Question

Rewrite the linear programming problem in standard form:

Maximize z = 6x1 + 4x2 + 9x3
Subject to: 3x1 − x2 + x3 ≥ 16
x1 + x2 + x3 ≤ 20,
4x1 − 2x3 ≥ 6,
4x2 + 2x3 ≥ 10,
2x1 + 5x2 − x3 ≤ 40,
x1 ≤ 4,
x3 ≥ 2
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0need correct answer

Answer 1

To rewrite the linear programming problem in standard form, we convert the inequality constraints to equations and introduce slack variables. The constraints in standard form are: 3x1 - x2 + x3 + s1 = 16, x1 + x2 + x3 + s2 = 20, 4x1 - 2x3 + s3 = 6, 4x2 + 2x3 + s4 = 10, 2x1 + 5x2 - x3 + s5 = 40, x1 + s6 = 4, x3 + s7 = 2. The objective function in standard form is: z = 6x1 + 4x2 + 9x3 + 0s1 + 0s2 + 0s3 + 0s4 + 0s5 + 0s6 + 0s7.

Step-by-step explanation:

To rewrite the linear programming problem in standard form, we need to convert all the inequality constraints to equations and introduce slack variables. Let's start with the first constraint: 3x1 - x2 + x3 ≥ 16. We can add a slack variable s1 to make it an equation: 3x1 - x2 + x3 + s1 = 16. We repeat this process for all the other constraints.

The constraints in standard form are:

• 3x1 - x2 + x3 + s1 = 16
• x1 + x2 + x3 + s2 = 20
• 4x1 - 2x3 + s3 = 6
• 4x2 + 2x3 + s4 = 10
• 2x1 + 5x2 - x3 + s5 = 40
• x1 + s6 = 4
• x3 + s7 = 2

We also need to make sure that all variables are non-negative: x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, s1 ≥ 0, s2 ≥ 0, s3 ≥ 0, s4 ≥ 0, s5 ≥ 0, s6 ≥ 0, s7 ≥ 0.

Finally, the objective function in standard form is: z = 6x1 + 4x2 + 9x3 + 0s1 + 0s2 + 0s3 + 0s4 + 0s5 + 0s6 + 0s7.