Estimating most productive scale size with stochastic data in data envelopment analysis M. Khodabakhshi ⁎ Department of Mathematics, Faculty of Science, Lorestan University, Khorram Abad, Iran abstract article info Article history: Accepted 4 March 2009 Keywords: Stochastic data Most productive scale size (mpss) Chance constraints Software Companies This article estimates most productive scale size in stochastic data envelopment analysis (DEA). Jahanshahloo and Khodabakhshi [Jahanshahloo, G.R. and Khodabakhshi, M., Using input–output orientation model for determining most productive scale size in DEA. Applied Mathematics and Computation 2003, 146(2–3), 849–855.] studied most productive scale size in classic data envelopment analysis. The classic data envelopment analysis requires that the values for all inputs and outputs be known exactly. However, this assumption may not be true, because data in many real applications cannot be precisely measured. One of the important methods to deal with imprecise data is considering stochastic data in DEA. Therefore, this research studies most productive scale size with considering stochastic data in DEA. To that end, input–output orientation model introduced in Jahanshahloo and Khodabakhshi [Jahanshahloo, G.R. and Khodabakhshi, M., Using input–output orientation model for determining most productive scale size in DEA. Applied Mathematics and Computation 2003, 146(2–3), 849–855.] is extended in stochastic data envelopment analysis. To solve the stochastic model, a deterministic equivalent is obtained. Although the deterministic equivalent is non-linear, it can be converted to a quadratic program. Furthermore, data of software companies is used to apply the proposed approach. Performance of software companies are evaluated based on their scale sizes in classic and stochastic data envelopment analysis. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Data envelopment analysis (DEA) initiated by Charnes et al. (1978), and the first model was called CCR model. This model, the CCR model, is a linear programming problem, and it is readily computable. The specific research standard of efficiency measurement for production units in the field of operational research took off with introducing the CCR model. For example, Banker et al. (1984) provided some models for estimating technical and scale inefficiencies in data envelopment analysis. They extended DEA by adding a convexity constraint to obtain a new model known as BCC model. BCC model is a variable returns to scale version of the CCR model. Tone (2001), also, introduced a non-radial model known as slack-based measure to evaluate efficiency of decision making units. The original models, CCR and BCC, in DEA only allow changes in the input combination of decision making units that are limited to the observed inputs of evaluating decision making units. Cooper et al. (2001), with a proper initiative on the data of textile industry of China for improving congestion management, increased labor input and reduced capital input and showed the new combination could have constructive results. The idea initiated by Cooper et al. (2001) motivated further work and new models in DEA, e.g., Jahanshahloo and Khodabakhshi (2004) and Khodabakhshi (2009). These models use more flexibility in changes of the used input combination to find the maximum possible output and can be useful to resources management. Since 1978 there has been a surge of research on DEA and many further models were introduced in the literature. See, for example, Andersen and Petersen (1993), Adler et al. (2002), Li et al. (2007) which are different approaches for ranking efficient units obtained by the original models in DEA. Note that although efficient units are benchmark for inefficient ones, they are not comparable among themselves with original DEA models. A thorough discussion on new development in DEA up to 1996 can be found in Cooper et al. (1996). The proposed model in Cooper et al. (1996) for determining most productive scale size has a fractional objective function. Jahanshahloo and Khodabakhshi (2003), also, provided an input–output orientation model to estimate most productive scale size units with linear objective function. Both models can determine most productive scale size units, while the later model is easier to solve than the first one because of their objective functions. One may refer to Cooper et al. (2000) and Thanassoulis (2001) which include most of the developments and extensions in DEA. One of the advantages of the DEA method is that it requires neither a priori weights nor explicit specification of functional relations among the multiple inputs and outputs. However, as one of the weaknesses, DEA does not allow stochastic variations in input–output data, such as measurement errors and data entry errors. Traditionally, the coefficients of data envelopment analysis (DEA) models, i.e., the data of inputs and outputs of the different decision making units (DMUs), are assumed to be measured with precision. On the other hand, as some authors point out (see, e.g., Liu, 1999), this is not always possible. To remove this weakness in the classic Economic Modelling 26 (2009) 968–973 ⁎ Tel.: +98 9126278846; fax: +98 661 2201333. E-mail address: mkhbakhshi@yahoo.com. 0264-9993/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2009.03.002 Contents lists available at ScienceDirect Economic Modelling journal homepage: www.elsevier.com/locate/econbase