# Optimal scale sizes and average-profit efficiency under uncertainty: A chance-constrained DEA approach

## TL;DR

This paper introduces a new method to evaluate profit efficiency under uncertainty, using a chance-constrained DEA approach to calculate optimal scale sizes.

## Contribution

A new definition of optimal scale size based on maximizing average profit efficiency is introduced.

## Key findings

- The average measure of profit efficiency is more accurate than comparative and profit efficiency measures.
- The proposed method was applied to postal areas in Iran, demonstrating its practical use.
- Profit efficiency models were adapted for stochastic environments to account for input and output errors.

## Abstract

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.