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
