Document Type : Article

**Authors**

^{1}
Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran

^{2}
Department of Business Systems and Analytics, Lindback Distinguished Chair of Information Systems and Decision Sciences, La Salle University, Philadelphia, PA 19141, USA.; Department of Business Information Systems, Faculty of Business Administration and Economics, University of Paderborn, D-33098 Paderborn, Germany.

**Abstract**

We present an integrated data envelopment analysis (DEA) and Malmquist productivity index (MPI) to evaluate the performance of decision making units (DMUs) by using a directional distance function with undesirable interval outputs. 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. We also present a case study in the banking industry to demonstrate applicability and efficacy of the proposed integrated approach.

**Keywords**

- Data envelopment analysis
- Malmquist productivity index
- Interval approach
- directional distance function
- undesirable outputs

**Main Subjects**

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Transactions on Industrial Engineering (E)

November and December 2019Pages 3819-3834