Discrete and integer valued inputs and outputs in data envelopment analysis

Standard axioms of free disposability, convexity and constant returns to scale employed in Data Envelopment Analysis (DEA) implicitly assume continuous, real-valued inputs and outputs. However, the implicit assumption of continuous data will never hold with exact precision in real world data. To address the discrete nature of data explicitly, various formulations of Integer DEA (IDEA) have been suggested. Unfortunately, the axiomatic foundations and the correct mathematical formulation of IDEA technology has caused considerable confusion in the literature. This chapter has three objectives. First, we re-examine the axiomatic foundations of IDEA, demonstrating that some IDEA formulations proposed in the literature fail to satisfy the axioms of free disposability of continuous inputs and outputs, and natural disposability of discrete inputs and outputs. Second, we critically examine alternative efficiency metrics available for IDEA. We complement the IDEA formulations for the radial input measure with the radial output measure and the directional distance function. We then critically discuss the additive efficiency metrics, demonstrating that the optimal slacks are not necessarily unique. Third, we consider estimation of the IDEA technology under stochastic noise, modeling inefficiency and noise as Poisson distributed random variables.