Relative partial efficiency: network and black box SBM DEA interpretations in multiplier form

In traditional black-box DEA when the ratio-based multiplier DEA model is estimated to obtain a technical efficiency score, the estimated multipliers (shadow prices) serve as the weights that maximize the ratio of the aggregation of weighted sum of outputs (virtual output) to that of inputs (virtual input) of the assessed DMU in comparison with the other decision making units (DMUs). With respect to the ratio-based multiplier model of non-radial slack-based measure (SBM), however, there does not exist such a nice efficiency interpretation. For the purpose of providing a reasonable efficiency interpretation for both black-box and network SBM models, this paper introduces a concept called relative partial efficiency (RPE). In the black box structure, RPEs are defined for each input-output pair and a multi objective programming is formed in order to maximize RPEs. Then, it is proved that its equivalent single objective programming problem is the same SBM multiplier DEA model. The obtained explicit efficiency interpretation coming from this novel concept is then generalized for the multiplier network SBM DEA model represented by Boloori (Comput Ind Eng 95:83-96,2016).