Questions: Using the big M method to find the maximum value. Maximize P=8x1+5x2+6x3 subject to 2x1+x2+2x3 <= 48 2x1+x2-2x3=5 x1, x2, x3 >= 0 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The maximum value P= at x1= x2= x3= B. There is no solution.

Using the big M method to find the maximum value.

Maximize P=8x1+5x2+6x3
subject to 2x1+x2+2x3 <= 48
2x1+x2-2x3=5
x1, x2, x3 >= 0

Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. The maximum value P= at x1= x2= x3=
B. There is no solution.

Transcript text: Using the big M method to find the maximum value. \[ \begin{array}{l} \text { Maximize } \quad \mathrm{P}=8 \mathrm{x}_{1}+5 \mathrm{x}_{2}+6 \mathrm{x}_{3} \\ \text { subject to } 2 x_{1}+x_{2}+2 x_{3} \quad \leq 48 \\ 2 x_{1}+x_{2}-2 x_{3}=5 \\ x_{1}, x_{2}, x_{3} \quad \geq 0 \end{array} \] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The maximum value $P=\square$ $\square$ at $x_{1}=$ $\square$ $x_{2}=$ $\square$ $x_{3}=$ $\square$ B. There is no solution.

Solution

Solution Steps

To solve this linear programming problem using the Big M method, we need to convert the given constraints into a standard form suitable for the simplex method. This involves introducing slack and artificial variables to handle inequalities and equalities, respectively. The Big M method uses a large constant $ M $ to penalize artificial variables in the objective function, ensuring they are driven to zero in the optimal solution. We then apply the simplex algorithm to find the maximum value of the objective function.

Step 1: Formulate the Problem

We are tasked with maximizing the objective function $ P = 8x_1 + 5x_2 + 6x_3 $ subject to the constraints: \[ \begin{align_} 2x_1 + x_2 + 2x_3 & \leq 48 \\ 2x_1 + x_2 - 2x_3 & = 5 \\ x_1, x_2, x_3 & \geq 0 \end{align_} \]

Step 2: Convert to Standard Form

To apply the Big M method, we convert the inequalities and equalities into standard form by introducing slack and artificial variables. The inequality becomes: \[ 2x_1 + x_2 + 2x_3 + s_1 = 48 \] where $ s_1 $ is a slack variable. The equality constraint remains as is, but we introduce an artificial variable $ a_1 $: \[ 2x_1 + x_2 - 2x_3 + a_1 = 5 \]

Step 3: Set Up the Objective Function

The modified objective function, incorporating the artificial variable with a large penalty $ M $, is: \[ P = 8x_1 + 5x_2 + 6x_3 - Ma_1 \]

Step 4: Solve the Problem

After applying the simplex method, we find the optimal solution. The maximum value of $ P $ is: \[ P = 197.0 \] with the corresponding values: \[ x_1 = 0.0, \quad x_2 = 26.5, \quad x_3 = 10.75 \]