The given linear programming problem can be solved using the Big-M method. Here, we'll go step-by-step to convert the inequalities into equalities by introducing slack, surplus, and artificial variables, and then formulate the initial simplex tableau.

Problem:

Objective function:

$\text{Maximize } Z = 3X_1 + 2X_2 + 3X_3$

Subject to constraints:

$2X_1 + X_2 + X_3 \leq 2$ $3X_1 + 4X_2 + 2X_3 \geq 8$ $X_1, X_2, X_3 \geq 0$

Step 1: Convert inequalities to equalities

1. First constraint: $2X_1 + X_2 + X_3 \leq 2$ Add a slack variable $S_1 \geq 0$ to convert it to an equality: $2X_1 + X_2 + X_3 + S_1 = 2$

2. Second constraint: $3X_1 + 4X_2 + 2X_3 \geq 8$ Subtract a surplus variable $S_2 \geq 0$ and add an artificial variable $A_1 \geq 0$: $3X_1 + 4X_2 + 2X_3 - S_2 + A_1 = 8$

Step 2: Define the new objective function

We introduce a penalty for artificial variables. The Big-M method assigns a large positive value $M$ (since we're maximizing, the penalty will be $-M$) for artificial variables in the objective function.

The new objective function becomes:

$\text{Maximize } Z = 3X_1 + 2X_2 + 3X_3 - M A_1$

Step 3: Set up the initial simplex tableau

Now, we can set up the initial tableau with the variables $X_1$, $X_2$, $X_3$, $S_1$, $S_2$, and $A_1$.

The system of equations is:

1. $2X_1 + X_2 + X_3 + S_1 = 2$
2. $3X_1 + 4X_2 + 2X_3 - S_2 + A_1 = 8$
3. Objective: $Z = 3X_1 + 2X_2 + 3X_3 - M A_1$

Tableau format:

\text{Basic} & X_1 & X_2 & X_3 & S_1 & S_2 & A_1 & \text{RHS} \\ \hline S_1 & 2 & 1 & 1 & 1 & 0 & 0 & 2 \\ A_1 & 3 & 4 & 2 & 0 & -1 & 1 & 8 \\ \hline Z & -3 & -2 & -3 & 0 & 0 & -M & 0 \\ \end{array}$$ ### Step 4: Solve the simplex tableau Now, the next step is performing the simplex iterations to solve the problem. We use the pivot operation to get an optimal solution by eliminating negative values in the objective row. Would you like me to continue with the simplex iterations for this problem? --- ### Relative questions: 1. What is the purpose of the Big-M method in linear programming? 2. How are slack, surplus, and artificial variables used in solving inequalities? 3. Why do we assign a large penalty $$M$$ to artificial variables? 4. Can the Big-M method be used for minimization problems as well? 5. How do we decide when to stop the simplex iterations? ### Tip: The Big-M method is helpful for handling equality and inequality constraints that involve artificial variables by penalizing them heavily in the objective function.