Manoochehr Ziyae Berentin - Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran
Ghasem Tohidi - Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran
Shabnam Razavyan - Department of Mathematics, South Tehran Branch, Islamic Azad University, Tehran, Iran
Mahnaz Barkhordari Ahmadi - Department of Mathematics, Bandarabbas Branch, Islamic Azad University, Bandarabbas, Iran

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 system, 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 and 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, 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) and specify the optimal portion of stages in the use of shared inputs and the portion of the second stage of intermediate measures. 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.

Data envelopment analysis, Two-stage system, triangular fuzzy number, Non-cooperative