A technology production company is optimizing its manufacturing process for five products: P$1,$ P$2,$ P$3,$ P$4,$ and P$5.$ Each product is manufactured using different amounts of three main resources: material $($M$),$ labor $($L$),$ and energy $($E$).$ The company's goal is to maximize the profits obtained from the production and sale of these products while respecting the available resource constraints. The company's data are as follows:

Profit per unit produced:

$-$P$1$:$$20$

$-$P$2$:$$25$

$-$P$3$:$$30$

$-$P$4$:$$40$

$-$P$5$:$$35$

Resource consumption per product unit $($in hours$/$materials$/$energy$)$:

P$1$: Material: $3,$ Labor: $2,$ Energy: $4$

P$2$: Material: $2,$ Labor: $3,$ Energy: $3$

P$3$: Material: $4,$ Labor: $5.$ Energy: $2$

P$4$: Material: $6.$ Labor: $3,$ Energy: $5$

P$5$: Material: $5,$ Labor: $4,$ Energy: $6$

Resource availability:

Available material: $50$ units

Available labor: $40$ hours

Available energy: $60$ nits

Additionally, to diversify the product portfolio, the company wants to manufacture at least one nit of each product and the number of P$3$ and P$4$ products together must not exceed $8$ units due to market constraints. Moreover, the total use of materials must not exceed $50$ nits; the total use of labor hours

must not exceed $40$ hours; and the total use of energy must not exceed $60$ nits.

a$.(12\mathrm{.}5$ points$).$ Formulate this problem as an I.P$.$

b$.(12\mathrm{.}5$ points$).$ Solve the following LP using Python and Gurobi. Report the Python code you used,

the optimal solution and the optimal value, or if the problem is unbounded or infeasible: