Efficiency of Extended Two-Stage Systems in Presence of Triangular Fuzzy Number

Data envelopment analysis (DEA) is an excellent method to evaluate the efficiency of homogeneous Decision Making Units (DMUs). This procedure considers the internal structure of DMUs as black box systems. In the real world, many systems are a combination of two stages that are connected with intermediate measures. In fact, intermediate measures are considered as the outputs of the first stage and the inputs of the second stage. To evaluate the efficiency of these systems, network DEA (NDEA) models are presented. In some cases, these systems may have shared inputs, and also part of the intermediate measures will be allocated as inputs of second-stage. We also frequently deal uncertain information such as stochastic data, fuzzy data and so on. Therefore, in practice, it is not easy to obtain the exact values of these inputs and assign them to each of the stages. Also, it is not possible to easily determine the use of the second stage of intermediate measures. Therefore, in this paper, we shall combine fuzzy DEA and NDEA models which introduce a model based on the multiplicative approach. To this end, we use the non-compensatory property of the multiplication operator and the   cut procedure. These models calculate the   cut interval of overall efficiency and efficiency of stages in the presence of triangular fuzzy numbers (TFNs). Also, specify the optimal portion of stages in the use of shared inputs and the portion of the second stage of intermediate measures is specified. Furthermore, the product of the upper (the lower) bound of the cut of the stages efficiency is considered as the upper (the lower) bound overall efficiency. Finally, we will illustrate the proposed models by using a numerical example extracted from the extant literature.