Linear Programming Examples

Example 1: Maximizing Profit with Graphical Method

1. Find the feasible region and optimum solution for this linear programming model, using the graphical method. What is the maximum profit?
Maximize Z = 6.5x1 + 10x2
Subject to:
2x1 + 4x2 ≤ 40
x1 + x2 ≤ 15
x1 ≥ 8 x1, x2 ≥ 0
2. Find the feasible region and optimum solution for this linear programming model, using the graphical method. What is the minimum cost?
Minimize Z = 8x1 + 6x2
Subject to:
4x1 +2x2 ≥ 20
-6x1 +4x2 ≤ 12
x1 +x2 ≥ 6 x1, x2 ≥ 0
3. Do the following LP problems have solutions? Why or why not?
3.1 Maximize: P = 300 x + 500ySubject to:
3x + 5y ≤ 30
x + y ≥ 18
x, y ≥ 0
2
3.2 Maximize: P = 300x + 500y
Subject to :
x ≤ 18
y ≥ 2
x, y ≥ 0
4. A jewellery store makes necklaces and bracelets from gold and platinum. The store has 18 ounces of gold and 20 ounces of platinum. Each necklace requires 3 ounces of gold and 2 ounces of platinum,
whereas each bracelet requires 2 ounces of gold and 4 ounces of platinum. The demand for bracelets is no more than four. A necklace earns $300 in profit and a bracelet, $400. The store wants to maximize profit.
a) Formulate a linear programming model for this problem.
b) Find the feasible region and optimum solution for this linear programming model, using the graphical method. What is the maximum profit?