A company produces two types of products, Product A and Product B$,$ using two machines, Machine X and Machine Y$.$ Each unit of Product A requires $3$ hours on Machine X and $2$ hours on Machine Y$,$ while each unit of Product B requires $1$ hour on Machine X and $4$ hours on Machine Y$.$ The company has $120$ hours available on Machine X and $100$ on Machine Y per week. Additionally, the company must satisfy the following constraints:

$•$ The company cannot produce more than $50$ units of Product A per week.

$•$ The profit per unit of Product A is $$50,$ and the profit per unit of Product B is $$40.$ The company wishes to determine how many units of each product to produce weekly to maximize total profit.

a$)(10$ points$)$ Write down a linear programming model for this problem. Explain each variable and constraint.

b$)(10$ points$)$ Use the graphical method to solve this model.

c$)(5$ points bonus$)$ Use the simplex algorithm. Show the entire process step by step and ensure your final answer is consistent with part b$.$ 

d$)(10$ points$)$ Use Python $($via the package we introduced in this course$)$ to solve this problem, and make sure your answer is consistent with part b$.$ Also, report the sensitivity analysis parameters such as slack and shadow prices for all constraints.

e$)(10$ points$)$ What constraints are binding? Interpret the shadow price value of each constraint. use python