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.     

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.     

关于要素不可控条件下的DEA模型的特性分析

在 DEA模型的实际应用中 ,往往要求某些要素的投入是不能改变的 ,即这些要素对于决策者是不可控的 .1 986年 ,Banker和 Morey为了克服此类问题 ,提出了具有固定不变投入的 DEA模型 ,被称为要素不可控条件下的 DEA模型 .详细研究了要素不可控条件下的 DEA模型 ,并且建立了要素不可控条件下的DEA(弱 )有效性与 (弱 ) Pareto解的等价关系     

On some generalization of the DEA models

Inadequate results may arise in some instances of DEA model applications. For example, a data envelopment analysis (DEA) model may show ‘a notoriously inefficient unit’ as an efficient one. In addition, too many efficient units may appear in some DEA models. An elegant and subtle approach was proposed to deal with these problems, which is based on incorporating domination cones in DEA models. Yu, Wei and Brockett suggested the generalized DEA (GDEA) model that unifies and extends most of the well-known DEA models based on using domination cones. In this paper, we propose a model that is more general than the GDEA model, on the one hand, as it covers situations that the GDEA model cannot describe. On the other hand, our model enables one to construct step-by-step any model from the family of the GDEA models by incorporating artificial units and rays in the space of inputs and outputs in the Banker, Charnes, Cooper (BCC) model, which makes the process of model construction visible and more understandable. Moreover, we show that any GDEA model can be approximated by some BCC model.     

Returns to scale in multiplicative models in data envelopment analysis

One class of models introduced in DEA is called multiplicative models, in which, as shown by Banker and Maindiratta (Manag. Sci. 32:126–135, 1986), the piecewise linear frontiers usually employed in DEA are replaced by a frontier that is piecewise Cobb-Douglas(=log  linear). Banker and Maindiratta (Manag. Sci. 32:126–135, 1986) introduced a model to identify the most productive scale size pattern, and Banker et al. (Eur. J. Oper. Res. 154:345–362, 2004) presented a two-stage method for the identification of returns to scale (RTS) in multiplicative models. In this paper it is shown that both the RTS situation and the MPSS pattern could be determined by a single model in one step. The new method is important in the computational point of view.     

DEA方法进行规模收益分析的几点注记

自从 Banker等人 ( 1 984)用 DEA方法进行规模收益分析 ,已有越来越多的学者也在进行相关的研究 .研究主要从两个方面进行 :基于 DEA输入模型的方法和基于 DEA输出模型的方法 .这两类方法所应用的模型、条件、结论都有所不同 ,但在应用中有些文章将它们混淆在一起 .本文针对应用中容易出现的错误 ,给出反例和正确的判别规模收益状况的充分必要条件 .并由规模收益的定义 ,建议使用 DEA—输出模型进行规模收益分析 .     

Relationships between Pareto optimality in multi-objective 0–1 linear programming and DEA efficiency

In a paper that appeared in EJOR, F.H.F. Liu, C.-C. Huang and Y.-L. Yen analyzed the relationships between data envelopment analysis (DEA) and multi-objective linear programming with binary variables (MOBLP). In particular they stated that a DEA efficient solution, in the sense of Banker, Charnes and Cooper (BCC), is a Pareto optimal solution of MOBLP. In this note, we show that this property is false, we establish the correct relations between these two notions, and we illustrate these statements by some examples.     

Data Envelopment Analysis with Preference Structure

It is important to consider the decision making unit (DMU)'s or decision maker's preference over the potential adjustments of various inputs and outputs when data envelopment analysis (DEA) is employed. On the basis of the so-called Russell measure, this paper develops some weighted non-radial CCR models by specifying a proper set of ‘preference weights’ that reflect the relative degree of desirability of the potential adjustments of current input or output levels. These input or output adjustments can be either less or greater than one; that is, the approach enables certain inputs actually to be increased, or certain outputs actually to be decreased. It is shown that the preference structure prescribes fixed weights (virtual multiplier bounds) or regions that invalidate some virtual multipliers and hence it generates preferred (efficient) input and output targets for each DMU. In addition to providing the preferred target, the approach gives a scalar efficiency score for each DMU to secure comparability. It is also shown how specific cases of our approach handle non-controllable factors in DEA and measure allocative and technical efficiency. Finally, the methodology is applied with the industrial performance of 14 open coastal cities and four special economic zones in 1991 in China. As applied here, the DEA/preference structure model refines the original DEA model's result and eliminates apparently efficient DMUs.     

An enhanced DEA Russell graph efficiency measure

The measurement of productive efficiency is an issue of great interest. Since Farrell (Farrell, M.J., 1957. Journal of Royal Statistical Society, Series A 120, 253) implemented the first measure of technical efficiency, many researchers have developed new measures or have extended the already existing ones. The beginning of Data Envelopment Analysis (DEA) meant a new way of empirically measuring productive efficiency. Under some specific technologies, Farrell's measure was implemented giving rise to the first DEA models, CCR (Charnes, A., Cooper, W.W., Rhodes, E., 1978. European Journal of Operational Research 2, 429) and BCC (Banker, R.D., Charnes, A., Cooper, W.W., 1984. Management Science, 1078). The fact that these measures only account for radial inefficiency has motivated the development of the so-called Global Efficiency Measures (GEMs) (Cooper, W.W., Pastor, J.T., 1995. Working Paper, Departamento de Estadı́stica e Investigación Operativa, Universidad de Alicante, Alicante, Spain). In this paper we propose a new GEM inspired by the Russell Graph Measure of Technical Efficiency which avoids the computational and interpretative difficulties with this latter measure. Additionally, the new measure satisfies some other desirable properties.     

