DEA cross-efficiency evaluation under variable returns to scale

Cross-efficiency evaluation in data envelopment analysis (DEA) has been developed under the assumption of constant returns to scale (CRS), and no valid attempts have been made to apply the cross-efficiency concept to the variable returns to scale (VRS) condition. This is due to the fact that negative VRS cross-efficiency arises for some decision-making units (DMUs). Since there exist many instances that require the use of the VRS DEA model, it is imperative to develop cross-efficiency measures under VRS. We show that negative VRS cross-efficiency is related to free production of outputs. We offer a geometric interpretation of the relationship between the CRS and VRS DEA models. We show that each DMU, via solving the VRS model, seeks an optimal bundle of weights with which its CRS-efficiency score, measured under a translated Cartesian coordinate system, is maximized. We propose that VRS cross-efficiency evaluation should be done via a series of CRS models under translated Cartesian coordinate systems. The current study offers a valid cross-efficiency approach under the assumption of VRS—one of the most common assumptions in performance evaluation done by DEA.     

On relations between DEA-risk models and stochastic dominance efficiency tests

In this paper, several concepts of portfolio efficiency testing are compared, based either on data envelopment analysis (DEA) or the second-order stochastic dominance (SSD) relation: constant return to scale DEA models, variable return to scale (VRS) DEA models, diversification-consistent DEA models, pairwise SSD efficiency tests, convex SSD efficiency tests and full SSD portfolio efficiency tests. Especially, the equivalence between VRS DEA model with binary weights and the SSD pairwise efficiency test is proved. DEA models equivalent to convex SSD efficiency tests and full SSD portfolio efficiency tests are also formulated. In the empirical application, the efficiency testing of 48 US representative industry portfolios using all considered DEA models and SSD tests is presented. The obtained efficiency sets are compared. A special attention is paid to the case of small number of the inputs and outputs. It is empirically shown that DEA models equivalent either to the convex SSD test or to the SSD portfolio efficiency test work well even with quite small number of inputs and outputs. However, the reduced VRS DEA model with binary weights is not able to identify all the pairwise SSD efficient portfolios.     

Bridging the gap between the constant and variable returns-to-scale models: selective proportionality in data envelopment analysis

In data envelopment analysis (DEA), the use of constant returns-to-scale (CRS) models requires the assumption of full proportionality between all inputs and outputs. Often such proportionality cannot be assumed, although there may be a subset of outputs proportional to a subset of inputs. By using the variable returns-to-scale (VRS) model, this information is effectively ignored and the efficiency of units is overestimated. This paper develops a hybrid approach that combines the assumption of CRS with respect to the selected sets of inputs and outputs, while preserving the VRS assumption with respect to the remaining indicators. The resulting hybrid returns-to-scale models exhibit better discrimination than the VRS model. In certain cases, their discrimination surpasses that of the CRS model, an example of which is given.     

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 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.     

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.     

A full-scale investigation of the directional returns to scale in data envelopment analysis

Returns to scale is considered as one of the important concepts in data envelopment analysis (DEA) which can be useful for deciding to increase or decrease the size of a particular decision making unit. Traditional returns to scale on the efficient surface of the production possibility set with variable returns to scale (VRS) technology is introduced as a ratio of proportional changes of output components to proportional changes of input components. However, a problem which may arise in the real world is the impossibility or undesirability of proportional change in the input or output components. One of the attempts which is made to solve the aforementioned problem is the work of Yang et al., 2014. They have introduced the “directional returns to scale” in the DEA framework and have proposed some procedures to estimate and measure it. In this paper, the introduced directional returns to scale is investigated from a new perspective based on the defining hyperplanes of the production possibility set with VRS technology. We propose some algebraic equations and linear programming models which in addition to measuring the directional returns to scale, they enable us to analyse it. Moreover, we introduce the concepts of the best input and output direction vectors for expansion of input components or compression of output components, respectively, and propose two linear programming models in order to obtain these directions. The presented equations and models are demonstrated using a case study and numerical examples.     

The impossibility of convex constant returns-to-scale production technologies with exogenously fixed factors

The extensions to the variable (VRS) and the constant (CRS) returns-to-scale models developed by Banker and Morey are considered among the main approaches to the incorporation of exogenously fixed factors in models of data envelopment analysis (DEA). Recently, Syrjänen showed that the Banker and Morey CRS technology is not convex. Taking into account that its subset VRS technology is explicitly assumed convex, this observation leads to difficulties with explaining the fundamental production assumptions of the CRS extension. Motivated by the example of Syrjänen, the contribution of this paper is twofold. First, we show that the nonconvex Banker and Morey CRS technology is nevertheless a suitable reference technology for the assessment of scale efficiency. Second, we ask if a convex technology could be constructed that would “correct” the nonconvexity of the CRS technology of Banker and Morey. The answer to this is negative: one consequence of assuming both convexity and ray unboundness with fixed exogenous factors is that we can always “mix-and-match” discretionary and nondiscretionary factors taken from different units, arriving at totally unrealistic production plans. This demonstrates that generally there exists no meaningful convex CRS technology with exogenously fixed factors that can be used in its own right, apart from its use as a reference technology in the measurement of scale efficiency.     

