Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie Expected efficiency based on directional distance function in data envelopment analysis Feng Yang a , Fangqing Wei a , Yongjun Li a , Ying Huang b , Yao Chen b,c, ⁎ a School of Management, University of Science and Technology of China, Hefei, Anhui 230026, PR China b Manning School of Business, University of Massachusetts, Lowell, MA 01854, USA c College for Auditing and Evaluation, Nanjing Audit University, Nanjing, Jiangsu 210017, PR China ARTICLE INFO Keywords: Expected efficiency Directional distance function Data envelopment analysis Decision making unit ABSTRACT Directional distance function (DDF), an evaluation technique that estimates relative efficiency of a decision making unit (DMU) along a pre-determined direction vector that is not restricted by the radial direction, has been widespread in productive efficiency research over the past two decades. A key challenge in DDF appli- cations, however, is to decide on an appropriate (or the best) direction along which to measure efficiency. To circumvent this issue, we build on the DDF model and propose expected efficiency in efficiency estimation. Expected efficiency is defined as the mean value of all relative efficiency scores of a DMU along all directions. When calculating the overall relative efficiency score of a DMU, the expected efficiency model incorporates all possible directions rather than choosing a particular direction. As such, the expected efficiency approach extends DDF from a single direction to all directions. Some benefits of the expected efficiency approach include (1) relieving a decision maker of the burden of determining a particular directional vector among many choices; (2) overcoming a decision maker’s subjectivity in the direction selection; (3) resolving the sensitivity issue caused by choosing different directions; and (4) ensuring that all DMUs are estimated in a consistent and equitable manner. Our study contributes to productive efficiency research and data envelopment analysis by introducing a new efficiency estimate that does not need to rely on one specific direction. Using two examples, we demonstrate the validity and the robustness of expected efficiency as an alternative efficiency estimate. 1. Introduction Efficiency evaluation is integral to effective business and operations management. Research on the measurement of productive efficiency has advanced after the seminal work by Farrell (1957). Among various efficiency evaluation methods, data envelopment analysis (DEA) is one of the most important tools and has been adopted for performance evaluation in the areas of operations management, economics, public affairs, finance, etc. (Liu, Lu, Lu, & Lin, 2013; Emrouznejad & Yang, 2018). First introduced by Charnes, Cooper, and Rhodes (1978), DEA is a nonparametric linear programming method that measures the relative efficiencies of a set of comparable entities called decision making units (DMUs) with multiple inputs and multiple outputs (Cook & Seiford, 2009; Cooper, Seiford, & Zhu, 2011). In traditional DEA models such as the CCR (Charnes et al., 1978) and BCC (Banker, Charnes, & Cooper, 1984) models, each DMU chooses its own weights, that is, the radial direction to the origin, to obtain the optimal efficiency score. Restricted by the radial direction, traditional DEA models have two shortcomings. First, because each DMU follows its own radial direction in estimation, DMUs are not evaluated on the same basis. Thus, evaluation results vary and rankings are largely inconsistent (Sun, Wu, & Guo, 2013). Second, because a set of weights that is favorable to one DMU is not necessarily favorable to other DMUs, one DMU may dominate other DMUs (Kao & Hung, 2005; Wang, Chin, & Leung, 2009) thus the eva- luation results may be unacceptable to other DMUs (Amin & Toloo, 2007; Wu, Chu, Sun, Zhu, & Liang, 2016). To avoid the restriction of the radial direction, Chambers, Chung, and Färe (1996) extended the DEA models to other non-radial direc- tions. Building on the distance function proposed by Shephard (1970) and Luenberger (1992). Chambers et al. (1996) proposed the direc- tional distance function (DDF) to calculate relative efficiency of DMUs along a predetermined direction. Using DDF, a decision maker now has the flexibility in choosing either the same directional vector for all DMUs or a specific vector for each DMU (Aparicio, Pastor, & Vidal, https://doi.org/10.1016/j.cie.2018.08.010 Received 8 December 2017; Received in revised form 2 August 2018; Accepted 6 August 2018 ⁎ Corresponding author at: College for Auditing and Evaluation, Nanjing Audit University, Nanjing, Jiangsu 210017, PR China. E-mail addresses: fengyang@ustc.edu.cn (F. Yang), wfq89072@mail.ustc.edu.cn (F. Wei), lionli@ustc.edu.cn (Y. Li), Ying_Huang1@uml.edu (Y. Huang), Yao_Chen@uml.edu (Y. Chen). Computers & Industrial Engineering 125 (2018) 33–45 Available online 10 August 2018 0360-8352/ © 2018 Elsevier Ltd. All rights reserved. T