Flow Disaggregation: Underdetermined Non-negative Linear Systems

Abstract

This article proposes and tests an iterative approximation algorithm to solve underdetermined non-negative linear systems of the form $A\textbf{x}=\textbf{b}$, where A is a binary matrix and all the elements of $\textbf{x}$ and $\textbf{b}$ are positive, while the system admits an infinite number of solutions. The algorithm converges (experimentally) to a solution based on the initial problem configuration, and is characterized by non-negativity and disaggregation invariance properties. The solution arises from the necessity to address the air transport problem of generating itinerary flows based on leg flows, and its effectiveness is demonstrated in this article. The algorithm exhibits positive results in this particular use case and is also applicable to other similar problems related to flows in areas such as transportation, migration, or economy.

The authors would like to express their sincere appreciation to the entire team who contributed to this project. Their invaluable support and dedication were instrumental in the successful completion of this research. Special thanks are due to Juan Villaseca, Rodrigo Álvarez, Albert Ruiz, Clara Argerich, Jesús Sanz (Airbus team), Iñaki A. Apaolaza, David Berrocal, Raúl Juan, Samuel Alonso, Jairo Calderón (AIR Institute team) whose contributions were particularly noteworthy.