Return to scale in two-stage networks based on two different approaches

One of the most important issues in data envelopment analysis is the determination of the return to scale class of decision-making units. One of the structures of decision-making units is the two-stage network structure. In this research, we obtain the return to scale class of decision-making units with two strategies. The first strategy is when each stage performs independently, and the input and output of each stage is important for it, the return to scale class is assigned corresponding to each stage and the overall process with this strategy. In the other case, we define the two-stage network structure as the leader and follower model, and the output of the first step is considered as the input of the second stage, in which case the return to scale class is determined based on the output ratio of each stage to the input of that stage. The second stage output to the first stage input ratio is used to determine the return to scale class of the final process. Determine the relationship between the return to scale class and the first and second stages, and provide sufficient conditions to determine the return to scale class. Finally, we illustrate the proposed approach with a numerical example.