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.     

Some remarks on the two-level DEA model

The two-level DEA model was introduced to increase the discriminational power of Data Envelopment Analysis (DEA) models. This nonlinear model was presented by Meng et al. (2008) [3], and then converted into a linear model by Kao (2008) [4].In this paper two subjects will be discussed: first, we show that the two-level DEA model is a special case of DEA models where weight restrictions are applied. Then, we express that the nonlinear model is equivalent to the conventional DEA model.     

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.     

DEA-based hypothesis tests for comparing two groups of decision making units

In this paper, we develop five statistical tests to compare the efficiencies of different groups of DMUs. We consider a data generating process (DGP) that models the deviation of the output from the best practice frontier as the sum of two components, a one-sided inefficiency term and a two-sided random noise term. We use simulation to evaluate the performance of the five tests against the Banker tests (Banker, 1993) that were designed for DGPs containing a single one-sided error term. It is found that while the Banker tests are very effective when efficiency dominates noise, the tests developed in this paper perform better than the Banker tests when noise levels are significant.     

The impact of supervisory monitoring on high-end retail sales productivity

Based on a two-stage analysis of a panel of data on 12 outlets of a high-end retailer for 24 months, we investigate how the level of supervisory monitoring affects retail sales productivity. In the first stage, we use Data Envelopment Analysis (DEA) to compute the relative productivity of retail outlets in using their labor and capital resources to generate store sales. In the second stage, we regress the logarithm of DEA scores on contextual variables to obtain consistent estimators of the impact of contextual variables on productivity (Banker and Natarajan in Operation Research 56:48–58, 2008). Contrary to agency theoretic prediction that supervisory monitoring leads to an increase in retail sales productivity, our empirical results indicate that the higher the level of supervisory monitoring, the lower is the retail sales productivity for high-end retail outlets.     

Farrell revisited–Visualizing properties of DEA production frontiers

The contributions of the paper are threefold: (i) compare with mathematical rigour the data envelopment analysis (DEA) model of Charnes, Cooper, and Rhodes and the Farrell model exhibiting constant returns to scale, (ii) reinterpret the contribution of Farrell and Fieldhouse that extended the analysis to variables returns to scale and establish the connection with the approach in Banker, Charnes, and Cooper, and (iii) provide graphical visualization of properties of the frontier function. Both papers by Farrell emphasized the importance of graphical visualization of non-parametric frontier functions, but, to our knowledge, this is seldom followed up in the literature. We use a graphical package (EffiVision) with a numerical representation of the frontier functions, representing the contemporary development of visualization. By making suitable cuts through the DEA frontier in multidimensional space, various graphical representations of features of economic interest can be done. Development of ray average cost function and scale elasticity plots are novel illustrations.     

An input relaxation measure of efficiency in stochastic data envelopment analysis

We introduce stochastic version of an input relaxation model in data envelopment analysis (DEA). The input relaxation model, recently developed in DEA, is useful to resource management [e.g. G.R. Jahanshahloo, M. Khodabakhshi, Suitable combination of inputs for improving outputs in DEA with determining input congestion, Appl. Math. Comput. 151(1) (2004) 263–273]. This model allows more changes in the input combinations of decision making units than those in the observed inputs of evaluating decision making units. Using this extra flexibility in input combinations we can find better outputs. We obtain a non-linear deterministic equivalent to this stochastic model. It is shown that under fairly general conditions this non-linear model can be replaced by an ordinary deterministic DEA model. The model is illustrated using a real data set.     

A new integrated DEA model for finding most BCC-efficient DMU

In many applications of widely recognized technique, DEA, finding the most efficient DMU is desirable for decision maker. Using basic DEA models, decision maker is not able to identify most efficient DMU. Amin and Toloo [Gholam R. Amin, M. Toloo, Finding the most efficient DMUs in DEA: an improved integrated model. Comput. Ind. Eng. 52 (2007) 71–77] introduced an integrated DEA model for finding most CCR-efficient DMU. In this paper, we propose a new integrated model for determining most BCC-efficient DMU by solving only one linear programming (LP). This model is useful for situations in which return to scale is variable, so has wider range of application than other models which find most CCR-efficient DMU. The applicability of the proposed integrated model is illustrated, using a real data set of a case study, which consists of 19 facility layout alternatives.     

Productivity change and innovation in Norwegian electricity distribution companies

Regulators of electricity distribution networks have typically applied Data Envelopment Analysis (DEA) to cross-section data for benchmarking purposes. However, the use of panel data to analyse the impact of regulatory policies on productivity change over time is less frequent. The main purpose of this paper is to construct a Malmquist productivity index to examine the recent productivity change experienced by Norwegian distribution companies between 2004 and 2007. The Malmquist index is decomposed in order to explore the sources of productivity change, and to identify the innovator companies that pushed the frontier forward each year. The input and output variables considered are those used by the Norwegian regulator. In order to reflect appropriately the exogenous conditions where the companies operate, the efficiency model used in this paper incorporates geography variables as outputs of the DEA model. Unlike the model used by the regulator, we included virtual weight restrictions in the DEA formulation to correct the biases in the DEA results that may be associated to a judicious choice of weights by some of the companies.