Slacks-based measures of efficiency in imprecise data envelopment analysis: An approach based on data envelopment analysis with double frontiers q Hossein Azizi a,⇑ , Sohrab Kordrostami a , Alireza Amirteimoori b a Department of Applied Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran b Department of Applied Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran article info Article history: Received 17 April 2013 Received in revised form 30 August 2014 Accepted 26 October 2014 Available online 4 November 2014 Keywords: Data envelopment analysis Imprecise data Optimistic efficiency interval Pessimistic efficiency interval Overall efficiency interval Ranking abstract Data envelopment analysis (DEA) is a mathematical approach for evaluating the efficiency of decision- making units (DMUs) that convert multiple inputs into multiple outputs. Traditional DEA models assume that all input and output data are known exactly. In many situations, however, some inputs and/or out- puts take imprecise data. In this paper, we present optimistic and pessimistic perspectives for obtaining an efficiency evaluation for the DMU under consideration with imprecise data. Additionally, slacks-based measures of efficiency are used for direct assessment of efficiency in the presence of imprecise data with slack values. Finally, the geometric average of the two efficiency values is used to determine the DMU with the best performance. A ranking approach based on degree of preference is used for ranking the effi- ciency intervals of the DMUs. Two numerical examples are used to show the application of the proposed DEA approach. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction Data envelopment analysis is a non-parametric technique for evaluating the relative efficiency of decision-making units (DMUs) that use multiple inputs to produce multiple outputs (Banker, Charnes, & Cooper, 1984; Charnes, Cooper, & Rhodes, 1978). DEA has been used in various environments and numerous applications (Amirteimoori, 2006; Amirteimoori & Mohaghegh Tabar, 2010; Chen, 2004; Jahanshahloo, Amirteimoori, & Kordrostami, 2004; Kao & Hwang, 2008; Khalili, Camanho, Portela, & Alirezaee, 2010; Lozano, Gutiérrez, & Moreno, 2013; Wang, Chin, & Jiang, 2011; Wu, An, Ali, & Liang, 2013). In traditional DEA models, the maxi- mum ratio of the weighted sum of outputs to the weighted sum of inputs is called the optimistic efficiency or the best relative effi- ciency. In other words, the traditional DEA determines a set of most favorable weights for each DMU under evaluation that maximizes the efficiency. If a DMU has an optimistic efficiency of 1, it is said to be DEA efficient or optimistic efficient; otherwise, it is DEA non-efficient or optimistic non-efficient. It is usually held that opti- mistic efficient DMUs have a better performance than optimistic non-efficient DMUs. When DEA identifies the efficient production frontier, it improves the performance of optimistic non-efficient DMUs by increasing their current output levels or decreasing their current input levels. On the other hand, if the minimum ratio of the weighted sum of outputs to the weighted sum of inputs is mea- sured, the resulting efficiency is called the pessimistic efficiency or the worst relative efficiency (Jahanshahloo & Afzalinejad, 2006; Liu & Chen, 2009; Paradi, Asmild, & Simak, 2004; Parkan & Wang, 2000). For each DMU under evaluation, this method determines a set of most unfavorable weights that minimizes the efficiency. If a DMU has a pessimistic efficiency of 1, that DMU is said to be pes- simistic inefficient or DEA inefficient; otherwise, it is called pessi- mistic non-inefficient or DEA non-inefficient. It is usually believed that pessimistic inefficient DMUs have a worse performance than pessimistic non-inefficient DMUs. The optimistic and pessimistic efficiencies measure the two extremes of performance for each DMU. Any evaluation method that considers only one of them is biased. For determining the overall performance of each DMU, it is necessary to consider both of these efficiencies simultaneously. The approach that determines the performance of each DMU relative to both optimistic and pes- simistic efficiencies is called the ‘‘DEA with double frontiers’’ approach (Wang & Chin, 2009; Wang & Chin, 2011). Entani, Maeda, and Tanaka (2002) were among the first researchers who measured the performance of DMUs from both optimistic and pessimistic perspectives. Their proposed DEA models were initially developed for crisp data and were then extended to interval and fuzzy data. Theoretically, their proposed http://dx.doi.org/10.1016/j.cie.2014.10.019 0360-8352/Ó 2014 Elsevier Ltd. All rights reserved. q This manuscript was processed by Area Editor Imed Kacem, Pr. ⇑ Corresponding author. E-mail address: azizhossein@gmail.com (H. Azizi). Computers & Industrial Engineering 79 (2015) 42–51 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie