Data envelopment analysis with embedded inputs and outputs

Applications of data envelopment analysis (DEA) often include inputs and outputs that are embedded in some other inputs or outputs. For example, in a school assessment, the sets of students achieving good academic results or students with special needs are subsets of the set of all students. In a hospital application, the set of specific or successful treatments is a subset of all treatments. Similarly, in many applications, labour costs are a part of overall costs. Conventional variable and constant returns-to-scale DEA models cannot incorporate such information. Using such standard DEA models may potentially lead to a situation in which, in the resulting projection of an inefficient decision making unit, the value of an input or output representing the whole set is less than the value of an input or output representing its subset, which is physically impossible. In this paper, we demonstrate how the information about embedded inputs and outputs can be incorporated in the DEA models. We further identify common scenarios in which such information is redundant and makes no difference to the efficiency assessment and scenarios in which such information needs to be incorporated in order to keep the efficient projections consistent with the identified embeddings.