# Braess' Paradox in City Planning

This paper was written for my undergraduate Calculus III class. The project was an instance where I applied the standard techniques of optimization from my multivariable calculus course to analyze a semi-realistic phenomenon— Braess' Paradox.

Braess’ Paradox is a counterintuitive phenomenon, in which the removal of an edge in a congested network actually results in improved flow. In the 1968 paper “Uber ein Paradoxon aus der Verkehrsplanung” (Braess, 1968), Dietrich Braess first proposed a mathematical framework for detecting this paradox in a network. When Braess’ paper appeared in 1968, the application of mathematics to traffic planning was still an relatively untapped vein of inquiry. Today, the framework continues to be analyzed by researchers and used by transportation specialists in the design of traffic networks. The key techniques that are relevant in this project are as follows:

- Lagrange multipliers
- Partial derivatives
- Critical points and the second derivative test for surfaces

The project outline is designed by Kenneth M Monks, and the assignment can be found here. Finally, my finished paper can be retrieved here.