On single-stage DEA models with weight restrictions

The literature on data envelopment analysis (DEA) often employs multiplier models that incorporate very small (theoretically infinitesimal) lower bounds on the input and output weights. Computational problems arising from the solution of such programs are well known. In this paper we identify an additional theoretical problem that may arise if such bounds are used in a multiplier model with weight restrictions. Namely, we show that the use of small lower bounds may lead to the identification of an efficient target with negative inputs. We suggest a corrected model that overcomes this problem

Solving DEA models in a single optimization stage: Can the non-Archimedean infinitesimal be replaced by a small finite epsilon?

Single-stage DEA models aim to assess the input or output radial efficiency of a decision making unit and potential mix inefficiency in a single optimization stage. This is achieved by incorporating the sum of input and output slacks, multiplied by a small (theoretically non-Archimedean infinitesimal) value epsilon in the envelopment model or, equivalently, by using this value as the lower bound on the input and output weights in the dual multiplier model. When this approach is used, it is common practice to select a very small value for epsilon. This is based on the expectation that, for a sufficiently small epsilon, the radial efficiency and optimal slacks obtained by solving the single-stage model should be approximately equal to their true values obtained by the two separate optimization stages. However, as well-known, selecting a small epsilon may lead to significant computational inaccuracies. In this paper we prove that there exists a threshold value, referred to as the effective bound, such that, if epsilon is smaller than this bound, the solution to the single-stage program is not approximate but precise (exactly the same as in the two-stage approach), provided there are no computational errors

Consistent weight restrictions in data envelopment analysis

It has recently been shown that the incorporation of weight restrictions in models of data envelopment analysis (DEA) may induce free or unlimited production of output vectors in the underlying production technology, which is expressly disallowed by standard production assumptions. This effect may either result in an infeasible multiplier model with weight restrictions or remain undetected by normal efficiency computations. The latter is potentially troubling because even if the efficiency scores appear unproblematic, they may still be assessed in an erroneous model of production technology. Two approaches to testing the existence of free and unlimited production have recently been developed: computational and analytical. While the latter is more straightforward than the former, its application is limited only to unlinked weight restrictions. In this paper we develop several new analytical conditions for a larger class of unlinked and linked weight restrictions

Consistency of returns-to-scale characterizations of production frontiers with respect to model specification

Returns-to-scale (RTS) characterizations and the underlying notion of scale elasticity are important characteristics of production frontiers, in both parametric and nonparametric methodologies of efficiency and productivity analysis. In practical applications of these methodologies, the model of technology is often experimented with and modified before it is finalized, which involves, for example, a change of the data set, incorporation, exclusion or aggregation of inputs and outputs, or experimentation with the production assumptions, or axioms, on which the model is based. While it is well-known how such modifications of technology affect the efficiency scores, their effect on the RTS characterization of the production frontier has not been sufficiently explored in the literature. In this paper we obtain several general results that clarify this issue

Optimal solutions of multiplier DEA models

Conventional models of data envelopment analysis (DEA) are based on the constant and variable returns-to-scale production technologies. Any optimal input and output weights of the multiplier DEA models based on these technologies are interpreted as being the most favourable for the decision making unit (DMU) under the assessment when the latter is benchmarked against the set of all observed DMUs. In this paper we consider a very large class of DEA models based on arbitrary polyhedral technologies, which includes almost all known convex DEA models. We highlight the fact that the conventional interpretation of the optimal input and output weights in such models is generally incorrect, which raises a question about the meaning of multiplier models. We address this question and prove that the optimal solutions of such models show the DMU under the assessment in the best light in comparison to the entire technology, but not necessarily in comparison to the set of observed DMUs. This result allows a clear and meaningful interpretation of the optimal solutions of multiplier models, including known models with a complex constraint structure whose interpretation has been problematic and left unaddressed in the existing literature

Cone extensions of polyhedral production technologies

In data envelopment analysis, cone extensions of production technologies are often used for the estimation of scale efficiency of decision making units. Furthermore, the non-increasing and non-decreasing returns-to-scale (NIRS and NDRS) technologies are often used for their returns-to-scale characterization. Although a number of new production technologies have recently been developed in the literature, their cone, NIRS and NDRS extensions have not always been fully explored. In this paper, we obtain general results that show how these extensions can be obtained, for an arbitrary polyhedral technology. We illustrate the usefulness of our results by examples