Transition from Economics to Industrial Engineering

Hello everyone!

I currently hold a bachelor's and a master's degree in Economics. I am currently enrolled in an Economic Theory Master's degree since my intention was getting better letters of recommendation for my PhD applications. It has(d) always been my idea to pursue a PhD in Economics.

For a few months already I have had in mind the idea of switching fields, changing the path of my professional career. To some extent, I don't think I enjoy Economics that much anymore. I like Mathematics and I have being self-studying for a couple of years already. The idea of an Applied Mathematics PhD seems a little crazy for me, so after doing some research I saw that the most natural field I could pursue is Operations Research or Industrial Engineering.

In terms of my transcript, so far I have only taken two classes in Real Analysis (last year). The first one used Basic Anlysis by Lebl and the second one used Real Analysis by Carothers (Part 1) and a few parts from Bartle's book to study the Riemann-Stieltjes integral and some of Fourier analysis. I aced both classes. I could still take a few more Math classes (if needed) that may improve my profile for PhD applications.

Under the assumption that I don't really care changing fields whatsoever, these are my questions:

1. What is Industrial Engineering mainly about? I would be most interested in things that have to do with Optimization. If you can suggest any other fields I should consider, please let me know.
2. I am open to working in academia or industry. I enjoy teaching a lot, and at my current job I do Economic research.
3. Do you know schools who would accept people with an Economics background? Schools who openly welcome people with other backgrounds? I am still going through several websites.
4. Overall, would it be hard to go from Economics to Industrial Engineering?

I appreciate any advise. Please let me know if I should add more details.