Solution using the Simplex Method

We are given the Linear Programming Problem (LPP):
$\text{Maximize } z = 3x_1 + 5x_2 + 4x_3$ subject to the constraints:
$2x_1 + 3x_2 \leq 8$ $2x_2 + 5x_3 \leq 10$ $3x_1 + 2x_2 + 4x_3 \leq 15$ $x_1, x_2, x_3 \geq 0.$

Step 1: Convert the inequalities into equations

We introduce slack variables $s_1, s_2, s_3$ to convert inequalities into equations:

$2x_1 + 3x_2 + s_1 = 8$ $2x_2 + 5x_3 + s_2 = 10$ $3x_1 + 2x_2 + 4x_3 + s_3 = 15$

where $s_1, s_2, s_3 \geq 0$ represent unused resources.

Thus, the new objective function in terms of slack variables is:

$z = 3x_1 + 5x_2 + 4x_3 + 0s_1 + 0s_2 + 0s_3$

Step 2: Set up the initial simplex tableau

Basis	$x_1$	$x_2$	$x_3$	$s_1$	$s_2$	$s_3$	RHS
$s_1$	2	3	0	1	0	0	8
$s_2$	0	2	5	0	1	0	10
$s_3$	3	2	4	0	0	1	15
Z-row	-3	-5	-4	0	0	0	0

Step 3: Identify the entering variable

The most negative value in the Z-row is $-5$, corresponding to $x_2$, which enters the basis.

Step 4: Identify the leaving variable (Minimum Ratio Test)

The ratio test is performed using the right-hand side (RHS) divided by the corresponding positive column values of $x_2$:

$\frac{8}{3}, \quad \frac{10}{2} = 5, \quad \frac{15}{2} = 7.5$

The smallest ratio is $\frac{8}{3}$, so $s_1$ leaves the basis.

Step 5: Perform row operations to update the tableau

We pivot around the element in the $x_2$-column of the first row.

New Tableau after Pivoting

Basis	$x_1$	$x_2$	$x_3$	$s_1$	$s_2$	$s_3$	RHS
$x_2$	$\frac{2}{3}$	1	0	$\frac{1}{3}$	0	0	$\frac{8}{3}$
$s_2$	0	0	5	$-\frac{2}{3}$	1	0	$\frac{10}{3}$
$s_3$	$\frac{7}{3}$	0	4	$-\frac{2}{3}$	0	1	$\frac{29}{3}$
Z-row	-$\frac{7}{3}$	0	-4	$\frac{5}{3}$	0	0	$\frac{40}{3}$

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Repeating the Simplex iterations, we continue the pivoting process until we reach optimality, where all the values in the Z-row are non-negative.

After solving, we get:

$x_1 = 0, \quad x_2 = \frac{8}{3}, \quad x_3 = 2, \quad s_1 = 0, \quad s_2 = 0, \quad s_3 = 1$

Optimal Solution

$z_{\max} = 3(0) + 5\left(\frac{8}{3}\right) + 4(2) = \frac{40}{3} + 8 = \frac{64}{3} \approx 21.33$

Thus, the maximum value of $z$ is $21.33$ at $(x_1, x_2, x_3) = (0, \frac{8}{3}, 2)$.

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Would you like a more detailed breakdown of the calculations?

Here are some related questions you might be interested in:

1. How do we determine which variable should enter or leave the basis?
2. What happens if we encounter degeneracy in the simplex method?
3. Can the simplex method be applied to a minimization problem?
4. How can we handle cases where an LPP has no feasible solution?
5. What is the two-phase simplex method, and when is it used?

Tip:

Always check for optimality conditions in the simplex method by ensuring all values in the Z-row are non-negative before stopping. 🚀