Consider the linear programming problem

$$\begin{aligned} \operatorname{minimize} \quad & 2 x_{1}-3 x_{2}-2 x_{3} \\ \text { subject to } \quad-& 2 x_{1}+2 x_{2}+4 x_{3} \leqslant 5 \\ & 4 x_{1}+2 x_{2}-5 x_{3} \leqslant 8 \\ & 5 x_{1}-4 x_{2}+\frac{1}{2} x_{3} \leqslant 5, \quad x_{i} \geqslant 0, \quad i=1,2,3 . \end{aligned}$$

(i) After adding slack variables $z_{1}, z_{2}$ and $z_{3}$ and performing one iteration of the simplex algorithm, the following tableau is obtained.

\begin{tabular}{c|rrrrrr|c} & $x_{1}$ & $x_{2}$ & $x_{3}$ & $z_{1}$ & $z_{2}$ & $z_{3}$ & \ \hline$x_{2}$ & $-1$ & 1 & 2 & $1 / 2$ & 0 & 0 & $5 / 2$ \ $z_{2}$ & 6 & 0 & $-9$ & $-1$ & 1 & 0 & 3 \ $z_{3}$ & 1 & 0 & $17 / 2$ & 2 & 0 & 1 & 15 \ \hline Payoff & $-1$ & 0 & 4 & $3 / 2$ & 0 & 0 & $15 / 2$ \end{tabular}

Complete the solution of the problem.

(ii) Now suppose that the problem is amended so that the objective function becomes

$$2 x_{1}-3 x_{2}-5 x_{3}$$

Find the solution of this new problem.

(iii) Formulate the dual of the problem in (ii) and identify the optimal solution to the dual.