方法并和其他模型的计算结果进行了比较.     

数据包络分析SBM超效率模型无可行解问题的两阶段求解法

数据包络分析是一种得到广泛应用的基于线性规划的非参数技术效率分析方法,其超效率模型是将被评价DMU从参照集中排除从而使求解得出的效率值可能大于1.超效率模型在文献中用于对有效的DMU进行排序、探测异常值、敏感性和稳定性分析、分析生产率变化的Malmquist模型、二人博弈模型等.超效率模型存在的一个缺陷是在规模收益可变的假设下会出现无可行解的问题.提出了一种基于两阶段求解的SBM超效率模型,并保持了与传统SBM超效率模型的兼容性:在传统的投入(产出)导向SBM超效率模型有可行解时,两阶段法获得的结果与之相同;在传统的投入(产出)导向SBM超效率模型无可行解时,两阶段超效率模型可以得出最接近投入(产出)导向定义的可行解.算例采用实际数据对方法进行了验证.     

Super-efficiency measurement under variable return to scale: an approach based on a new directional distance function

A modified super-efficiency model based on the directional distance function (DDF) has recently been developed in order to tackle the infeasibility issue in the two exceptions of the Nerlove–Luenberger (N–L) super-efficiency model under variable return to scale (VRS). However, we find that model does not fully eliminate the infeasibility issue. This paper chooses an appropriate reference bundle in the DDF so that the resulting DDF-based VRS super-efficiency model is always feasible. The proposed new model successfully addresses the infeasibility issue of conventional VRS super-efficiency models and fully eliminates the infeasibility issue in the two exceptions of the VRS N–L super-efficiency model. Additional advantages of the new model include: it is unit-invariant and does not need to predetermine any parameter. Theoretical analyses and numerical examples support the practicality and superiority of our model when compared with other super-efficiency models.     

Sensitivity analysis of efficient units in the presence of non-discretionary inputs

Discretionary models of data envelopment analysis (DEA) assume that all inputs and outputs can be varied at the discretion of management or other users. In any realistic situation, however, there may exist “exogenously fixed” or non-discretionary factors that are beyond the control of a DMU’s management, which also need to be considered. This paper discusses and reviews the use of super-efficiency approach in data envelopment analysis (DEA) sensitivity analyses when some inputs are exogenously fixed. Super-efficiency data envelopment analysis (DEA) model is obtained when a decision making unit (DMU) under evaluation is excluded from the reference set. In this paper by means of modified Banker and Morey’s (BM hereafter) model [R.D. Banker, R. Morey, Efficiency analysis for exogenously fixed inputs and outputs, Operations Research 34 (1986) 513–521], in which the test DMU is excluded from the reference set, we are able to determine what perturbations of discretionary data can be tolerated before frontier DMUs become nonfrontier.     

The linear formulation of the ZSG-DEA models with different production technologies

The zero sum gains data envelopment analysis models (ZSG-DEA models) are non-linear. In this paper, we first show that the ZSG-DEA models can be transformed to linear or parametric linear models and discuss the feasible domains of the parameters. Second, we show that the linear formulations of ZSG-DEA models under the equal output reduction strategy and the proportional output reduction strategy in a single output case are equivalent to the output-oriented super-efficiency model under variable returns-to-scale (VRS) assumption. As a matter of course, the models may encounter infeasibility. Third, we propose the linear transformations of ZSG-DEA models under constant returns-to-scale (CRS) assumption and compare them with the VRS models. In the end, we evaluate the participant countries at the Olympic Games by the linear equivalent models with multiple outputs under different weight restrictions. Our results are compared with the efficiencies obtained from the original ZSG-DEA model with an aggregated output under both CRS and VRS assumptions. It is found that the original method with aggregated output tends to underestimate the efficiencies of DMUs.     

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.     

A distance-based measure of super efficiency in data envelopment analysis: an application to gas companies

Conventional data envelopment analysis (DEA) assists decision makers in distinguishing between efficient and inefficient decision making units (DMUs) in a homogeneous group. Standard DEA models can not provide more information about efficient units. Super-efficiency DEA models can be used in ranking the performance of efficient DMUs and overcome this obstacle. Because of the possible infeasibility, the use of super efficiency models has been restricted. This research proposes a methodology to determine a distance-based measure of super-efficiency. The proposed methodology overcomes the infeasibility problem of the existing ranking methodologies. The applicability of the proposed model is illustrated in the context of the analysis of gas companies?? performance.     

Super-efficiency in DEA by effectiveness of each unit in society

One of the most important topics in management science is determining the efficiency of Decision Making Units (DMUs). The Data Envelopment Analysis (DEA) technique is employed for this purpose. In many DEA models, the best performance of a DMU is indicated by an efficiency score of one. There is often more than one DMU with this efficiency score. To rank and compare efficient units, many methods have been introduced under the name of super-efficiency methods. Among these methods, one can mention Andersen and Petersen’s (1993) [1] super-efficiency model, and the slack-based measure introduced by Tone (2002) [4]. Each of the methods proposed for ranking efficient DMUs has its own advantages and shortcomings. In this paper, we present a super-efficiency method by which units that are more effective and useful in society have better ranks. In fact, in order to determine super-efficiency by this method, the effectiveness of each unit in society is considered rather than the cross-comparison of the units. To do so, we divide the inputs and outputs into two groups, desirable and undesirable, at the discretion of the manager, and assign weights to each input and output. Then we determine the rank of each DMU according to the weights and the desirability of inputs and outputs.     

Finding the efficiency score and RTS characteristic of DMUs by means of identifying the efficient frontier in DEA

Data envelopment analysis (DEA) is basically a linear programming based technique used for measuring the relative performance of organizational units, referred to as decision-making units (DMUs), where the presence of multiple inputs and outputs makes comparisons difficult. The ability of identifying frontier DMUs prior to the DEA calculation is of extreme importance to an effective and efficient DEA computation. In this paper, a method for identifying the efficient frontier is introduced. Then, the efficiency score and returns to scale (RTS) characteristic of DMUs will be produced by means of the equation of efficient frontier.     

Improving envelopment in Data Envelopment Analysis under variable returns to scale

In a Data Envelopment Analysis model, some of the weights used to compute the efficiency of a unit can have zero or negligible value despite of the importance of the corresponding input or output. This paper offers an approach to preventing inputs and outputs from being ignored in the DEA assessment under the multiple input and output VRS environment, building on an approach introduced in Allen and Thanassoulis (2004) for single input multiple output CRS cases. The proposed method is based on the idea of introducing unobserved DMUs created by adjusting input and output levels of certain observed relatively efficient DMUs, in a manner which reflects a combination of technical information and the decision maker’s value judgements. In contrast to many alternative techniques used to constrain weights and/or improve envelopment in DEA, this approach allows one to impose local information on production trade-offs, which are in line with the general VRS technology. The suggested procedure is illustrated using real data.