Classifying inputs and outputs in data envelopment analysis

In conventional data envelopment analysis it is assumed that the input versus output status of each of the chosen performance measures is known. In some situations, however, certain performance measures can play either input or output roles. We refer to these performance measures as flexible measures. This paper presents a modification of the standard constant returns to scale DEA model to accommodate such flexible measures. Both an individual DMU model and an aggregate model are suggested as methodologies for deriving the most appropriate designations for flexible measures. We illustrate the application of these models in two practical problem settings.     

Super-efficiency DEA in the presence of infeasibility

It is well known that super-efficiency data envelopment analysis (DEA) approach can be infeasible under the condition of variable returns to scale (VRS). By extending of the work of Chen (2005), the current study develops a two-stage process for calculating super-efficiency scores regardless whether the standard VRS super-efficiency mode is feasible or not. The proposed approach examines whether the standard VRS super-efficiency DEA model is infeasible. When the model is feasible, our approach yields super-efficiency scores that are identical to those arising from the original model. For efficient DMUs that are infeasible under the super-efficiency model, our approach yields super-efficiency scores that characterize input savings and/or output surpluses. The current study also shows that infeasibility may imply that an efficient DMU does not exhibit super-efficiency in inputs or outputs. When infeasibility occurs, it can be necessary that (i) both inputs and outputs be decreased to reach the frontier formed by the remaining DMUs under the input-orientation and (ii) both inputs and outputs be increased to reach the frontier formed by the remaining DMUs under the output-orientation. The newly developed approach is illustrated with numerical examples.     

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.     

Super-efficiency DEA in the presence of infeasibility: One model approach

A two-stage procedure is developed by Lee et al. (2011) [European Journal of Operational Research doi:10.1016/j.ejor.2011.01.022] to address the infeasibility issue in super-efficiency data envelopment analysis (DEA) models. We point out that their two-stage procedure can be solved in a single DEA-based model.     

Technical efficiency estimation with multiple inputs and multiple outputs using regression analysis

Regression and linear programming provide the basis for popular techniques for estimating technical efficiency. Regression-based approaches are typically parametric and can be both deterministic or stochastic where the later allows for measurement error. In contrast, linear programming models are nonparametric and allow multiple inputs and outputs. The purported disadvantage of the regression-based models is the inability to allow multiple outputs without additional data on input prices. In this paper, deterministic cross-sectional and stochastic panel data regression models that allow multiple inputs and outputs are developed. Notably, technical efficiency can be estimated using regression models characterized by multiple input, multiple output environments without input price data. We provide multiple examples including a Monte Carlo analysis.     

Improvements by analyzing the efficient frontier in DEA

In this paper, we suggest four types of improvements for making inefficient DMUs efficient in the CCR model with the minimal change of input and output values. Moreover, we propose an algorithm for calculating such improvements by applying quadratic programming techniques. Furthermore, since all equations constructing the efficient frontiers of the CCR and BCC models are necessary to execute the algorithm, we present a procedure for calculating them.     

Enhancing the inclusion of non-discretionary inputs in DEA

The aim of this paper is to provide an alternative approach for estimating efficiency when a set of decision-making units uses non-discretionary inputs in the productive process. To test the influence of these variables, our proposal uses a multi-stage approach based on Tobit regressions. In order to avoid potential bias, a bootstrap procedure is used to estimate these regressions. This methodology allows enhancing other models previously proposed to introduce non-controllable inputs in data envelopment analysis (DEA) overcoming, thus, some of their main shortcomings. We illustrate our framework with an empirical application on Spanish high schools where non-controllable factors play a major role to explain educational achievements.     

Decomposing the efficient frontier of the DEA production possibility set into a smallest number of convex polyhedrons by mixed integer programming

This paper extends the works by Olesen and Petersen (2003), Russell and Schworm (2006) and Cooper et al. (2007) about describing the efficient frontier of a production possibility set by the intersection of a finite number of closed halfspaces, in several ways. First, we decompose the efficient frontier into a smallest number of convex polyhedrons, or equivalently into a new class of efficient faces, called maximal efficient faces (MEFs). Second, we show how to identify all MEFs even if full dimensional efficient faces do not exist. Third, by applying the MEF decomposition to various real-world data sets, we demonstrate the validity of the MEF decomposition and how it can contribute to the DEA literature. Finally, we illustrate how to use the identified MEFs in practice.     

Robustness of the efficient DMUs in data envelopment analysis

By means of modified versions of CCR model based on evaluation of a decision making unit (DMU) relative to a reference set grouped by all other DMUs, sensitivity analysis of the CCR model in data envelopment analysis (DEA) is studied in this paper. The methods for sensitivity analysis are linear programming problems whose optimal values yield particular regions of stability. Sufficient and necessary conditions for upward variations of inputs and for downward variations of outputs of an (extremely) efficient DMU which remains efficient are provided. The approach does not require calculation of the basic solutions and of the inverse of the corresponding optimal basis matrix. The approach is illustrated by two numerical examples.     

Nonparametric measurement of productive efficiency in the presence of regulation

During the late 1970s and 1980s, volatility in the demand for natural gas in the United States created havoc in the industry's transmission sector. Managers of firms were presented with the task of improving productive efficiency in an uncertain and regulated industry. We measure the relative performance of 20 pipelines, using Data Envelopment Analysis (DEA) to produce a regulation constrained minimum cost frontier. With the results we construct a Fisher Productivity Index (FPI). Results show that despite productive efficiency decline for most years, most firms experienced improvement in technical efficiency and technical progress in most years. Most of the productive efficiency decline was due to scale diseconomies.     

