Question $3$

$20$

$3\mathrm{.}1$ A firm that makes three products and has three machines available as resources constructs the following LP problem:

Maximize profit $=4x_1+4x_2+7x_3$

subject $to1x_1+7x_2+4x_{3}100(hourson$ machine $1)$

$2x_1+1x_2+7x_{3}110(hourson$ machine $2)$

$8x_1+4x_2+1x_{3}110(hourson$ machine $3)$

Solve this LP problem by answering the following questions:

$3\mathrm{.}1\mathrm{.}1$ Before the third iteration of the simplex method, which machine still has unused time available?

$3\mathrm{.}1\mathrm{.}2$ When the final solution is reached, is there any unused time available on any of the three machines?

$3\mathrm{.}1\mathrm{.}3$ What would it be worth to the firm to make an additional hour of time available on the third machine?

$3\mathrm{.}1\mathrm{.}4$ How much would the firm's profit increase if an extra $10$ hours of time were made available on the second machine at no extra cost?