Thomas Kratzer is the purchasing manager for the headquarters of a large insurance company chain with a central inventory operation. Thomas’s fastest moving inventory item has a demand of 6000 units per year. The cost of each unit is $100.00, and the inventory carrying cost is $10.00 per unit per year. The average ordering cost is $30.00 per order. It takes about 5 days for an order to arrive, and demand for 1 week is 120 units (this is a corporate operation, there are 250 working days per year).

a. What is the EOQ?
b. What is the average inventory if the EOQ is used?
c. What is the optimal number of orders per year?
d. What is the optimal number of days in between any two orders?
e. What is the annual cost of ordering and holding and holding inventory?
f. What is the total annual inventory cost, including cost of the 6,000 units?

a. What is the EOQ? = 189.74 units

Step (1): Determine the Annual Set-up Cost
*Annual set-up cost = (# of orders placed per year) x (Setup or order cost per order)
= Annual Demand
# of units in each order ¡Á (Setup or order cost per order)
= (D/Q) ¡Á(S)
= (6000/Q) x (30)
Step (2): Annual holding cost = Average inventory level x Holding cost per unit per year
= (Order Quantity/2) (Holding cost per unit per year)
= (Q/2) ($10.00)

Step (3):
Optimal order quantity is found when annual setup cost equals annual holding cost:
(D/Q) x (S) = (Q/2) x (H)
(6,000/Q) x (30) = (Q/2) (10)
=(2)(6,000)(30) = Q2 (10)
Q2 = [(2 ¡Á6,000 ¡Á30)/($10)] = 36,000
Q = ¡Ì([(2 ¡Á6,000 ¡Á30)/(10)]) = ¡Ì36,000
Q = 189.736 ¡Ö 189.74 units
EOQ = 189.74 units

b. What is the average inventory if the EOQ is used?
Average inventory level = (Order Quantity/2)
= (189.74) /2 = 94.87
Average Inventory level =94.87 units

c. What is the optimal number of orders per year?
N= ( Demand/ order quantity) = (6000/ 189.736)=31.62
N = 31.62
The optimal number of orders per year = 31.62

d. What is the optimal number of days in between any two orders?
T = (Number of Working Days per year) / (optimal number of orders)
T = 250 days per year / 31.62 = 7.906
T= 7.91
The optimal number of days in between any two orders = 7.91

e. What is the annual cost of ordering and holding inventory?
(Q) x (H)
(189.736 units) x ($10) =$1,897.36
¡Ö $1,897

The annual cost of ordering and holding the inventory = $1,897

f. What is the total annual inventory cost, including cost of the 6,000 units?
TC = setup cost + holding cost
TC = (Dyear/Q) (S) + (Q/2) (H)
TC = (6,000/189.74) ($30.00) + (189.74/2) ($10.00)
TC = $948.67 + $948.7
TC = 1,897.37 ¡Ö $1,897
Purchase cost = (6,000 units) x ($100/unit) = $600,000
Total annual inventory cost = $600,000 + $1,897 = $601,897

Total annual inventory cost = $601,897

$601,897 is correct!

For the annual holding cost the formula is (Q/2) * H, therefore We would have

(189.763/2) * 10 = $948.815 per year

For the annual ordering cost, the formula goes (D/Q) * S where s is ordering costs

So (6000/189.74) * 30 = 948.66659639506693369874565194477
or more simmply $948.67

a. The EOQ (Economic Order Quantity) can be calculated using the formula: EOQ = √[(2DS) / H], where D is the annual demand, S is the ordering cost, and H is the carrying cost.

In this case, D = 6000 units, S = $30.00 per order, and H = $10.00 per unit per year.

EOQ = √[(2 * 6000 * 30) / 10] = √(360,000 / 10) = √36,000 = 189.74 (approximately 190 units)

b. The average inventory if the EOQ is used can be calculated as Average Inventory = (EOQ / 2).

Average Inventory = 190 / 2 = 95 units.

c. The optimal number of orders per year can be calculated as Number of Orders = (D / EOQ).

Number of Orders = 6000 / 190 = 31.58 (approximately 32 orders)

d. The optimal number of days in between any two orders can be calculated as Time Between Orders = (Working Days per Year / Number of Orders).

Time Between Orders = 250 / 32 = 7.81 (approximately 8 days)

e. The annual cost of ordering and holding inventory can be calculated as Cost of Ordering and Holding = (D * S / EOQ) + (EOQ / 2 * H)

Cost of Ordering and Holding = (6000 * 30 / 190) + (190 / 2 * 10) = 950 + 950 = $1900.00

f. The total annual inventory cost, including the cost of the 6000 units, can be calculated as Total Annual Inventory Cost = (D * H) + Cost of Ordering and Holding.

Total Annual Inventory Cost = (6000 * 10) + 1900 = 60,000 + 1900 = $61,900.00

To answer the given questions, we will use the Economic Order Quantity (EOQ) model, which helps determine the optimal order quantity to minimize inventory costs.

a. To calculate the EOQ, we'll use the formula:
EOQ = sqrt((2 * Annual Demand * Ordering Cost) / Carrying Cost per Unit)
EOQ = sqrt((2 * 6000 * $30) / $10)
EOQ = sqrt(360,000 / $10)
EOQ = sqrt(36,000)
EOQ ≈ 189.74 units

b. The average inventory level when using EOQ can be calculated as:
Average Inventory = EOQ / 2
Average Inventory = 189.74 / 2
Average Inventory ≈ 94.87 units

c. The optimal number of orders per year can be determined by dividing the Annual Demand by the EOQ:
Optimal Number of Orders = Annual Demand / EOQ
Optimal Number of Orders = 6000 / 189.74
Optimal Number of Orders ≈ 31.61 orders

d. The optimal number of days in between any two orders can be calculated by dividing the Number of Working Days per Year by the Optimal Number of Orders:
Optimal Number of Days = Number of Working Days per Year / Optimal Number of Orders
Optimal Number of Days = 250 / 31.61
Optimal Number of Days ≈ 7.91 days

e. The annual cost of ordering and holding inventory can be calculated by multiplying the Number of Orders per Year by the Ordering Cost and adding the Carrying Cost per Unit multiplied by the Average Inventory:
Annual Cost = (Number of Orders per Year * Ordering Cost) + (Average Inventory * Carrying Cost per Unit)
Annual Cost = (31.61 * $30) + (94.87 * $10)
Annual Cost ≈ $948.30 + $948.70
Annual Cost ≈ $1,896.00

f. The total annual inventory cost, including the cost of the 6,000 units, can be calculated by multiplying the Average Inventory by the Carrying Cost per Unit, and adding the Annual Cost:
Total Annual Inventory Cost = (Average Inventory * Carrying Cost per Unit) + Annual Cost
Total Annual Inventory Cost = (94.87 * $10) + $1,896.00
Total Annual Inventory Cost ≈ $948.70 + $1,896.00
Total Annual Inventory Cost ≈ $2,844.70

Therefore:
a. The EOQ is approximately 190 units.
b. The average inventory, when EOQ is used, is around 95 units.
c. The optimal number of orders per year is roughly 32 orders.
d. The optimal number of days in between any two orders is approximately 8 days.
e. The annual cost of ordering and holding inventory is approximately $1,896.00.
f. The total annual inventory cost, including the cost of the 6,000 units, is approximately $2,844.70.

ANNUAL COST OF ORDERING=60000/189.74*30=$9483.67

ANNUAL COST OF CARRYING=189.74/2*10=$949