José L. Ruiz

AN ENHANCED DEA RUSSELL GRAPH EFFICIENCY MEASURE

Abstract 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, Farrells 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 Estadistica e Investigacion 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.

A fuzzy mathematical programming approach to the assessment of efficiency with DEA models

In many real applications, the data of production processes cannot be precisely measured. This is particularly worrying when assessing efficiency with frontier-type models, such as data envelopment analysis (DEA) models, since they are very sensitive to possible data errors. For this reason, the possibility of having available a methodology that allows the analyst to deal with imprecise data becomes an issue of great interest in these contexts. To that end, we develop some fuzzy versions of the classical DEA models (in particular, the BCC model) by using some ranking methods based on the comparison of α-cuts. The resulting auxiliary crisp problems can be solved by the usual DEA software. We show, in a numerical example, how our models become specially useful for detecting sensitive decision-making units. Our approaches can be seen as an extension of the DEA methodology that provides users and practitioners with models which represent some real life processes more appropriately.

A Statistical Test for Nested Radial Dea Models

Some problems in economics, operations research, and engineering may be approached by means of a pair of radial DEA models that are nested, i.e., that the set of constraints of one of them is included in that of the other. In this paper we have focused on analyzing the marginal role of a given variable, called candidate, with respect to the efficiency measured by means of a DEA model. First, we have defined a new efficiency contribution measure ECM, which finally compares the efficiency scores of the two radial DEA models differing in the candidate. This can be either one input or one output. Then,based on ECM, we have also approached the problem from a statistical point of view. To be precise, we have developed a statistical test that allows us to evaluate the significance of the observed efficiency contribution of the candidate. Eventually, solving this test may provide some useful insights in order to decide the incorporation or the deletion of a variable into/from a given DEA model, on the basis of the information supplied by the data. Two procedures for progressive selection of variables were designed by sequentially applying the test: a forward selection and a backward elimination. These can be very helpful in the initial selection of variables when building a radial DEA model.

CHOOSING WEIGHTS FROM ALTERNATIVE OPTIMAL SOLUTIONS OF DUAL MULTIPLIER MODELS IN DEA

In this paper we propose a two-step procedure to be used for the selection of the weights that we obtain from the multiplier model in a DEA efficiency analysis. It is well known that optimal solutions of the envelopment formulation for extreme efficient units are often highly degenerate and, consequently, have alternate optima for the weights. Different optimal weights may then be obtained depending, for instance, on the software used. The idea behind the procedure we present is to explore the set of alternate optima in order to help make a choice of optimal weights. The selection of weights for a given extreme efficient point is connected with the dimension of the efficient facets of the frontier. Our approach makes it possible to select the weights associated with the facets of higher dimension that this unit generates and, in particular, it selects those weights associated with a full dimensional efficient facet (FDEF) if any. In this sense the weights provided by our procedure will have the maximum support from the production possibility set. We also look for weights that maximize the relative value of the inputs and outputs included in the efficiency analysis in a sense to be described in this article.

A statistical test for detecting influential observations in DEA

This paper deals with the problem of detecting influential observations in deterministic nonparametric DEA models. The technique we present is intended to classify for a further analysis those sample observations considerably affecting the measured efficiency for the remaining units. Then, the analyst will have to check whether these observations are contaminated by data errors or not. This approach also allows to determine when efficiency changes due to the presence of a given unit in the sample are statistically significant. Thus, ours is a statistical alternative to approach the problem of detecting influential observations in deterministic nonparametric DEA models.

On the choice of weights profiles in cross-efficiency evaluations

In this paper, we propose a new approach to cross-efficiency evaluation that focuses on the choice of the weights profiles to be used in the calculation of the cross-efficiency scores. It has been claimed in the literature that cross-efficiency eliminates unrealistic weighting schemes in the sense that their effects are cancelled out in the summary that the cross-efficiency evaluation makes. The idea of our approach here is to try to avoid these unreasonable weights instead of expecting that their effects are cancelled out in the amalgamation of weights that is made. To do it, we extend the ideas of the multiplier bound approach to the assessment of efficiency without slacks in Ramon et al. (2010) to its use in cross-efficiency evaluations. The models used look for the profiles with the least dissimilar weights, and also guarantee non-zero weights. In particular, this approach allows the inefficient DMUs to make a choice of weights that prevent them from using unrealistic weighting schemes. We use some examples of the literature to illustrate the performance of this approach and discuss some issues of interest regarding the choice of weights in cross-efficiency evaluations.

Variables With Negative Values In Dea

In this chapter we present an overview of the different existing approaches dealing with the treatment of negative data in DEA. We discuss both the classical approaches and the most recent contributions to this problem. The focus is mainly on issues such as translation invariance and units invariance of the variables, classification invariance of the units, as well as efficiency measurement and target setting

Common sets of weights as summaries of DEA profiles of weights: With an application to the ranking of professional tennis players

In this paper, we propose a DEA approach aimed at deriving a common set of weights (CSW) to be used to the ranking of decision making units (DMUs). The idea of this approach is to minimize the deviations of the CSW from the DEA profiles of weights without zeros of the efficient DMUs. This minimization reduces in particular the differences between the DEA profiles of weights that are chosen, so the CSW proposed is a representative summary of such DEA weights profiles. We use several norms to the measurement of such differences. As a result, the CSWs derived are actually different summaries of the chosen DEA profiles of weights like their average profile of their median profile. This approach is illustrated with an application to the ranking of professional tennis players.

On the DEA total weight flexibility and the aggregation in cross-efficiency evaluations

This paper discusses the DEA total weight flexibility in the context of the cross-efficiency evaluation. The DMUs in DEA are often assessed with unrealistic weighting schemes in their attempt to achieve the best ratings in their self-evaluation. We claim here that in a peer-appraisal like the cross-efficiency evaluation the cross-efficiencies provided by such weights cannot play the same role as those obtained with more reasonable weights. To address this issue, we propose to calculate the cross-efficiency scores by means of a weighted average of cross-efficiencies, instead of with the usual arithmetic mean, so the aggregation weights reflect the disequilibrium in the profiles of DEA weights that are used. Thus, the cross-efficiencies provided by profiles with large differences in their weights, especially those obtained with zero weights, would be attached lower aggregation weights (less importance) than those provided by more balanced profiles of weights.

Ranking ranges in cross-efficiency evaluations

The existence of alternate optima for the DEA weights may reduce the usefulness of the cross-efficiency evaluation, since the ranking provided depends on the choice of weights that the different DMUs make. In this paper, we develop a procedure to carry out the cross-efficiency evaluation without the need to make any specific choice of DEA weights. The proposed procedure takes into consideration all the possible choices of weights that all the DMUs can make, and yields for each unit a range for its possible rankings instead of a single ranking. This range is determined by the best and the worst rankings that would result in the best and the worst scenarios of each unit across all the DEA weights of all the DMUs. This approach might identify good/bad performers, as those that rank at the top/bottom irrespective of the weights that are chosen, or units that outperform others in all the scenarios. In addition, it may be used to analyze the stability of the ranking provided by the standard cross-efficiency evaluation.