Data Envelopment Analysis (DEA) is a beneficial mathematical programming method to measure relative efficiencies. In conventional DEA models, Decision Making Units (DMUs) are usually considered as black boxes. Also, the efficiency of DMUs is evaluated in the presence of the specified inputs and outputs. Nevertheless, in real-world applications, there are situations in which the performance of multi-stage processes like supply chains with forward and reverse flows must be measured such that some of the intervening factors, called proportional dual-role factors, are presented that one part of each proportional dual-role factor plays the input role and the other plays the output role. To address this issue, the current study proposes radial and non-radial DEA models for evaluating the overall and stage efficiencies of the closed-loop supply chains when there are proportional dual-role factors. To illustrate, a proportional dual-role factor is divided into portions of the input of the first stage and the output of the second stage such that the optimal overall and stage efficiency scores of closed-loop supply chain are obtained. A case study is used to illustrate the proposed approach. The experimental results obtained from real world data show the convincing performance of our proposed method.
In classic data envelopment analysis models, two-stage network structures are studied in cases in which the input/output data set are deterministic. In many real applications, however, we face uncertainty. This paper proposes a two-stage network DEA model when the input/output data are stochastic. A stochastic two-stage network DEA model is formulated based on the chance-constrained programming. Linearization techniques and the assumption of single underlying factor of the data are used to construct the equivalent deterministic linear programming model. The relationship between the stochastic efficiency of each stage and stochastic centralized efficiency of the whole process, at different confidence levels, is discussed. To illustrate the real applicability of the proposed approach, a real case on 16 commercial banks in China is given.