Optimising the design of headphones

Problem definition

You work on the R&D department of Banshee Ltd., a company that manufactures headphones. Through a customer survey you have determined that the level of satisfaction of your headphones is given by the following expression:

\(z = 5 + ln(x_1) + x_2^2 + x_1*x_2\)

where \(x_1\) is the thickness of the diaphragm in microns and \(x_2\) is the radius of the diaphragm in millimeters. Due to design constraints, the volume of the driver, which can be estimated as \(0.2*\pi*x_2*x_1\) cannot be larger than 0.172 cubic millimeters. The properties of the material you use makes it unfeasible to manufacture a diaphragm with a thickness lower than \(0.3\) microns or higher than \(0.9\) microns.

a Write a Non-Linear Programming (NLP) problem to find the optimal design specifications (thickness and radius of diaphragm) for Banshee Ltd.

b Write down the Lagrangian and the Kuhn Tucker conditions

c A solver gives you the following solution (\(x_1 = 0.9\), \(x_2 = 0.304\)). Determine if it is a local or global maximum