To solve the linear programming problem (LPP) using the simplex method, we need to follow a series of steps. Here’s how we can do it:

Step 1: Convert the inequalities to equalities by adding slack variables.

Given constraints:

1. $36x_1 + 6x_2 \geq 108$
2. $3x_1 + 12x_2 \geq 36$
3. $20x_1 + 10x_2 \geq 100$

Rewriting them as equalities with slack variables:

1. $36x_1 + 6x_2 - s_1 = 108$
2. $3x_1 + 12x_2 - s_2 = 36$
3. $20x_1 + 10x_2 - s_3 = 100$

Also, $x_1, x_2, s_1, s_2, s_3 \geq 0$.

Step 2: Set up the initial simplex tableau.

The objective function $z = 20x_1 + 40x_2$ can be rewritten in terms of minimizing $z$ as:

$z - 20x_1 - 40x_2 = 0$

Initial simplex tableau:

& x_1 & x_2 & s_1 & s_2 & s_3 & z & \text{RHS} \\ \hline s_1 & 36 & 6 & -1 & 0 & 0 & 0 & 108 \\ s_2 & 3 & 12 & 0 & -1 & 0 & 0 & 36 \\ s_3 & 20 & 10 & 0 & 0 & -1 & 0 & 100 \\ \hline z & -20 & -40 & 0 & 0 & 0 & 1 & 0 \\ \end{array}$$ **Step 3: Perform the simplex iterations.** To identify the pivot element, we look for the most negative value in the bottom row of the tableau. Here, it’s $$-40$$ (under $$x_2$$). Next, we find the pivot row by dividing the RHS values by their corresponding positive $$x_2$$ coefficients: $$\frac{108}{6} = 18, \quad \frac{36}{12} = 3, \quad \frac{100}{10} = 10$$ The smallest ratio is 3, so the pivot row is the second row (corresponding to $$s_2$$). **Step 4: Pivot to form a new tableau.** The pivot element is 12 in row 2, column 2. We now perform row operations to make all other entries in the pivot column zero and the pivot element one. $$\begin{array}{c|cccccc|c} & x_1 & x_2 & s_1 & s_2 & s_3 & z & \text{RHS} \\ \hline s_1 & 36 & 6 & -1 & 0 & 0 & 0 & 108 \\ x_2 & \frac{1}{4} & 1 & 0 & -\frac{1}{12} & 0 & 0 & 3 \\ s_3 & 15 & 0 & 0 & \frac{5}{6} & -1 & 0 & 70 \\ \hline z & -10 & 0 & 0 & \frac{10}{3} & 0 & 1 & 120 \\ \end{array}$$ **Step 5: Check for optimality.** All the coefficients in the bottom row (excluding the RHS column) are non-negative, indicating that the current solution is optimal. **Step 6: Read the solution.** The values for $$x_1$$ and $$x_2$$ can be found from the tableau: - $$x_1 = 0$$ - $$x_2 = 3$$ The slack variables: - $$s_1 = 108 - 6 \cdot 3 = 90$$ - $$s_2 = 0$$ - $$s_3 = 70 - 0 = 70$$ The minimum value of $$z$$: $$z = 120$$ Therefore, the solution to the LPP is $$x_1 = 0$$, $$x_2 = 3$$, and the minimum value of $$z$$ is 120. Would you like further details or have any questions? Here are some additional questions you might find interesting: 1. How do you solve a linear programming problem graphically? 2. What are the applications of the simplex method in real life? 3. Can you explain the duality theorem in linear programming? 4. How do sensitivity analysis and shadow prices relate to linear programming? 5. What are the differences between the simplex method and the interior-point method? **Tip:** Always double-check your pivot operations during each step of the simplex method to ensure accuracy.