Two-stage DEA model with additional input in the second stage and part of intermediate products as final output

• We propose a two-stage DEA model that the intermediate product is freely distributed.
• We get the upper and lower efficiencies of each stage using leader–follower game.
• The non-linear programming model is transformed to a parametric linear programming.
• A heuristic search algorithm is designed to obtain the cooperative efficiency.
• A set of appropriate proportion for intermediate products is obtained.

Data envelopment analysis (DEA) has been proved to be an excellent approach for measuring performance of decision making units (DMUs) that use multiple inputs to generate multiple outputs. In many real world scenarios, inputs or outputs may be shared among various activities. This paper proposes a two-stage DEA model with additional input in the second stage and part of intermediate products as final output. We first discuss the non-cooperative condition in order to determine the upper and lower bounds of the efficiencies of sub-DMUs in different stages. A parametric transformation is described to solve the non-linear programming of the overall cooperative efficiency model. An application is provided.