Answer :
To solve the given linear programming problem using the Simplex method, we need to convert all inequalities into equalities. This is achieved by introducing slack variables. Let's go through the steps in detail:
### Step 1: Introduce Slack Variables
Given constraints:
1. [tex]\( 4x_1 + 3x_2 \leq 7 \)[/tex]
2. [tex]\( x_1 + 4x_2 \leq 8 \)[/tex]
We introduce slack variables [tex]\( s_1 \)[/tex] and [tex]\( s_2 \)[/tex] to convert these inequalities into equalities. Slack variables measure the difference between the left-hand side and the right-hand side of the inequality, ensuring the constraints are met as equalities.
1. [tex]\( 4x_1 + 3x_2 + s_1 = 7 \)[/tex]
2. [tex]\( x_1 + 4x_2 + s_2 = 8 \)[/tex]
Here, [tex]\( s_1 \geq 0 \)[/tex] and [tex]\( s_2 \geq 0 \)[/tex].
### Step 2: Set Up the Objective Function
The objective function to maximize is:
[tex]\[ z = 3x_1 + x_2 \][/tex]
We need to express this in a form suitable for the simplex tableau. In the tableau form, the coefficients of the slack variables in the objective function are zero.
### Step 3: Construct the Initial Simplex Tableau
The simplex tableau includes the coefficients of the variables from the constraints and the objective function. Each row corresponds to one constraint, and the last row corresponds to the negation of the objective function (since we are maximizing).
The initial Simplex tableau is constructed as follows:
| Basic Variables | [tex]\(x_1\)[/tex] | [tex]\(x_2\)[/tex] | [tex]\(s_1\)[/tex] | [tex]\(s_2\)[/tex] | RHS |
|-------------------|--------|--------|--------|--------|------|
| Slack variable [tex]\(s_1\)[/tex] | 4 | 3 | 1 | 0 | 7 |
| Slack variable [tex]\(s_2\)[/tex] | 1 | 4 | 0 | 1 | 8 |
| Objective Function row | -3 | -1 | 0 | 0 | 0 |
However, in the context of adjusting the tableau for the Simplex algorithm specifically (as we translate this programatically in Python), we structure it as described below.
Creating the initial simplex tableau:
```plaintext
| x_1 | x_2 | s_1 | s_2 | RHS |
|--------|--------|--------|--------|------|
| 4 | 3 | 1 | 0 | 7 |
| 1 | 4 | 0 | 1 | 8 |
| -3 | -1 | 0 | 0 | 0 |
```
This final tabular form correctly represents our problem ready to be tackled by the Simplex method.
Thus, the initial simplex tableau is:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x_1 & x_2 & s_1 & s_2 & \text{RHS} \\ \hline 4 & 3 & 1 & 0 & 7 \\ 1 & 4 & 0 & 1 & 8 \\ 3 & 1 & 0 & 0 & 0 \\ \hline \end{array} \][/tex]
This tableau sets up the problem for application of the Simplex algorithm to find the optimal solution.
### Step 1: Introduce Slack Variables
Given constraints:
1. [tex]\( 4x_1 + 3x_2 \leq 7 \)[/tex]
2. [tex]\( x_1 + 4x_2 \leq 8 \)[/tex]
We introduce slack variables [tex]\( s_1 \)[/tex] and [tex]\( s_2 \)[/tex] to convert these inequalities into equalities. Slack variables measure the difference between the left-hand side and the right-hand side of the inequality, ensuring the constraints are met as equalities.
1. [tex]\( 4x_1 + 3x_2 + s_1 = 7 \)[/tex]
2. [tex]\( x_1 + 4x_2 + s_2 = 8 \)[/tex]
Here, [tex]\( s_1 \geq 0 \)[/tex] and [tex]\( s_2 \geq 0 \)[/tex].
### Step 2: Set Up the Objective Function
The objective function to maximize is:
[tex]\[ z = 3x_1 + x_2 \][/tex]
We need to express this in a form suitable for the simplex tableau. In the tableau form, the coefficients of the slack variables in the objective function are zero.
### Step 3: Construct the Initial Simplex Tableau
The simplex tableau includes the coefficients of the variables from the constraints and the objective function. Each row corresponds to one constraint, and the last row corresponds to the negation of the objective function (since we are maximizing).
The initial Simplex tableau is constructed as follows:
| Basic Variables | [tex]\(x_1\)[/tex] | [tex]\(x_2\)[/tex] | [tex]\(s_1\)[/tex] | [tex]\(s_2\)[/tex] | RHS |
|-------------------|--------|--------|--------|--------|------|
| Slack variable [tex]\(s_1\)[/tex] | 4 | 3 | 1 | 0 | 7 |
| Slack variable [tex]\(s_2\)[/tex] | 1 | 4 | 0 | 1 | 8 |
| Objective Function row | -3 | -1 | 0 | 0 | 0 |
However, in the context of adjusting the tableau for the Simplex algorithm specifically (as we translate this programatically in Python), we structure it as described below.
Creating the initial simplex tableau:
```plaintext
| x_1 | x_2 | s_1 | s_2 | RHS |
|--------|--------|--------|--------|------|
| 4 | 3 | 1 | 0 | 7 |
| 1 | 4 | 0 | 1 | 8 |
| -3 | -1 | 0 | 0 | 0 |
```
This final tabular form correctly represents our problem ready to be tackled by the Simplex method.
Thus, the initial simplex tableau is:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x_1 & x_2 & s_1 & s_2 & \text{RHS} \\ \hline 4 & 3 & 1 & 0 & 7 \\ 1 & 4 & 0 & 1 & 8 \\ 3 & 1 & 0 & 0 & 0 \\ \hline \end{array} \][/tex]
This tableau sets up the problem for application of the Simplex algorithm to find the optimal solution.
