Expert Systems With Applications 99 (2018) 213–230 Contents lists available at ScienceDirect Expert Systems With Applications journal homepage: www.elsevier.com/locate/eswa A novel two-stage DEA production model with freely distributed initial inputs and shared intermediate outputs Mohammad Izadikhah a , Madjid Tavana b,c,∗ , Debora Di Caprio d,e , Francisco J. Santos-Arteaga f a Department of Mathematics, College of Science, Arak Branch, Islamic Azad University, Arak, Iran b Business Systems and Analytics Department, Distinguished Chair of Business Analytics, La Salle University, Philadelphia, PA 19141, USA c Business Information Systems Department, Faculty of Business Administration and Economics, University of Paderborn, D-33098 Paderborn, Germany d Department of Mathematics and Statistics, York University, Toronto, M3J 1P3, Canada e Polo Tecnologico IISS G. Galilei, Via Cadorna 14, 39100, Bolzano, Italy f Faculty of Economics and Management, Free University of Bolzano, Bolzano, Italy a r t i c l e i n f o Article history: Received 21 June 2017 Revised 9 October 2017 Accepted 1 November 2017 Available online 10 November 2017 Keywords: Data envelopment analysis Intermediate measure Efficiency Shared flow Two-stage process a b s t r a c t Conventional data envelopment analysis (DEA) models treat the decision-making units (DMUs) as black- boxes: inputs enter the system and outputs exit the system, with no consideration for the intermediate steps characterizing the DMUs. As a result, intermediate measures are lost in the process of changing the inputs to outputs and it becomes difficult, if not impossible, to provide individual DMU managers with specific information on what part of a DMU is responsible for the overall inefficiency. This study defines a two-stage DEA model, where each DMU is composed of two sub-DMUs in series, the intermediate prod- ucts by the sub-DMU in Stage 1 are partly consumed by the sub-DMU in Stage 2, and the initial inputs of the DMU can be freely allocated in both stages. Also, there are additional inputs directly consumed in Stage 2 while part of the outputs of Stage 1 are final outputs. We develop four new linear models to de- termine the upper and lower bounds of the efficiencies of the two sub-DMUs in a non-cooperative setting and a linear model to calculate the overall efficiency of DMU in a cooperative setting. That is, the overall efficiency of a DMU is modelled in a cooperative setting via upper and lower bounds obtained in the non-cooperative one. The proposed two-stage DEA method allows for important applications to several management areas. A case study in the banking industry is presented to demonstrate the applicability and exhibit the efficacy of the proposed models. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction Data envelopment analysis (DEA) is an effective non-parametric evaluation method for measuring the relative efficiency of a set of decision making units (DMUs) each of which uses multiple in- puts to produce multiple outputs. In the traditional DEA methods, DMUs are treated as black-boxes, that is, the internal structure of the DMUs is often ignored. As a result, the focus of the investi- gation is on the single operational processes with a set of initial inputs and final outputs that is unable of pinpointing the sources of inefficiency within the DMUs (Lewis, Mallikarjun, & Sexton, 2013; Wang, Huang, Wu, & Liu, 2014). On the other hand, in the ∗ Corresponding author at: Business Systems and Analytics Department, Distin- guished Chair of Business Analytics, La Salle University, Philadelphia, PA 19141, USA. E-mail addresses: m-izadikhah@iau-arak.ac.ir (M. Izadikhah), tavana@lasalle.edu (M. Tavana), dicaper@mathstat.yorku.ca, debora.dicaprio@istruzione.it (D. Di Caprio), fsantosarteaga@unibz.it (F.J. Santos-Arteaga). two-stage DEA models, DMUs are modelled as systems composed of two sub-DMUs in series where the outputs of the sub-DMU in Stage 1, known as intermediate measures/products/outputs, are considered as inputs of the sub-DMU in Stage 2. As a result, a two- stage DEA model allows one to further investigate the structure of a DMU and its processes and, hence, to identify the misallocation of the inputs among the sub-DMUs (Du, Liang, Chen, Cook, & Zhu, 2011; Ebrahimnejad, Tavana, Lotfi, Shahverdi, & Yousefpour, 2014; Yu, Shi, & Song, 2013). Despite appearing as the simplest multi-stage approach to ef- ficiency evaluation, two-stage DEA models are the building blocks for the study of series systems whose DMUs consist of multiple sub-DMUs operating through procedures of different complexity. The work of Charnes et al. (1986) on army recruitment using a two-stage approach was the first study to discuss the loss of infor- mation intrinsic to single-stage models. In recent years, many re- searchers have studied several applications of two-stage DEA mod- els to diverse decision making situations such as healthcare man- https://doi.org/10.1016/j.eswa.2017.11.005 0957-4174/© 2017 Elsevier Ltd. All rights reserved.