Lexicographic hyperbolic DEA

The hyperbolic distance function (HDF) reduces all inputs and increases all outputs simultaneously and at the same rate. Although the corresponding data envelopment analysis (DEA) model is non-linear, for constant returns to scale it can be linearised and for variable returns to scale an efficient iterative approach based on the directional distance function (DDF) model can be used. However, HDF does not necessarily project onto an efficient target. To remedy this, lexicographic hyperbolic DEA (LexHDEA) is proposed in this article. Thus, before solving the HDF model, the input or output dimensions that can be improved are determined. A reduced HDF model is then solved, looking for improvements only in these dimensions. If the corresponding target is efficient, then no further steps are necessary. Otherwise, a reduced HDF model that improves only those dimensions that can be further improved is solved. If this improved target is efficient the process stops. Otherwise the process is repeated until eventually the efficient frontier is reached. In addition to guaranteeing an efficient target the proposed approach also computes an efficiency measure that has indication of efficiency and units invariance. The proposed approach can be extended to handle a preference structure, non-discretionary variables and undesirable outputs.