Measurement of the worst practice of decision-making units in the presence of non-discretionary factors and imprecise data

The objective of the present paper is to propose a novel pair of data envelopment analysis (DEA) models for measurement of relative efficiencies of decision-making units (DMUs) in the presence of non-discretionary factors and imprecise data. Compared to traditional DEA, the proposed interval DEA approach measures the efficiency of each DMU relative to the inefficiency frontier, also called the input frontier, and is called the worst relative efficiency or pessimistic efficiency. On the other hand, in traditional DEA, the efficiency of each DMU is measured relative to the efficiency frontier and is called the best relative efficiency or optimistic efficiency. The pair of proposed interval DEA models takes into account the crisp, ordinal, and interval data, as well as non-discretionary factors, simultaneously for measurement of relative efficiencies of DMUs. Two numeric examples will be provided to illustrate the applicability of the interval DEA models.     

An analytic derivation of admissible efficient frontier with borrowing

The admissible efficient portfolio selection problem for risky assets has been discussed by Zhang and Nie. In this paper, the admissible efficient portfolio model is proposed under the assumption that there exists the borrowing (money or a risk free asset) case. The admissible efficient frontiers are developed by the spreads of expected return and risk from admissible errors. The analytic forms of the admissible efficient frontiers when short sales are not allowed on all risky assets are derived from two cases: the borrowing with an upper bound constraint, or without an upper bound constraint. The influence on the admissible efficient frontier is explained under the different interest rates of the borrowing. The differences between the results with the borrowing and the results without the borrowing is revealed by a real numerical example.     

Ranking efficient decision making units in data envelopment analysis based on reference frontier share

Data envelopment analysis (DEA) is a powerful technique for performance evaluation of decision making units (DMUs). Ranking efficient DMUs based on a rational analysis is an issue that yet needs further research. The impact of each efficient DMU in evaluation of inefficient DMUs can be considered as additional information to discriminating among efficient DMUs. The concept of reference frontier share is introduced in which the share of each efficient DMU in construction of the reference frontier for evaluating inefficient DMUs is considered. For this purpose a model for measuring the reference frontier share of each efficient DMU associated with each inefficient one is proposed and then a total measure is provided based on which the ranking is made. The new approach has the capability for ranking extreme and non-extreme efficient DMUs. Further, it has no problem in dealing with negative data. These facts are verified by theorems, discussions and numerical examples.     

不存在无风险资产的投资组合灵敏度分析

本文研究了M－V证券投资组合灵敏度分析方法。考虑了不存在无险资产时证券预睡益率和协方差矩阵存在扰动的情形，给出了最优投资组合有效边缘的漂移方程及组合扩展路径。     

Super-efficiency in stochastic data envelopment analysis: An input relaxation approach

This paper addresses super-efficiency issue based on input relaxation model in stochastic data envelopment analysis. The proposed model is not limited to using the input amounts of evaluating DMU, and one can obtain a total ordering of units by using this method. The input relaxation super-efficiency model is developed in stochastic data envelopment analysis, and its deterministic equivalent, also, is derived which is a nonlinear program. Moreover, it is shown that the deterministic equivalent of the stochastic super-efficiency model can be converted to a quadratic program. As an empirical example, the proposed method is applied to the data of textile industry of China to rank efficient units. Finally, when allowable limits of data variations for evaluating DMU are permitted, the sensitivity analysis of the proposed model is discussed.     

Generating strong defining hyperplanes of the production possibility set in data envelopment analysis

In data envelopment analysis (DEA), identification of the strong defining hyperplanes of the empirical production possibility set (PPS) is important, because they can be used for determining rates of change of outputs with change in inputs. Also, efficient hyperplanes determine the nature of returns to scale. The present work proposes a method for generating all linearly independent strong defining hyperplanes (LISDHs) of the PPS passing through a specific decision making unit (DMU). To this end, corresponding to each efficient unit, a perturbed inefficient unit will be defined and, using at most m+s linear programs, all LISDHs passing through the DMU will be determined, where m and s are the numbers of inputs and outputs, respectively.     

Heuristic algorithms for the cardinality constrained efficient frontier

This paper examines the application of genetic algorithm, tabu search and simulated annealing metaheuristic approaches to finding the cardinality constrained efficient frontier that arises in financial portfolio optimisation. We consider the mean-variance model of Markowitz as extended to include the discrete restrictions of buy-in thresholds and cardinality constraints. Computational results are reported for publicly available data sets drawn from seven major market indices involving up to 1318 assets. Our results are compared with previous results given in the literature illustrating the effectiveness of the proposed metaheuristics in terms of solution quality and computation time.     

Forward search outlier detection in data envelopment analysis

In this paper we tackle the problem of outlier detection in data envelopment analysis (DEA). We propose a procedure where we merge the super-efficiency DEA and the forward search. Since DEA provides efficiency scores which are not parameters to fit the model to the data, we introduce a distance, to be monitored along the search. This distance is obtained through the integration of a regression model and the super-efficiency DEA. We simulate a Cobb-Douglas production function and we compare the super-efficiency DEA and the forward search analysis in both uncontaminated and contaminated settings. For inference about outliers, we exploit envelopes obtained through Monte Carlo simulations.     

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.     

On classifying inputs and outputs in DEA: A revised model

Cook and Zhu [Cook, W.D., Zhu, J., 2007. Classifying inputs and outputs in data envelopment analysis. European Journal of Operational Research 180, 692–699] introduced a new method to determine whether a measure is an input or an output. In practice, however, their method may produce incorrect efficiency scores due to a computational problem as result of introducing a large positive number to the model. This note introduces a revised model that does not need such a large positive number.