Estimating the revenue efficiency of entities being evaluated is crucial as it provides valuable information about organizations, assuming that the output prices are known. This research introduces a new definition of optimal scale size (OSS) based on maximizing the average revenue efficiency (ARE). Additionally, the ARE is defined for both convex and non-convex sets, independent of returns to scale and the assumption that the vector of input-output prices of units is uniform. Moreover, to address the presence of uncertain data in real-world applications, the introduced ARE model is extended to evaluate systems with random inputs and outputs, along with approaches for its calculation. Finally, the proposed method is applied in an experimental example, calculating the ARE for a dataset of postal areas in Iran.IntroductionThe concept of optimal scale size has been extensively studied in the field of data envelopment analysis. Cesaroni and Giovannola's research on non-convex FDH technology reveals that the optimal scale size is a point in the production possibility set that minimizes average cost efficiency. Average cost efficiency, a new measure combining scale and allocation efficiencies, provides a more accurate performance assessment compared to cost and scale efficiencies. When evaluating units with known output prices instead of input prices, assessing revenue efficiency can offer more valuable insights. This paper extends the research on cost evaluation to revenue evaluation. It introduces the concepts of average revenue efficiency and optimal scale size based on revenue maximization. The optimal scale size based on revenue maximization is defined as the point in the production possibility set that maximizes the average radial income for the unit under investigation. Average revenue efficiency serves as an evaluation measure of unit revenue, surpassing revenue and scale efficiencies in accuracy. The paper examines methods for calculating average revenue efficiency in both convex and non-convex technologies. It demonstrates that the average revenue efficiency model in convex technology with variable returns to scale is equivalent to the revenue model with constant returns to scale. Furthermore, the relationship between optimal scale size points based on revenue maximization and the most productive scale size is determined. Next, the paper presents the average revenue efficiency model for stochastic sets with the presence of stochastic data. An experimental example is used to calculate the average revenue efficiency and obtain the optimal scale size for a set of postal areas in Iran.Materials and MethodsThe study builds upon Cesaroni and Giovannola's method for calculating average cost efficiency and optimal scale size to develop models for average revenue efficiency and optimal scale size based on revenue. It also utilizes chance-constrained probabilistic models with a deterministic objective function in DEA literature to present average revenue efficiency for stochastic sets. The model is transformed from stochastic to deterministic and then converted into a linear model using the error structure method.Discussion and ResultsThis paper introduces average revenue efficiency and optimal revenue scale size, demonstrating the equivalence between the average revenue efficiency models in convex technology with variable returns to scale and those with constant returns to scale. It also presents the average revenue efficiency model for stochastic sets, enabling the calculation of average revenue efficiency and optimal revenue scale size for units with random inputs and outputs.ConclusionIn many real-world scenarios, particularly when output prices are known, evaluating revenue efficiency holds greater significance than cost efficiency. This study develops the concepts of average cost efficiency and optimal scale size for revenue evaluation, expanding upon the existing literature on data envelopment analysis. The paper demonstrates how average revenue efficiency can be calculated as a valuable and accurate measure of efficiency in convex and non-convex technologies, without making assumptions about returns to scale. By assuming the randomness of input and output variables and employing chance-constrained models, a quadratic deterministic model is presented to calculate average revenue efficiency. It is then transformed into a linear model assuming uncorrelated variables, enabling the determination of average revenue efficiency and optimal scale size based on revenue maximization for random units. The proposed models are applied to a real-world sample, evaluating the average revenue efficiency of twelve postal units. The results highlight the models' ability to provide a more accurate evaluation of revenue efficiency and identify the best revenue scale size as the reference for inefficient units
