To rewrite the given linear programming problem using slack variables and determine the initial simplex tableau, follow these steps:

1. Problem Formulation

Given the problem:

Maximize:
$P = x_1 + x_2$

Subject to:
$2x_1 + x_2 \leq 4$
$x_1 + 2x_2 \leq 3$
$x_1, x_2 \geq 0$

2. Introduce Slack Variables

For each inequality constraint, we introduce a slack variable to convert the inequalities into equalities:

• The first constraint $2x_1 + x_2 \leq 4$ becomes: $2x_1 + x_2 + s_1 = 4$ where $s_1$ is the slack variable associated with the first constraint.

• The second constraint $x_1 + 2x_2 \leq 3$ becomes: $x_1 + 2x_2 + s_2 = 3$ where $s_2$ is the slack variable associated with the second constraint.

3. Rewriting the Objective Function

The objective function remains the same: $P = x_1 + x_2$

However, for the simplex method, we typically express it in terms of a maximization form with 0 on the right side: $\text{Maximize } Z = x_1 + x_2$
and the constraints become:

1. $2x_1 + x_2 + s_1 = 4$
2. $x_1 + 2x_2 + s_2 = 3$

4. Initial Simplex Tableau

Now we can set up the initial simplex tableau. The columns correspond to the variables $x_1$, $x_2$, $s_1$, $s_2$, and the right-hand side (RHS) values (the constants from the equations):

Basic Var	$x_1$	$x_2$	$s_1$	$s_2$	RHS
$s_1$	2	1	1	0	4
$s_2$	1	2	0	1	3
Z	-1	-1	0	0	0

Explanation:

• The basic variables $s_1$ and $s_2$ are initially in the solution.
• The coefficients for the objective function row (Z) are negated because we are solving for a maximization problem.
• The right-hand side (RHS) column contains the constants from the constraints.

Conclusion:

This is the correct formulation of the problem using slack variables, and the initial simplex tableau has been set up. If you'd like to proceed with the simplex method or have any further questions about this, feel free to ask!

Follow-up Questions:

1. How do you perform the simplex method from this initial tableau?
2. How do you interpret the simplex tableau at each iteration?
3. What are the conditions for the optimal solution in the simplex method?
4. What happens if there are multiple optimal solutions in the simplex method?
5. How do we handle a degenerate solution in the simplex method?

Tip:

Remember that introducing slack variables is essential to convert inequalities into equalities for the simplex method to work properly.