$3\mathrm{.}1-11.*$ The Omega Manufacturing Company has discontinued the production of a certain unprofitable product line. This act

created considerable excess production capacity. Management is considering devoting this excess capacity to one or more of three products; call them products $1,2,$ and $3.$ The available capacity on the machines that might limit output is summarized in the following table:

$\backslash$table$[[$Machine Type,$\backslash$table$[[$Available Time$],[($Machine Hours per Week$)]]],[$Milling machine,$500],[$Lathe$,350],[$Grinder$,150]]$

The nu

$3\mathrm{.}1-11.*$ The Omega Manufacturing Company has discontinued the production of a certain unprofitable product line. This act spectiv

Productivity coefficient $($in machine hours per unit$)$

$\backslash$table$[[$Machine Type,Product $1,$Product $2,$Product $3],[$Milling machine,$9,3,5],[$Lathe$,5,4,0],[$Grinder$,3,0,2]]$

The sales department indicates that the sales potential for products $1$ and $2$ exceeds the maximum production rate and that the sales potential for product $3$ is $20$ units per week. The unit profit would be $$50,$$$20,$ and $$25,$ respectively, on products $1,2,$ and $3.$ The objective is to determine how much of each product Omega should produce to maximize profit.

$($a$)$ Formulate a linear programming model for this problem.

c $($b$)$ Use a computer to solve this model by the simplex method.