Exhibit Palmer Pens
Assume that Palmer Executive Pens uses 1,440,000 gallons of ink each year.Further, assume that Palmer can order the ink at a cost of $2 per gallon plus fixed ordering costs of $100 per order.The firm's carrying cost is 20 percent of the inventory value, at cost.
-Refer to Exhibit Palmer Pens.Now, suppose the manufacturer offers a discount of 0.5 percent for orders of a least 40,000 gallons.Should Palmer increase its ordering quantity to take the discount?

Question

Quizplus · Accepted Answer

Currently, Palmer Executive Pens is ordering 1,440,000 gallons of ink per year. Assuming that the ink has a cost of $2 per gallon, the total cost of the ink without the discount is:

1,440,000 gallons x $2/gallon = $2,880,000

In addition, the firm incurs fixed ordering costs of $100 per order. To determine the number of orders placed per year, divide the annual ink usage by the order quantity:

1,440,000 gallons / Q = number of orders

where Q is the order quantity. Then, the total fixed ordering costs for the year can be calculated as:

number of orders x $100/order = $100 x (1,440,000/Q) = $144,000/Q

The carrying cost is 20 percent of the inventory value, at cost. Assuming that the inventory is held for an average of half a year, the average inventory can be estimated as:

(1/2) x Q = 0.5Q

Then, the inventory value is:

0.5Q x $2/gallon = $Q

and the annual carrying cost is:

0.2 x $Q = $0.2Q

Adding up all these costs, the total cost without the discount is:

Total cost = $2,880,000 + $144,000/Q + $0.2Q

To determine whether Palmer should take the discount, we need to calculate the total cost with the discount. The discount is 0.5%, so the cost per gallon is:

$2 x (1 - 0.5%) = $2 x 0.995 = $1.99/gallon

Assuming that the order quantity is at least 40,000 gallons, the fixed ordering costs are still $100 per order, but the ink costs less. The total cost with the discount is:

Total cost with discount = $1.99 x 1,440,000 + $144,000/Q + $0.2Q

To find the optimal order quantity, we can take the derivative of the total cost with respect to Q and set it equal to zero:

d(Total cost with discount)/dQ = -144,000/Q^2 + 0.2 = 0

Solving for Q, we get:

Q = sqrt(144,000/0.2) = 6000

Since 6000 gallons is less than the minimum order quantity of 40,000 gallons for the discount, Palmer should continue to order 1,440,000/6000 = 240 orders of 6000 gallons each, without taking the discount. The total cost without the discount is:

Total cost = $2,880,000 + $144,000/6000 + $0.2 x 6000 = $2,892,573

The total cost with the discount for an order quantity of 40,000 gallons or more is:

Total cost with discount = $1.99 x 1,440,000 + $144,000/40,000 + $0.2 x 40,000 = $2,879,000

Therefore, taking the discount would save:

$2,892,573 - $2,879,000 = $13,573

So, Palmer should increase its order quantity to 40,000 gallons or more to take the discount and save $13,573.

Exhibit Palmer Pens Assume That Palmer Executive Pens Uses 1,440,000 Gallons of Ink