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We classify the contributions of DEA literature assessing Decision Making Units (DMUs) whose internal structure is known. Starting from an elementary framework, we define the main research areas as shared flow, multilevel and network models, depending on the assumptions they are subject to. For each model category, the principal mathematical formulations are introduced along with their main variants, extensions and applications. We also discuss the results of aggregating efficiency measures and of considering DMUs as submitted to a central authority that imposes constraints or targets on them. A common feature among the several models is that the efficiency evaluation of the DMU depends on the efficiency values of its subunits thereby increasing the discrimination power of DEA methodology with respect to the black box approach.Keywords Efficiency evaluation · Data envelopment analysis · Networks · Hierarchy · Multi-stage production processes Data Envelopment Analysis (DEA) has been a standard tool for evaluating the relative efficiencies of Decision Making Units (DMUs) since the paper of Charnes et al. (1978) based on the seminal work of Farrell (1957). Some underlying assumptions are common to classical DEA models. The efficiency of a DMU is defined as the weighted ratio of the outputs (products or outcomes) yielded by the DMU over the inputs (resources used or consumed). All DMUs considered are homogeneous, i.e., they all have the same types of inputs and outputs, and are independent, i.e., no constraint binds input and output levels of a DMU with the inputs and outputs of other DMUs. Furthermore, DMUs are seen as black boxes, i.e., their internal structures are not considered. As a consequence, generally, there is no clear L. Castelli ( ) · W. Ukovich

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