Measuring performance of a three-stage structure using data envelopment analysis and Stackelberg game

In this paper, we consider a three-stage network comprised of a leader and two followers in respect to the additional desirable and undesirable inputs and outputs. We utilize the non-cooperative approach multiplicative model to measure the efficiency of the overall system and the performances of decision-making units (DMUs) from both, the optimistic and pessimistic views. Moreover, we utilize the concept of a goal programming and define a kind of cooperation between the leader and followers, so that the objectives of the managers are capable of being inserted in the models. In actual fact, a kind of collaboration is considered in a non-cooperative game. The non-cooperative models from these view cannot be converted into linear models. Therefore, a heuristic method is proposed to convert the nonlinear models into linear models. After obtaining the efficiencies based on the double-frontier view, the DMUs are ranked and classified into three clusters by the k-means algorithm. Finally, this paper considers a genuine world example, in relevance to production planning and inventory control, for model application and analyzes it from the double-frontier view. The proposed models are simulations of a factory in a real world, with a production area as leader and a warehouse and a delivery point as two followers. This factory has been regarded as a dynamic network with a time period of 24 intervals.