Stochastic ranking and dominance in DEA

Data Envelopment Analysis (DEA) requires deterministic input/output data for efficiency evaluation of a set of Decision Making Units (DMUs). When there are more than one set of input/output data for each DMU, however, such requirement is infeasible. Stochastic DEA (SDEA), where input/output data are assumed to be stochastic, is a natural approach for such applications. Performance evaluation of DMUs in SDEA naturally calls for ranking methods that can account for stochastic fluctuations of the input/output, and hence the efficiency score. None of the proposed methods in the current literature incorporates all the information in the efficiency score distributions for ranking DMUs. To fill this gap, we introduce two ranking methods, a partial and a linear, for performance evaluation in SDEA using the reliability function of the efficiency scores. Our proposed partial ranking is based on the notion of stochastic ordering, while our linear ordering is a weighted average of the reliability function of the efficiency scores. Special cases of our proposed linear ranking method include mean and median ordering in SDEA. Our proposed partial ordering provides a notion for stochastic dominance using which one can define a natural notion of admissibility as a minimal performance requirement. We demonstrate how the proposed ranking methods can be implemented and illustrate the methods using the Grundfeld data, analyzed using both parametric and non-parametric approaches.