Efficiency evaluation of units is often a question of prime interest in many areas of application ranging from banking, business, economy to health care and many others. Efficiency analysis concerns with the performance of each unit in transforming their inputs into quantities of outputs. The efficiency is measured based on the deviation of the position of a specific Decision Making Units (DMUs) from the efficient frontier (production function). Data envelopment analysis (DEA) is a mathematical method for evaluating the relative efficiency of a set of homogeneous DMUs using linear programming. In many real applications, the observed data might be subject to uncertainty or might have been collected over several time periods such as monthly returns of hedge funds, number of patients visited a hospital in a week, and so on. In such cases, the efficiency of DMUs are also a random variable. We are the first to study the structure of efficiency distribution of DMUs. We show that the efficiency score does not have a continuous distribution even if the random input and output variables distributions are continuous. The efficiency score distribution has, in fact, a point mass decomposition at 1. We then propose several ranking methods for stochastic DMUs using the efficiency distribution and then, will explore their relationship between our proposed ranking methods.
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