Let's solve the Linear Programming problem using the Simplex Method step by step.

Problem:

Maximize $Z = 6x + 3y$

Subject to constraints:

1. $2x + y \leq 8$
2. $3x + 3y \leq 18$
3. $y \leq 3$
4. $x, y \geq 0$

Step 1: Convert constraints to equalities using slack variables

Introduce slack variables $s_1, s_2, s_3$ for each inequality:

1. $2x + y + s_1 = 8$
2. $3x + 3y + s_2 = 18$
3. $y + s_3 = 3$

Now, rewrite the objective function and constraints in standard form:

• Maximize $Z = 6x + 3y + 0s_1 + 0s_2 + 0s_3$
• Subject to: 2x + y + s_1 &= 8 \\ 3x + 3y + s_2 &= 18 \\ y + s_3 &= 3 \\ x, y, s_1, s_2, s_3 &\geq 0 \end{aligned}$$

Step 2: Initial Simplex Table

The initial simplex table is as follows:

Step 3: Simplex Iteration 1

Step 3.1: Identify entering and leaving variables

• Entering variable: The column with the most negative value in the bottom row of the objective function. Here, $x$ ($-6$) is the entering variable.
• Leaving variable: Determine the row using the minimum ratio test: $\text{RHS} / \text{Pivot column}$. $\text{Ratios: } \frac{8}{2} = 4, \frac{18}{3} = 6, \frac{3}{0} = \infty$
  • The minimum ratio is 4 (row 1), so $s_1$ is the leaving variable.

Step 3.2: Perform row operations to make the pivot element 1

• Pivot element: $2$ (row 1, column 1).

Divide row 1 by 2: $R1 = R1 / 2 \quad \Rightarrow \quad \text{New row 1: } [1, 0.5, 0.5, 0, 0, 4]$

Perform row operations on the other rows to make the pivot column 0: $R2 = R2 - 3 \cdot R1, \quad R3 = R3 - 0 \cdot R1, \quad Z = Z + 6 \cdot R1$

New tableau:

Step 4: Simplex Iteration 2

Step 4.1: Identify entering and leaving variables

• Entering variable: $y$ ($0.5$ in row 1).
• Leaving variable: Perform the ratio test. After adjustments, we find $y$ will continue improving the solution.

At this stage, we see that the iterations stabilize.

Final Solution:

• $x = 4$, $y = 0$
• Maximum $Z = 24$

Would you like more details or want to discuss a specific step?

Related Questions:

1. What is the geometric interpretation of the Simplex Method?
2. How do slack variables help in converting inequalities into equalities?
3. Can you describe the two-phase simplex method?
4. What is the difference between primal and dual linear programming?
5. What are some real-world applications of linear programming?

Tip: Always perform the minimum ratio test carefully to avoid errors in identifying the leaving variable.