A new non-oriented model for classifying flexible measures in DEA

In conventional data envelopment analysis (DEA), measures are classified as either input or output. However, in some real cases there are variables which act as both input and output and are known as flexible measures. Most of the previous suggested models for determining the status of flexible measures are oriented. One important issue of these models is that unlike standard DEA, even under constant returns to scale the input- and output-oriented model may produce different efficiency scores. Also, can be expected a flexible measure is selected as an input variable in one model but an output variable in the other model. In addition, in all of the previous studies did not point to variable returns to scale (VRS), but the VRS assumption is prevailed on many real applications. To deal with these issues, this study proposes a new non-oriented model that not only selects the status of each flexible measure as an input or output but also determines returns to scale status. Then, the aggregate model and an extension with the negative data related to the proposed approach are presented.     

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.     

互联网金融背景下我国上市商业银行的效率实证研究

本文通过数据包络分析(DEA)中的CRS模型和VRS模型,对我国16家上市商业银行于2007至2014年在互联网金融背景下的效率进行了测度。根据16家商业银行的综合效率、纯技术效率、规模效率的变动情况,来分析互联网金融对商业银行造成影响和冲击。结果表明,商业银行的整体效率波动与互联网金融环境的变化基本一致,并且,商业银行的规模效率变化与综合技术效率变化保持一致,股份制商业银行的规模效率一直高于国有商业银行。     

Performance assessment of secondary schools: the snapshot of a country taken by DEA

This paper describes a performance assessment of Portuguese secondary schools using data envelopment analysis (DEA). The assessment adopts a perspective where schools are viewed as promoting students achievement given their characteristics in terms of academic abilities and socio-economic background. Our sample comprised all secondary schools in Portugal with both basic and secondary education levels. Two types of DEA analysis are performed: one using an output-oriented model that restricts output (exam scores) weights to be linked to the number of students that have done that exam in the school, and the other using a model that restricts factor weights to be equal for all schools. In this model the weight restrictions are linked to the total number of exams done nationally. The first model is well suited for identifying worst performing schools and to assess schools that may specialize in certain subjects, whereas the latter is best suited for improving discrimination between best performing schools when pursuing the identification of benchmarks, as well as to construct performance rankings.     

DEA model with shared resources and efficiency decomposition

Data envelopment analysis (DEA) has proved to be an excellent approach for measuring performance of decision making units (DMUs) that use multiple inputs to generate multiple outputs. In many real world scenarios, DMUs have a two-stage network process with shared input resources used in both stages of operations. For example, in hospital operations, some of the input resources such as equipment, personnel, and information technology are used in the first stage to generate medical record to track treatments, tests, drug dosages, and costs. The same set of resources used by first stage activities are used to generate the second-stage patient services. Patient services also use the services generated by the first stage operations of housekeeping, medical records, and laundry. These DMUs have not only inputs and outputs, but also intermediate measures that exist in-between the two-stage operations. The distinguishing characteristic is that some of the inputs to the first stage are shared by both the first and second stage, but some of the shared inputs cannot be conveniently split up and allocated to the operations of the two stages. Recognizing this distinction is critical for these types of DEA applications because measuring the efficiency of the production for first-stage outputs can be misleading and can understate the efficiency if DEA fails to consider that some of the inputs generate other second-stage outputs. The current paper develops a set of DEA models for measuring the performance of two-stage network processes with non splittable shared inputs. An additive efficiency decomposition for the two-stage network process is presented. The models are developed under the assumption of variable returns to scale (VRS), but can be readily applied under the assumption of constant returns to scale (CRS). An application is provided.     

Impacts of random noise and specification on estimates of capacity derived from data envelopment analysis

Data envelopment analysis (DEA) is widely used to estimate the efficiency of firms and has also been proposed as a tool to measure technical capacity and capacity utilization (CU). Random variation in output data can lead to downward bias in DEA estimates of efficiency and, consequently, upward bias in estimates of technical capacity. This can be particularly problematic for industries such as agriculture, aquaculture and fisheries where the production process is inherently stochastic due to environmental influences. This research uses Monte Carlo simulations to investigate possible biases in DEA estimates of technically efficient output and capacity output attributable to noisy data and investigates the impact of using a model specification that allows for variable returns to scale (VRS). We demonstrate a simple method of reducing noise induced bias when panel data is available. We find that DEA capacity estimates are highly sensitive to noise and model specification. Analogous conclusions can be drawn regarding DEA estimates of average efficiency.     

