Equivalent multiplier and envelopment DEA models for measuring efficiency under general network structures

Data envelopment analysis (DEA) has been shown to be a very useful mathematical programming tool to measure the relative efficiency of decision making units (DMUs), especially when the so-called internal network structure of the production process is taken into account. Under a network structure, however, two standard directions of modeling the production process may generally lead to a pair of multiplier and envelopment DEA models so that the outcomes are not necessarily equivalent, i.e. a network duality problem occurs. Although, the duality problem has recently been addressed for specific cases of network structures, for more complex structures, DEA models have only been able to be developed by following either the envelopment form or multiplier form. Investigating this duality problem, this paper also proposes DEA models for general network structures with two additional properties. Due to the first property, all factors in a general network structure including main inputs/outputs and/or intermediate inputs/outputs can be shared among the divisions while the second property assumes that a factor in a structure may be considered as both intermediate input/output and main input/output simultaneously. We will show that the proposed network DEA models cannot only deal with the already existing general network structures in the literature, but are also represented by dual multiplier and envelopment linear programming-based problems by which consistent outcomes can be obtained. A comprehensive numerical example will be presented to explain the properties and features of the suggested models.