Question

The Wheel Fun corporation produces roller skates, skateboards, and inline skates, and also sells replacement components. They can combine 8 regular wheels, 2 plates, and 2 boots to make a pair of roller skates, or they can combine 4 regular wheels, 2 plates, and a board to make a skateboard. They can combine 8 large wheels, 2 plates, and 2 boots to make inline skates. They can sell regular or large wheels for $2 profit; they can sell one boot for $6 profit; they can sell boards for a $5 profit, and they cannot sell plates for a profit. Selling a pair of roller skates yields a $40 profit, selling a skateboard yields a $20 profit, and selling inline skates yields a $30 profit. Their current stock is 600 regular wheels, 60 large wheels, 500 plates, 80 boots, and 30 boards. Ignoring integrality concerns, formulate a linear program to decide what Wheel Fun should sell, to maximize profit from its stock of components.

Answer 1

The Wheel Fun corporation produces roller skates, skateboards, and inline skates, and also sells replacement components. They can combine 8 regular wheels, 2 plates, and 2 boots to make a pair of roller skates, or they can combine 4 regular wheels, 2 plates, and a board to make a skateboard. They can combine 8 large wheels, 2 plates, and 2 boots to make inline skates. They can sell regular or large wheels for $2 profit; they can sell one boot for $6 profit; they can sell boards for a $5 profit, and they cannot sell plates for a profit. Selling a pair of roller skates yields a $40 profit, selling a skateboard yields a $20 profit, and selling inline skates yields a $30 profit. Their current stock is 600 regular wheels, 60 large wheels, 500 plates, 80 boots, and 30 boards. Ignoring integrality concerns, formulate a linear program to decide what Wheel Fun should sell, to maximize profit from its stock of components.

Ans:

The corporation sells following items

Roller skates - 8 regular wheels, 2 plates and 2 boots -> $40 profit

Skateboards - 4 regular wheels, 2 plates and 1 board -> $20 profit

Inline skates - 8 large wheels, 2 plates and 2 boots -> $30 profit

Profit/item

Regular wheels - $2

Large wheels - $2

Boot - $6

Boards - $5

Plates - $0

Stock: 600 regular wheels, 60 large wheels, 500 plates, 80 boots, 30 boards

Let fun wheel make x, y and z quantities of roller skates, skateboards and inline skates

Now we have to make roller skates, skateboards and inline such that the total profit is maximized

The total profit for these items = 40x + 20y + 30z

Further, total regular wheels used = 8x + 4y <= 600

Total large wheels used = 8z <= 60

Total plates used = 2x + 2y + 2z <= 500

Total boots used = 2x + 2z <= 80

Total boards used = y <= 30

Also, any remainig inventory of regular wheels, large, boots and boards will be sold at profit of 2, 2, 6 and 5 per unit respectively

Remaining regular wheels = 600 - (8x + 4y)

Remaining large wheels = 60 - 8z

Remaining boots = 80 - 2x - 2z

Remaining boards = 30 - y

So the LP is formulated as given below:

Maximise total profit = 40x + 20y + 30z + 2(600 - 8x - 4y) + 2(60-8z) + 6(80-2x-2z) + 5(30-y)

= 40x + 20y + 30z + 1200 - 16x - 8y + 120 - 16z + 480 -12x - 12z + 150 - 5y

= 12x + 7y + 2z + 1950

Subject to

8x + 4y <= 600

8z <= 60

2x + 2y + 2z <= 500

2x + 2z <= 80

y <= 30

x,y,z >= 0