On stochastic data envelopment analysis
SDEA is a part of DEA methodology in which stochastic models based on the possibility of random variations in input–output data are studied. There are different approaches to stochastic DEA (see, for example, Banker, 1988, 1993; Ray, 2004, Chapter 12, pp 307–326; and the survey paper Grosskopf, 1996). Chance-constrained programming formulations for stochastic characterizations of efficiency and dominance in DEA were studied by Cooper et al (1998). In a study by Cooper et al (2002), ‘ordinary DEA formulations are replaced with stochastic counterparts in the form of a series of chance constrained programming models’, with ‘emphasis on technical efficiencies and inefficiencies which do not require costs or prices, but which are nevertheless basic in that the achievement of technical efficiency is necessary for the attainment of ‘allocative’, ‘cost’ and ‘other types of efficiencies’. Desai et al (2005), considering chance-constrained formulations of DEA that allow random variations in the data, ‘suggest that, in keeping with the tradition of DEA, the simulation approach allows users to explicitly consider different data generating processes and allows for greater flexibility in implementing DEA under stochastic variations in data’. There are chance-constrained programming approaches to congestion in stochastic DEA in a study by Cooper et al (2004). Congestion in stochastic DEA based on the input relaxation approach was studied by Asgharian et al (2010). The deterministic equivalent to the stochastic congestion model was obtained, and when allowable limits of data variations for evaluating DMU were permitted, sensitivity analysis was studied. Huang and Li (1996) developed stochastic models in DEA with the possibility of random variations in input–output data. Dominance structures on the DEA envelopment side are used to incorporate the model builder's preferences and to discriminate efficiencies among DMUs. The efficiency measure can be characterized by solving a chance-constrained programming problem. The relationships between the general stochastic DEA models and the conventional DEA models are discussed. Stochastic DEA models with different types of input–output disturbances were studied by Huang and Li (2001). An input relaxation measure of efficiency in stochastic DEA was studied by Khodabakhshi and Asgharian (2009). They introduced a stochastic version of an input relaxation model in DEA, which ‘allows more changes in the input combinations of DMUs than those in the observed inputs of evaluating DMU’. A non-linear deterministic equivalent to this stochastic model under fairly general conditions can be replaced by an ordinary deterministic DEA model. The model is illustrated using a real data set. Khodabakhshi (2009) studied the estimating of the most productive scale size with stochastic data in DEA. To solve the stochastic model, a deterministic non-linear equivalent is obtained, which can be converted to a quadratic program. Using the proposed approach, the performance of software companies is evaluated. A chance-constrained additive input relaxation model in stochastic DEA was studied by Khodabakhshi (2010a). An input-oriented super-efficiency measure in stochastic DEA was studied by Khodabakhshi et al (2010). Sensitivity analysis of the proposed super-efficiency model is discussed. The results are illustrated by evaluating chief executive officers of US public banks and thrifts. For an output-oriented super-efficiency measure in stochastic DEA when considering Iranian electricity distribution companies, see Khodabakhshi (2010b). In a study by Li (1998), stochastic DEA models were developed and the stochastic efficiency measure of a DMU was defined through joint probabilistic comparisons of inputs and outputs with other DMUs. The measure can be characterized by solving a chance-constrained programming problem. An analysis of stochastic variable returns to scale was also developed. Morita and Seiford (1999) ‘consider an efficiency analysis of DMUs by using inputs and outputs data with stochastic variations, and discuss some stochastic measures of efficiency taking into account of the measurement error. The most interesting characteristic is reliability and robustness of the efficiency result’. Ruggiero (2004) considers ‘panel data models of efficiency estimation. One DEA model that has been used averages cross-sectional efficiency estimates across time and has been shown to work relatively well. It is shown that this approach leads to biased efficiency estimates and provides an alternative model that corrects this problem. The approaches are compared using simulated data for illustrative purposes’. Sengupta (1987) studied DEA for efficiency measurement in a stochastic case with some empirical applications in the measurement of the efficiency of public schools to test the sensitivity and robustness of efficiency ranking and measurement. Sengupta (1998) developed ‘pre- and post-optimality analysis of input–output data as a practical method of Farell-type efficiency measurement. Filtering of noise and specifying the optimal output distributions provide the basic tools, which are applied to time series data for international airlines’. In a study by Sengupta (2000a), ‘the nonparametric approach of efficiency analysis is generalized in the stochastic case, when the input prices and capital adjustment costs vary. An empirical application illustrates the economic applications’. One of the features of the study by Sengupta (2000b) is the consideration of ‘the stochastic aspects of DEA efficiency in terms of the stochastic dominance approach’. ‘Various applied aspects of dynamic and stochastic efficiency theory are also explained, so that the DEA model can be empirically implemented in diverse economic situations’. Kao and Liu (2009) studied measuring the efficiency of Taiwan commercial banks using stochastic DEA. The results obtained show that ‘when multiple observations are available for each DMU, the stochastic-data approach produces more reliable and informative results than the average-data and interval-data approaches do’. Sueyoshi (2000) presented a stochastic DEA model and reformulated it in a manner that the stochastic model can incorporate future information. The proposed approach was applied to plan the restructuring strategy of a Japanese petroleum company. Watson et al (2011) proposed the application of a simulation approach to stochastic DEA and contributed to the existing finance literature by examining whether the ratings of the Morningstar in Australia provide useful information for an investor by way of investigating the efficiency of domestic Australian equity funds. The benefit of the paper for investors and fund managers is the improved efficiency of stochastic DEA as a tool for measuring the efficiency of DMUs within investment fund markets. Kousmanen and Johnson (2010) show ‘that DEA can be alternatively interpreted as a nonparametric least-squares regression subject to shape constraints on the frontier and sign constraints on residuals’. The DEA model that is observed is a standard multiple input and single output model. They also developed a non-parametric variant of the corrected ordinary least-squares method and show that this method is consistent and asymptotically unbiased. Collier et al (2011) developed ‘deterministic cross sectional and stochastic panel data regression models that allow multiple inputs and outputs’. They also show how technical efficiency can be estimated without input price data using regression models. Kousmanen and Kortelainen (2012) examined ‘an encompassing semiparametric frontier model that combines the DEA-type non-parametric frontier, which satisfies monotonicity and concavity, with the Stochastic Frontier Analysis (SFA)-style homoscedastic composite error term’. A new two-stage method is proposed to estimate this model.