Question

Soylent Diet Planning. Recall the LP from Problem 1.a of homework 1:

Maximize 0.1xB + 0.25xᵣ+ 0.2xY
s.t. 25xB + 110xᵣ + 7xY + 72xY ≥ 1500
0.1xB + 2.2xR + 0.65xY ≤ 20
6.3xB + 1.8xR + 4.1xY ≥ 100
xB + xR + xY = 30
xB, xR, xY ≥ 0

rewrite the LP in standard equality form

Answer 1

To rewrite the linear programming (LP) in standard equality form, introduce slack variables to convert the inequality constraints to equality constraints. The LP is rewritten as: Maximize 0.1xB + 0.25xᵣ+ 0.2xY, subject to: 25xB + 110xᵣ + 7xY + 72xY + s₁ = 1500, 0.1xB + 2.2xR + 0.65xY + s₂ = 20, 6.3xB + 1.8xR + 4.1xY + s₃ = 100, xB + xR + xY = 30, and xB, xR, xY, s₁, s₂, s₃ ≥ 0.

Step-by-step explanation:

To rewrite the linear programming (LP) in standard equality form, we need to convert the inequality constraints to equality constraints. We can do this by introducing slack variables. Let's rewrite the LP:

Maximize 0.1xB + 0.25xᵣ+ 0.2xY

Subject to:

• 25xB + 110xᵣ + 7xY + 72xY + s₁ = 1500
• 0.1xB + 2.2xR + 0.65xY + s₂ = 20
• 6.3xB + 1.8xR + 4.1xY + s₃ = 100
• xB + xR + xY = 30
• xB, xR, xY, s₁, s₂, s₃ ≥ 0

In the above equations, s₁, s₂, and s₃ are slack variables introduced to convert the inequalities to equalities.