To solve the linear programming problem using the Simplex method, we aim to maximize the objective function:

$Z = 20x + 50y$

subject to the following constraints:

$x + y \leq 30 \quad \text{(Constraint 1)}$ $x + 2y \leq 40 \quad \text{(Constraint 2)}$ $x \geq 0, \quad y \geq 0 \quad \text{(Non-negativity constraints)}$

Step 1: Convert the inequalities into equalities by adding slack variables

We introduce slack variables $s_1$ and $s_2$ to convert the inequalities into equalities:

$x + y + s_1 = 30$ $x + 2y + s_2 = 40$

Here, $s_1$ and $s_2$ represent the unused resources in each constraint. So, the system becomes:

$x + y + s_1 = 30$ $x + 2y + s_2 = 40$

Step 2: Set up the initial simplex tableau

We rewrite the objective function in standard form for the tableau. The objective function can be written as:

$Z - 20x - 50y = 0$

Now we set up the initial tableau, including the slack variables:

\text{Basic} & x & y & s_1 & s_2 & Z & \text{RHS} \\ \hline s_1 & 1 & 1 & 1 & 0 & 0 & 30 \\ s_2 & 1 & 2 & 0 & 1 & 0 & 40 \\ Z & -20 & -50 & 0 & 0 & 1 & 0 \\ \end{array}$$ ### Step 3: Apply the Simplex Method #### Iteration 1: Identify the entering and leaving variables - **Entering variable:** The most negative coefficient in the $$Z$$-row is $$-50$$ (corresponding to $$y$$), so $$y$$ is the entering variable. - **Leaving variable:** We calculate the ratio of the right-hand side to the coefficients of $$y$$ in each row: - For row 1: $$\frac{30}{1} = 30$$ - For row 2: $$\frac{40}{2} = 20$$ The smallest ratio is 20, so $$s_2$$ is the leaving variable. #### Iteration 2: Pivot and update the tableau We pivot around the element in row 2, column $$y$$, and update the tableau. After performing the pivot operation, the new tableau becomes: $$\begin{array}{c|ccccc|c} \text{Basic} & x & y & s_1 & s_2 & Z & \text{RHS} \\ \hline s_1 & 0 & -1 & 1 & -1 & 0 & 10 \\ y & 0.5 & 1 & 0 & 0.5 & 0 & 20 \\ Z & 5 & 0 & 0 & 25 & 1 & 1000 \\ \end{array}$$ ### Step 4: Check for optimality There are no more negative coefficients in the $$Z$$-row, so the current solution is optimal. ### Step 5: Interpret the solution The solution is: - $$y = 20$$ - $$x = 10$$ (found by substituting $$y = 20$$ into the equation $$x + 2y = 40$$) - The maximum value of $$Z = 1000$$ ### Final Answer: - $$x = 10$$ - $$y = 20$$ - Maximum $$Z = 1000$$ Would you like more details or have any questions? Here are 5 related questions to explore further: 1. What happens if you change the objective function to $$Z = 30x + 40y$$? 2. How does the Simplex method handle problems with more than two constraints? 3. Can you explain the significance of slack variables in the Simplex method? 4. How would you handle a minimization problem using the Simplex method? 5. What is the geometric interpretation of the Simplex method? **Tip:** Always check for negative coefficients in the objective function row to determine if the current solution is optimal.