Problem $1$: Integer Programming $(25$ points$)$

An investment firm is looking to invest in a portfolio of projects. They only have $\backslash (\backslash$$ $10\backslash )$ million to invest, and want to maximize the expected return from their projects. The table below shows the capital requirements for each project, as well as the expected return.

Additionally,

$-$ If project E is chosen, project F must be chosen.

$-$ If project A is chosen, project C cannot be chosen.

$-$ The firm has to choose at least $3$ projects.

Formulate this problem as an integer programming model. Clearly define your decision variables, objective function, and constraints.