Measuring the Robustness of Empirical Efficiency Valuations

We study the robustness of empirical efficiency valuations of production processes in an extended Farrell model. Based on input and output data, an empirical efficiency status---efficient or inefficient---is assigned to each of the processes. This status may change if the data of the observed processes change. As illustrated by a capacity planning problem for hospitals in Germany, the need arises to gauge the robustness of empirical efficiency valuations. The example suggests to gauge the robustness of the efficiency valuation for a process with respect to perturbations of prespecified elements of the data. A natural measure of robustness is the minimal perturbation, in terms of a suitable distance function, of the chosen data elements that is necessary to change the efficiency status of the process under investigation. Farrell's (1957) efficiency score is an example of such a robustness measure. We give further examples of relevant data perturbations for which the robustness measure can be computed efficiently. We then focus on weighted maximum norm distance functions, such as the maximal absolute or percentage deviation, but allow for independent perturbations of the elements of an arbitrary a priori fixed subset of the data. In this setting, the robustness measure is naturally related to a certain threshold value for a linear monotone one-parameter family of perturbations and can be calculated by means of a linear programming--based bisection method. Closed form solutions in terms of Farrell's efficiency score are obtained for specific perturbations. Following the theoretical developments, we revisit the hospital capacity planning problem to illustrate the managerial relevance of our techniques.