$(35$ points$)$ As a financial advisor, your role is to help clients achieve their savings goals while minimizing the cost of their total investment. Today, a couple has approached you to create an investment portfolio consisting of stocks from Company A and Company B$.$ Their goal is to make at least $$20$ in profit by the end of the year, in addition to recovering their initial investment. You know that for each dollar invested in Stock A$,$ they will receive $$1\mathrm{.}50$ at the end of the year, while each dollar invested in Stock B will yield $$1\mathrm{.}20.$ Due to regulatory constraints, they are allowed to invest at most $1\mathrm{.}5$ times more in Stock A than in Stock B$.$ Additionally, the couple prefers Stock B for its lower risk, despite its lower return, and has requested that at least $$5$ be invested in Stock B$.$

The total budget available for their investment is $$100.$ Your task is to determine the minimum$-$cost portfolio that meets all these requirements.

$($a$)(10$ points$)$ Define the decision variables for this problem. What is its gradient?

$1$

$($b$)(5$ points$)$ Plot the gradient and the feasible region $($clearly indicating all the constraints and "shading" the feasible region$).$ Solve this problem using the graphical method, indicating the values of all the variables and the objective function associated with the optimal solution.

$($c$)(10$ points$)$ Solve the problem using Excel Solver and indicate the values of all the variables and the objective function associated with the optimal solution. Compare your results with part c$.$ Include a snapshot of your Excel model $($Excel cells and the Solver window$).$

$($d$)(10$ points$)$ Solve the problem using Gurobi$/$Python$.$ Compare your results with parts b and c$.$ Include a snapshot of your Gurobi$/$Python code and obtained results.