A bargaining game model for measuring efficiency of two-stage network DEA with non-discretionary inputs

Conventional data envelopment analysis (DEA) is a method for measuring the efficiency of decision-making units (DMUs). Recently, to measure the efficiency of sub-DMUs (Stages), several network DEA models have been developed, in which the results of network DEA models not only provide the overall efficiency of the whole system but also provide the efficiency of the individual stages. This study develops a bargaining game model for measuring the efficiency of DMUs that have a two-stage network structure with non-discretionary inputs, that the model as a method of dealing with the conflict arising from the intermediate measures. Under the Nash bargaining game theory, the two stages in the network DEA are considered as players and network DEA model is a cooperative game model. Here, the non-discretionary additional inputs in the second stage make changes in the cooperative game model, so that managers of units cannot change the value of non-discretionary inputs in measuring the efficiency of the bargaining game model, and this causes the desired and expected output of the managers not to be produced. In addition, it can be stated that the presence of such inputs is capable, significantly affecting the system efficiency score and stages. So that the existence of the inputs in the measuring efficiency of decision-making units reduces the efficiency score of cooperative game. In this study, linearizing the model in the presence of the non-discretionary input is a new idea in the bargaining game model. A numerical example shows the applicability of the new model.