Multiplicative aggregation of division efficiencies in network data envelopment analysis

Network systems have two basic structures, series and parallel, and a network system can be transformed into a series system of subsystems, where each has a parallel structure composed of a number of divisions. The efficiency of the system can thus be expressed as the product of those of the subsystems, and the efficiency of each subsystem is a weighted average of those of its component divisions, under a relational data envelopment analysis (DEA) model. A previous study showed that the efficiency of a network system can be expressed as an additive aggregation of those of the divisions adjusted by a factor, and the former is bounded from above by the latter. This paper shows that the efficiency of the system can be expressed as a multiplicative aggregation of those of the divisions multiplied by a factor of greater than one. The system efficiency is thus bounded from below by the multiplicative aggregation of the division efficiencies. An example is used to show how to calculate the efficiency bounds, and how to identify the divisions that have stronger effects on the performance of the system.