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 decisionmaking 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 this 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.