Returns to scale of two-stage production process

• We propose new network DEA models for production system with two stages.
• Raise necessary and sufficient conditions for distinguishing the status of RTS.
• Gain a method to calculate the changing quantity of the initial input.
• Determine the relationships of RTS between the sub-stage and the whole process.
• The proposed method can be extended to production system with multi-stage structures.

Network data envelopment analysis (network DEA) is a non-parametric approach for measuring the relative efficiencies of decision making units (DMUs) with multi-stage structures. With the application of network DEA techniques, this research studies two-stage production process, and proposes definitions of returns to scale (RTS) as well as the necessary and sufficient conditions for distinguishing the status of RTS of each stage under the framework of network DEA models in two cases. The first case is that the initial input takes changes only to maximize the ratio between the intermediate output and the initial input, and the final output changes following the corresponding change of intermediate product, i.e., Stage 1 is “leader” and Stage 2 is “follower”; the second case is that the initial input changes so as to obtain the maximal sum of ratio between the output and the input of the two stages. In these two cases, a new method of calculating the changing quantity of the initial input is proposed, and the relationships of RTS between the sub-stage and the entire production process are determined consequently. Numerical examples illustrate the theoretical results.