Identifying the Global Reference Set in DEA: An Application to the Determination of Returns to Scale

In data envelopment analysis (DEA), any set of observed decision making units (DMUs) that produce a projection point of an inefficient DMU is called a reference set of this DMU. Based on this definition, however, the concept of reference set is not mathematically well defined in the non-radial DEA setting. This is because a given projection point may be generated by multiple unary reference sets, and different projection points may result in multiple maximal reference sets. In this chapter, first, we address this issue by differentiating between the uniquely found reference set, called the global reference set (GRS), and the unary and maximal types of the reference set for which the multiplicity issue may occur. Second, to identify the GRS, we propose a general linear programming based approach that is computationally more efficient than its alternatives. Third, we define the returns to scale (RTS) of an inefficient DMU at its projection point that is produced by all—but not some—of the units in its GRS. By this definition, the notion of RTS is unambiguous, since the GRS is unique and the projection points generated by all the possible reference units all exhibit the same type of RTS. Fourth, using a non-radial DEA model, we develop two precise multiplier- and envelopment-based methods to determine RTS possibilities of the DMUs. To demonstrate the ready applicability of our approach, we finally conduct an empirical analysis based on a real-life data set.