High School

Use the simplex method to maximize

\[ P = 4x_1 + 7x_2 + 2x_3 \]

subject to the constraints:

\[ x_1 + 5x_2 + 5x_3 \leq 38 \]
\[ x_2 + 7x_3 \leq 7 \]
\[ 4x_3 \leq 10 \]
\[ x_1, x_2, x_3 \geq 0 \]

Answer :

By applying the simplex method to the given linear programming problem, we can maximize the objective function P = 4x1 + 7x2 + 2x3 subject to a set of constraints.

The simplex method is an iterative algorithm used to solve linear programming problems. Let's consider the given problem: maximize P = 4x1 + 7x2 + 2x3 subject to the following constraints:

-x1 - x2 + x3 + 5x2 + 5x3 ≤ 38,

7 + 4x3 ≤ 10,

x1, x2, x3 ≥ 0.

First, we convert the problem into standard form by introducing slack variables and forming the initial tableau. The initial tableau is:

Cb x1 x2 x3 S1 S2 RHS

P 4 7 2 0 0 0

S1 -1 -1 1 5 5 38

S2 0 0 4 0 0 3

The pivot element is chosen as the most negative coefficient in the objective row (P). In this case, it is -1, corresponding to the x1 variable. By performing row operations, we pivot on x1 and obtain a new tableau:

Cb x1 x2 x3 S1 S2 RHS

S1 0 -6 2 5 6 46

P 0 11 6 -5 -5 38

S2 0 0 4 0 0 3

Next, we select the most negative coefficient in the objective row (P) as the pivot element, which is -5 corresponding to the S1 variable. By performing row operations, we pivot on S1 and obtain the following tableau:

Cb x1 x2 x3 S1 S2 RHS

S2 0 6 -2 -5 -6 8

P 0 5 4 0 1 16.2

S1 0 0 4 0 0 3

Since there are no negative coefficients in the objective row (P), the solution is optimal. The optimal solution is x1 = 16.2, x2 = 0, x3 = 0, with the maximum value of P = 72.2. Thus, the objective function is maximized under the given constraints.

Learn more about simplex method here:

https://brainly.com/question/32298193

#SPJ11