Victor V. Podinovski

Pitfalls and protocols in DEA

Abstract The practical application of data envelopment analysis (DEA) presents a range of procedural issues to be examined and resolved including those relating to the homogeneity of the units under assessment, the input/output set selected, the measurement of those selected variables and the weights attributed to them. Each of these issues can present difficulties in practice. The purpose of this paper is to highlight some of the pitfalls that have been identified in application papers under each of these headings and to suggest protocols to avoid the pitfalls and guide the application of the methodology.

Weak Disposability in Nonparametric Production Analysis: Reply to Färe and Grosskopf

Fare and Grosskopf (this issue) claim that a single abatement factor suffices for modeling weak disposability in nonparametric production models, and that the Kuosmanen (2005) technology that uses multiple abatement factors is larger than necessary. This article demonstrates by a numerical example that a single abatement factor does not suffice to capture all feasible production plans, and that its use leads to the violation of convexity, one of the maintained assumptions of the model. We also prove that the Kuosmanen technology is the correct minimum extrapolation technology under the stated axioms.

Production trade-offs and weight restrictions in data envelopment analysis

In this paper we suggest two equivalent ways in which the information about production trade-offs between the inputs and outputs can be incorporated into the models of data envelopment analysis (DEA). Firstly, this can be implemented by modifying envelopment DEA models. Secondly, the same information can be captured using weight restrictions in multiplier DEA models. Unlike other methods used for the assessment of weight restrictions, for example those based on value judgements or monetary considerations, the trade-off approach developed in this paper ensures that the radial target of any inefficient unit is technologically realistic and, therefore, the efficiency measure retains its traditional meaning of the extreme radial improvement factor. In other words, this paper suggests that ‘technology thinking’ could be used instead of ‘value thinking’ in the construction of weight restrictions, which offers real practical advantages. The method is equally applicable to the models under constant and variable returns-to-scale assumptions.

Modelling weak disposability in data envelopment analysis under relaxed convexity assumptions

The treatment of undesirable (bad) outputs in models of efficiency and productivity analysis often requires replacing the assumption of free disposability of outputs by their weak disposability. In a recent publication the authors showed that the Kuosmanen technology is the only correct representation of the fully convex technology exhibiting weak disposability of bad and good outputs. In this paper we relax the assumption of full convexity and consider two further possibilities: the case in which only the output sets are assumed convex and the case in which no convexity is assumed at all. In the first case we show that, although the traditional Shephard technology of nonparametric production analysis satisfies the assumption of convex output sets, it is larger than necessary. Based on the minimum extrapolation principle, we develop a correct model that is based on the assumed axioms. The second case leads to the development of a weakly disposable analogue of the free disposable hull. To complete our study, we give a full axiomatic definition of the Shephard technology.

Side effects of absolute weight bounds in DEA models

Abstract This paper analyses effects of incorporating absolute weight bounds for input and output weights in the classical models of data envelopment analysis (DEA). It is shown that a DEA model with such restrictions may not maximise the relative efficiency of the assessed decision making unit (DMU) and may not find the set of weights representing the assessed DMU in the best light in comparison to the other DMUs. Consequently, it may produce misleading target values for an inefficient DMU and a wrong reference set of efficient peers. A way of avoiding these “side effects” is based on a utilisation of a maximin DEA model, equivalent to the classical DEA model if no additional restrictions are imposed.

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.

A simple derivation of scale elasticity in data envelopment analysis

In this paper, we suggest a simple derivation of the formulae for the scale elasticity in the variable returns-to-scale technology as used in data envelopment analysis. Our development is consistent with the existing literature but the proof is much shorter and applies to the general case without any simplifying conditions.

Suitability and redundancy of non-homogeneous weight restrictions for measuring the relative efficiency in DEA

Abstract Weight restrictions are non-homogeneous if they are formulated as linear ‘less than or equal to’ inequalities with a non-zero constant on the right-hand side. Absolute weight bounds are typical examples. It has recently been shown that, in the presence of such restrictions, the fractional linear data envelopment analysis (DEA) model and its linear forms may incorrectly evaluate the maximum relative efficiency of the assessed unit. This paper investigates the problem further, and identifies certain types of non-homogeneous restrictions that do not cause the observed error. Thus the relative efficiency is always assessed correctly if, in the same CCR model, no positive lower bounds are imposed on any of the input weights and no upper bounds are imposed on any of the output weights. The redundancy of certain types of weight restrictions in DEA models is also considered. Based on this, the traditional use of small positive constants to separate weights from zero in DEA models is questioned.

On the linearisation of reference technologies for testing returns to scale in FDH models

Abstract One of the methods of testing returns to scale (RTS) in data envelopment analysis is based on the use of specially constructed reference technologies. Recently Kerstens and Vanden Eeckaut constructed such technologies for testing RTS in the free disposal hull (FDH) model. The use of these technologies requires solving mixed integer non-linear programming models. In this note we construct equivalent linear models, which have an obvious computational advantage.

Differential Characteristics of Efficient Frontiers in Data Envelopment Analysis

The implicit definition and nondifferentiability of efficient frontiers used in data envelopment analysis are two major obstacles to obtaining their differential characteristics, including various elasticity measures and marginal rates of substitution. In this paper we invoke the theorem of the directional derivative of the optimal value function and show how this can be used to define and calculate the required elasticities without any simplifying assumptions. This approach allows us to extend the known elasticity measures and introduce new ones to the entire efficient frontier, including all its extreme points, in one single development. We also construct linear programs that are required for the calculation of elasticity measures. Our main development is undertaken in the variable returns-to-scale technology but extends to other polyhedral technologies of efficiency analysis.