Question

The Janie Gioffre Drapery Company makes three types of draperies at two different locations. At location I, it can make 10 pairs of deluxe drapes, 20 pairs of better drapes, and 13 pairs of standard drapes per day. At location II, it can make 20 pairs of deluxe drapes, 50 pairs of better drapes, and 6 pairs of standard drapes per day. The company has orders for 2,000 pairs of deluxe drapes, 4,200 pairs of better drapes, and 1,200 pairs of standard drapes. The daily costs are $600 per day at location I and $825 per day at location II.

a. Formulate linear programming model based on the above problem that minimize the cost. (7 marks)

b. How many days should Janie schedule at each location in order to fill the orders at minimum cost? (Solve using the corner point method) (10 marks)

c. Find the minimum cost. (3 marks)

Answer 1

The optimal solution is to schedule 75 days at location I and 54 days at location II, resulting in a minimum cost of $89,550 to fulfill the drapery orders.

Certainly! Let's go through the calculations step by step:

a. Linear Programming Model:

The objective function is Z = 600x1 + 825x2, and the constraints are:

10x1 + 20x2 ≥ 2000 (Deluxe drapes)

20x1 + 50x2 ≥ 4200 (Better drapes)

13x1 + 6x2 ≥ 1200 (Standard drapes)

x1, x2 ≥ 0 (Non-negativity)

b. Corner Point Method:

To find the corner points, solve the systems of equations formed by pairs of constraints. The corner points are the intersections.

Corner Point 1:

Solve:

10x1 + 20x2 = 2000

20x1 + 50x2 = 4200

Solving, we get x1 = 120 and x2 = 76.

Corner Point 2:

Solve:

20x1 + 50x2 = 4200

13x1 + 6x2 = 1200

Solving, we get x1 = 75 and x2 = 54.

Corner Point 3:

Solve:

10x1 + 20x2 = 2000

13x1 + 6x2 = 1200

Solving, we get x1 = 60 and x2 = 70.

c. Minimum Cost:

Evaluate Z = 600x1 + 825x2 at each corner point:

For Corner Point 1: Z = 600(120) + 825(76) = 115,200 + 62,700 = 177,900

For Corner Point 2: Z = 600(75) + 825(54) = 45,000 + 44,550 = 89,550

For Corner Point 3: Z = 600(60) + 825(70) = 36,000 + 57,750 = 93,750

The minimum cost occurs at Corner Point 2, where x1 = 75 and x2 = 54, with a minimum cost of $89,550