Question 1
1.1 Consider the following mathematical programming model:
Minimise $z = x_1 + x_2 - x_3$
subject to
$\frac{x_1 + x_2}{x_1 - x_2 - x_3} \le 0.5$
$x_1 - 3x_3 = -2x_2$
$-2 \le x_1 + x_2 - x_3 \le 8$
and
$x_1, x_2, x_3 \ge 0.$
Transform this model to a standard LP model.
1.2 Consider the following LP model in algebraic representation form
Maximise $z = \sum_{j=1}^3 c_jx_j + \sum_{j=1}^3 h_j[\sum_{k=1}^3 (x_k - d_k)]$
subject to
$\sum_{j=1}^3 x_{ij} \le m_i$
$\sum_{k=1}^3 (x_k - d_k) \ge 0$
and
$x_j \ge 0$ ($j = 1, 2, 3$),
Here $x_j$ are decisions variables. In addition $c_j$, $h_j$, $d_j$ and $m_j$ are parameters that represent
entries of the following vectors
$[c_j] = \begin{bmatrix} 60 \\ 45 \\ 52 \end{bmatrix}$, $[h_j] = \begin{bmatrix} 30 \\ 24 \\ 28 \end{bmatrix}$, $[d_j] = \begin{bmatrix} 1000 \\ 1200 \\ 950 \end{bmatrix}$ and $[m_j] = \begin{bmatrix} 1200 \\ 1350 \\ 1100 \end{bmatrix}$.
Put this LP model in its standard form
