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Introduction to Optimization Problem 1: Solve the following LPP using the Simplex method. Objective:
Maximize
$$
F(x_1, x_2, x_3) = 9x_1 + 10x_2 + 16x_3
$$
Constraints:
$$
\begin{align*}
&18x_1 + 15x_2 + 12x_3 \leq 360, \
&6x_1 + 4x_2 + 8x_3 \leq 192, \
&5x_1 + 3x_2 + 3x_3 \leq 180, \
&x_1, x_2, x_3 \geq 0.
\end{align*}
$$
Step 1: Rewrite in Standard Form Convert inequalities to equalities by introducing slack variables $s_1, s_2, s_3$ .
Objective:
Maximize
$$
F(x_1, x_2, x_3) = 9x_1 + 10x_2 + 16x_3
$$
Constraints:
$$
\begin{align*}
&18x_1 + 15x_2 + 12x_3 + s_1 = 360, \
&6x_1 + 4x_2 + 8x_3 + s_2 = 192, \
&5x_1 + 3x_2 + 3x_3 + s_3 = 180, \
&x_1, x_2, x_3, s_1, s_2, s_3 \geq 0.
\end{align*}
$$
Basic
$z$ $s_1$ $s_2$ $s_3$
$x_1$ $-9$ 18
6
5
$x_2$ $-10$ 15
4
3
$x_3$ $-16$ 12
8
3
$s_1$ $0$ 1
0
0
$s_2$ $0$ 0
1
0
$s_3$ $0$ 0
0
1
Solution: $[0, 360, 192, 180]$
Iteration 1: Entering Variable $x_3$ , Leaving Variable $s_2$ Compute $\theta = \frac{b_i}{a_{i3}}$ :
$$
\theta = \min\left(\frac{360}{12}, \frac{192}{8}, \frac{180}{3}\right) = 24.
$$
New Basic Solution:
$x_3$ enters, $s_2$ leaves.
Basic
$z$ $s_1$ $x_3$ $s_3$
$x_1$ $-9 + 4$ 18
6/8
5
$x_2$ $-10 + 2$ 15
4/8
3
$s_1$ $0$ 1
0
0
$x_3$ $0$ 12
1
3
$s_2$ $0$ 0
8/8 = 1
0
$s_3$ $0$ 0
0
1
Solution: $[72, 24, 192 - 24 \cdot 3, 108]$
Iteration 2: Entering Variable $x_2$ , Leaving Variable $s_1$ Compute $\theta = \min\left(\frac{72}{9}, \frac{24}{\frac{1}{2}}, \frac{108}{\frac{3}{2}}\right) = 8$ .
New Basic Solution:
$x_2$ enters, $s_1$ leaves.
Basic
$z$ $x_2$ $x_3$ $s_3$
$x_1$ $9$ $\frac{5}{3}$ $\frac{11}{3}$ $\frac{3}{4} - \frac{1}{2}$
$x_2$ $0$ $1$ $0$ $\frac{3}{2}$
$s_1$ $0$ $0$ $0$ $-\frac{1}{6}$
$x_3$ $0$ $0$ $1$ $0$
$s_2$ $-3/2$ $\frac{5}{3}$ $\frac{-17}{3}$ $\frac{4}{3}$
$s_3$ $0$ $0$ $0$ $1$
Solution: $[400, 8, 20, 96]$
$$
x_1 = 0, \quad x_2 = 8, \quad x_3 = 20.
$$
Maximized value of $F(x) = 400$ .
Problem 2: Solve the following LPP using Simplex method. Objective:
Maximize
$$
F(x_1, x_2, x_3, x_4, x_5, x_6) = 2x_1 + 3x_2 - x_4
$$
Constraints:
$$
\begin{align*}
&2x_1 - x_2 - 2x_4 + x_5 = 16, \
&3x_1 + 2x_2 + x_3 - 3x_4 = 18, \
&-x_1 + 3x_2 + 4x_4 + x_6 = 24, \
&x_i \geq 0.
\end{align*}
$$
Basic variables: $x_5, x_3, x_2$ .
Basic
$z$ $x_5$ $x_3$ $x_2$
$x_1$ $-3$ $\frac{5}{3}$ $\frac{11}{3}$ $-\frac{1}{3}$
$x_2$ $0$ $0$ $0$ $1$
$x_3$ $9$ $\frac{-5}{3}$ $1$ $-\frac{2}{3}$
Solution: $[0, 16, 18, 24]$
Basic variables: $x_5, x_1, x_2$ .
Basic
$z$ $x_5$ $x_1$ $x_2$
$x_1$ $0$ $\frac{4}{3}$ $1$ $-\frac{1}{3}$
$x_2$ $0$ $0$ $0$ $1$
Solution: $[282/11, 254/11, 6/11, 90/11]$
$$
(x_1, x_2, x_3, x_4, x_5, x_6) = \left(\frac{