$3\mathrm{.}1-13.*$ 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$)]]],[$Miling machine,$500],[$Lathe$,350],[$Grinder$,150]]$

The number of machine hours required for each unit of the respective products is

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

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

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.