Linear Programming: Exercise Sheet $1$

$1$ Solve the following linear programming problem using graphical methods. Find $x_1$ and $x_2$ to maximise $x_1+2x_2$ subject to the constraints

$x_1+6x_{2}12$

$3x_1+5x_{2}15$

$x_{1}0,x_{2}0$

$2$ Repeat question $1$ in the case where

a$)$ the objective function becomes $2x_1+x_2$

b$)$ the constraints become

$-x_1+6x_{2}12$

$-3x_1+5x_{2}15$

$x_{1}0,x_{2}0$

with the original objective function.

$3$ Solve the following linear programming problem using graphical methods. Find $y_1$ and $y_2$ to minimise $y_1+y_2$ subject to the constraints

$y_1+2y_{2}3$

$2y_1+y_{2}5$

$y_{2}0$

$4$ Maximise $a{x}_1+x_2$ subject to the constraints

$4x_1+2x_{2}12$

$-x_1+x_{2}1$

$x_1+2x_{2}4$

$x_{1}0,x_{2}0$

Find the value as a function of $a.$

$5$ Repeat question $1$ using the simplex method in two dimensions and confirm that you get the same result.

$6$ Repeat question $3$ using the simplex method in two dimensions and confirm that you get the same result.

$1$