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.     

Chapter 12 New uses of DEA and statistical regressions for efficiency evaluation and estimation — with an illustrative application to public secondary schools in Texas

This paper examines new combinations of Data Envelopment Analysis (DEA) and statistical approaches that can be used to evaluate efficiency within a multiple-input multiple-output framework. Using data on five outputs and eight inputs for 638 public secondary schools in Texas, unsatisfactory results are obtained initially from both Ordinary Least Squares (OLS) and Stochastic Frontier (SF) regressions run separately using one output variable at-a-time. Canonical correlation analysis is then used to aggregate the multiple outputs into a single aggregate output, after which separate regressions are estimated for the subsets of schools identified as efficient and inefficient by DEA. Satisfactory results are finally obtained by a joint use of DEA and statistical regressions in the following manner. DEA is first used to identify the subset of DEA-efficient schools. The entire collection of schools is then comprehended in a single regression with dummy variables used to distinguish between DEA-efficient and DEA-inefficient schools. The input coefficients are positive for the efficient schools and negative and statistically significant for the inefficient schools. These results are consistent with what might be expected from economic theory and are informative for educational policy uses. They also extend the treatments of production functions usually found in the econometrics literature to obtain one regression relation that can be used to evaluate both efficient and inefficient behavior.     

A linear programming framework for free disposal hull technologies and cost functions: Primal and dual models

The free disposal hull (FDH) model, introduced by Deprins et al. [The Performance of Public Enterprises Concepts and Measurements, Elsevier, 1984], is based on a representation of the production technology given by observed production plans, imposing strong disposability of inputs and outputs but without the convexity assumption. In its traditional form, the FDH model assumes implicitly variable returns to scale (VRS) and the model was solved by a mixed integer linear program (MILP). The MILP structure is often used to compare the FDH model to data envelopment analysis (DEA) models although an equivalent FDH LP model exists (see Agrell and Tind [Journal of Productivity Analysis 16 (2) (2001) 129]). More recently, specific returns to scale (RTS) assumptions have been introduced in FDH models by Kerstens and Vanden Eeckaut [European Journal of Operational Research 113 (1999) 206], including non-increasing, non-decreasing, or constant returns to scale (NIRS, NDRS, and CRS, respectively). Podinovski [European Journal of Operational Research 152 (2004) 800] showed that the related technical efficiency measures can be computed by mixed integer linear programs. In this paper, the modeling proposed here goes one step further by introducing a complete LP framework to deal with all previous FDH models.     

Slack-adjusted DEA for time series analysis: Performance measurement of Japanese electric power generation industry in 1984–1993

Using a new slack-adjusted data envelopment analysis (SA-DEA) model which explicitly incorporates an influence of slacks into its efficiency measurement, this study discusses a use of various efficiencies and index measures for DEA dynamic analysis. An analytical formulation to determine the type of return to scale (RTS) is proposed for the new DEA model. This paper mathematically discusses when multiple solutions occur on RTS and how to deal with such a difficulty. As an important case study, this paper applies the proposed DEA approach to examine the performance of Japanese electric power generation companies from 1984 to 1993. Two policy implications are suggested for guiding the Japanese electric power industry.     

Graphing Calculator Use in Algebra Teaching

This study examines graphing calculator technology availability, characteristics of teachers who use it, teacher attitudes, and how use reflects changes to algebra curriculum and instructional practices. Algebra I and Algebra II teachers in 75 high school and junior high/middle schools in a diverse region of a northwestern state were surveyed. Forty of the 75 schools (53%) returned a total of 109 individual surveys. Results indicated that: (1) While 78% of teachers have some access to the technology, only 28% use it regularly. (2) Statistically significant relationships exist between use and age, years of experience, teaching assignment, and teaching level. (3) Respondents view graphical solution methods as secondary to symbolic methods. (4) Teachers are more receptive to using technology to supplement rather than expand the curriculum.     

The measurement of relative efficiency using data envelopment analysis with assurance regions that link inputs and outputs

The most popular weight restrictions are assurance regions (ARs), which impose ratios between weights to be within certain ranges. ARs can be categorized into two types: ARs type I (ARI) and ARs type II (ARII). ARI specify bounds on ratios between input weights or between output weights, whilst ARII specify bounds on ratios that link input to output weights. DEA models with ARI successfully maximize relative efficiency, but in the presence of ARII the DEA models may under-estimate relative efficiency or may become infeasible. In this paper we discuss the problems that can occur in the presence of ARII and propose a new nonlinear model that overcomes the limitations discussed. Also, the dual model is described, which enables the assessment of relative efficiency when trade-offs between inputs and outputs are specified. The application of the model developed is illustrated in the efficiency assessment of Portuguese secondary schools.