Consider the following linear programming problem,

$$\begin{array}{lrl} \operatorname{minimize} \quad(3-p) x_{1}+p x_{2} & \\ \text { subject to } & 2 x_{1}+x_{2} & \geqslant 8 \\ x_{1}+3 x_{2} & \geqslant 9 \\ x_{1} & \leqslant 6 \\ x_{1}, x_{2} & \geqslant 0 \end{array}$$

Formulate the problem in a suitable way for solution by the two-phase simplex method.

Using the two-phase simplex method, show that if $2 \leqslant p \leqslant \frac{9}{4}$ then the optimal solution has objective function value $9-p$, while if $\frac{9}{4}<p \leqslant 3$ the optimal objective function value is $18-5 p$.