The standard deviation of data for 2016–2018.

When the costs of the inputs and outputs of the units under evaluation are known, the evaluation of the profit efficiency of the units is one of the most significant evaluations that can provide valuable information about them. In this research, first, a new definition of the optimal scale size based on the maximization of the average measure of profit efficiency is presented. The average measure of profit efficiency develops the concept of economic efficiency measure by introducing a more accurate measure of efficiency compared to the measure of comparative and profit efficiency. It has been shown that the average measure of profit efficiency in a convex space is equivalent to the measure of profit efficiency in constant returns to scale technology, and then, some models are presented to calculate profit efficiency in a stochastic environment, to increase the ability of profit models in real examples by considering the calculation errors of inputs and outputs. Finally, the proposed method is used in an empirical example to calculate the average profit efficiency of a set of postal areas in Iran.