# A New Mathematical Model for the Efficiency Calculation

## TL;DR

This paper introduces a novel mathematical model for calculating efficiency using differential equations to determine optimal input combinations, challenging traditional sample-based benchmarking methods.

## Contribution

It proposes a new approach that replaces sample comparison with a differential equation system to find the most efficient input mix.

## Key findings

- The model effectively determines optimal inputs without relying on sample data.
- Numerical example demonstrates the model's practical application.
- Challenges the traditional benchmark approach in efficiency analysis.

## Abstract

During the past sixty years, a lot of effort has been made regarding the productive efficiency. Such endeavours provided an extensive bibliography on this subject, culminating in two main methods, named the Stochastic Frontier Analysis (parametric) and Data Envelopment Analysis (non-parametric). The literature states this methodology also as the benchmark approach, since the techniques compare the sample upon a chosen `more-efficient' reference. This article intends to disrupt such premise, suggesting a mathematical model that relies on the optimal input combination, provided by a differential equation system instead of an observable sample. A numerical example is given, illustrating the application of our model's features.