Ranking Invariant DEA Based Efficiency Measures

Public service spending is an important issue with great economic, social and political ramifications. Consequently, correct measurements with respect to productivity evaluation of such expenditures is of paramount significance. Motivated by evaluating public services from a non-parametric production efficiency perspective and the computational irregularities this may entail, we present refined bounds for the crucial non-Archimedean infinitesimal ({\it aka} epsilon). In non-parametric efficiency models the epsilon plays a key role as a multiplication factor to the sum of input and output slacks in the objective function, equivalently, it is used as a lower bound for the input and output weights in productivity multiplier models. Selecting a value for epsilon is non-trivial since: it has to be sufficiently small to guarantee the envelopment model is bounded (or the multiplier model feasible) yet large enough to provide managerial insight and not cause computational problems; is highly context specific, depending on the input and output metrics; sensitive to underlying assumptions with respect to constant or variable returns-to-scale; and may lead to drastically different relative efficiency rankings. To guarantee the relative ranking of the evaluated units remain consistent we provide two bounds for the epsilon. The first,{\it positive efficiency guarantee}, ensures the obtained efficiency measures are positive and well-defined, and the second, {\it ranking invariance guarantee}, is a refinement such that the relative efficiency rankings are provably consistent. We illustrate our bounds and their implications using data from twelve public healthcare centers