A Malmquist productivity index with the directional distance function and uncertain data

In the present study, by using the directional distance function with undesirable interval outputs, the malmquist productivity index (mpi) and integrated data envelopment analysis (dea) are employed for evaluating the function of decision-making units (dmus). the mpi calculation is performed to compare the efficiency of the dmus in distinct time periods. the uncertainty inherent in real-world problems is considered by using the best- and worst-case scenarios, defining an interval for the mpi, and reflecting the dmus’ advancement or regress. the optimal solution of the robust model lies in the efficiency interval, i.e., it is always equal to or less than the optimal solution in the optimistic case and equal to or greater than the optimal solution in the pessimistic case. this study also presented a case study in the banking industry to demonstrate the applicability and efficacy of the proposed integrated approach.