Malmquist–Luenberger Productivity Index for a Two-Stage Structure in the Presence of Undesirable Outputs and Uncertainty

Network Data Envelopment Analysis (NDEA) models assess the processes of the underlying system at a certain moment and disregard the dynamic effects in the production process. Hence, distorted efficiency evaluation is gained that might give misleading information to decision-making units (DMUs). Malmquist–Luenberger Productivity Index (MPI) assesses efficiency changes over time, which are measured as the product of recovery and frontier-shift terms, both coming from the DEA framework. In this study, a form of MPI involving network structure for evaluating DMUs in the presence of uncertainty and undesirable outputs in two periods of time is presented. To cope with uncertainty, we use the stochastic p-robust approach and the weak disposability of Kuosmanen (American Journal Agricultural Economics 87 (4):1077–1082, 2005) proposed to take care of undesirable outputs. The proposed fractional models for stages and overall system are linearized by applying the Charnes and Cooper transformation. Finally, the proposed models are applied to evaluate the efficiency of 11 petroleum wells to identify the main factors determining their productivity, utilizing the data from the 2020 to 2021 period. The results show that the management of resource consumption, especially equipment and capital, is not appropriate and investment is inadequate. Although the depreciation rate of capital facilities in this industry is high, the purpose of the investment is not to upgrade the level of technology.