Question

Bruno Fruscalzo decided to start a small production facility in Sydney to sell gelato to the local restaurants. His local milk supplier charges $0.5 per kg of milk plus a $15 delivery fee (that is independent of the amount ordered). Bruno’s holding cost is $0.03 per kg per month. He needs 9250 kgs of milk per month.

a. Suppose Bruno orders 9750 kgs each time. What is his average inventory (kgs)? 4,875 kgs
(Round your answer to 1 decimal place.)
b. Suppose Bruno orders 6250 kgs each time. How many orders does he place with his supplier each year? 1.0 orders
c. How many kgs should Bruno order from his supplier with each order to minimize the sum of ordering and holding costs? 15 kgs
(Do not round intermediate calculations. Round your answer to 2 decimal places.)
d. If Bruno’s storage vessel can hold only 2000 kgs of milk, what would be Bruno’s annual ordering and holding costs?
(Do not round intermediate calculations. Round your answer to 2 decimal places.)
e. If Bruno’s storage vessel can hold only 5750 kgs of milk, what would be Bruno’s annual ordering and holding costs?
(Do not round intermediate calculations. Round your answer to 2 decimal places.)
f. Bruno’s supplier’s truck can carry 20,000 kgs of milk. The supplier does not want to deliver to more than 3 customers with each truck. Thus, the supplier requires a minimum order quantity of 6,500 kgs. If Bruno orders the minimum amount, what would be the sum of his annual ordering and holding costs? Assume he has a storage vessel large enough to hold 6,500 kgs.
(Round your answer to 3 decimal places.)
g. Bruno’s supplier offers a 4% discount when a customer orders a full truck, which is 20,000 kgs. Assume Bruno can store that quantity and the product will not spoil. If Bruno orders a full truck, what would be the inventory holding and ordering cost incurred per kg of milk? Assume the discount also applies to the holding cost.
1 decimal places required.

Answer 1

a. 4875 kg

b. 11.76 orders per year

c. 3041.38 kg

d. $56,692.50

e. $56,824.57

f. $56,926.154

g. 51.2¢

Explanation:

You want a variety of quantities and costs associated with the scenario that 9250 kg of milk is used per month, orders cost $0.50 per kg plus a $15 delivery fee, holding cost is $0.03 per kg per month.

A. Average inventory

Assuming inventory is replenished at precisely the moment it becomes zero, and that it is used at a uniform rate, the average inventory is half the order quantity.

For an order quantity of 9750 kg, the average inventory is 4875 kg.

B. Orders per year

The quantity of milk used per year is (9250 kg/mo)(12 mo) = 111,000 kg. Then the number of orders per year at 6250 kg/order is ...

(111,000 kg/yr)/(6250 kg/order) = 17.76 orders/yr

C. Minimal holding cost

For an order quantity of x kg, the average inventory is (x/2) kg. The time that inventory is held is (x/9250) months, so the inventory holding cost for an order quantity of x is ...

($0.03/(kg·mo))·(x kg)·(x/9250 mo) = 0.015/9250·x²

The sum of the holding cost and ordering cost will be a minimum when the holding cost is equal to the $15 delivery fee. This requires ...

(0.015/9250)x² = 15

x² = (9250/0.015)·15 = 9250000

x = √(9250000) ≈ 3041.38

The sum of ordering and holding cost is minimized for an order quantity of 3041.38 kg.

D – F Annual costs

The ordering and holding cost per kg will be ...

(annual kg) × ((holding cost per kg) + (ordering cost per kg))

= 111000 × (0.015/9250x + 0.50 + 15/x)

In the attached calculator display, the second factor of this expression is defined as Y0(x). So, the annual costs are ...

D. 110000·Y0(2000) = $56,692.50 . . . . . cost for orders of 2000 kg

E. 110000·Y0(5750) = $56,824.57 . . . . . cost for orders of 5750 kg

F. 110000·Y0(6500) = $56,926.154 . . . . . cost for orders of 6500 kg

G. Discounted cost

The cost per kg will be Y0(20000). Discounted by 4%, it will be 0.96·Y0(20000). Rounding this to one decimal place, it will be $0.5, essentially identical to the cost per kg without the holding and delivery costs added.

In order to make this a more useful number, we can round it to one decimal place expressed as cents:

The discounted cost per kg will be 51.2 cents.

__

Additional comment

We have assumed that that "ordering and holding costs" include the product cost of $0.50 per kg. If the intention is to just include the delivery fee and the holding cost, then $55,500 can be subtracted from the annual costs, and $0.50 = 50¢ can be subtracted from the per kg cost.

We're not quite sure why the rounding requirements are different for the different order quantities, but we have made that accommodation.

The reason for reporting part (g) cost in cents is described above.

In a real business, inventory would be replaced before it hit zero, so there would be some residual inventory incurring holding costs. We have ignored that here.

That delivery and holding costs are the same when their sum is minimum is not hard to prove. The sum is of the form ax +b/x. The derivative of the sum is zero when x=√(b/a), so each cost is √(ab).

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