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 6,100 units per year. The cost of each unit is $101, and the inventory carrying cost is $8 per unit per year. The average ordering cost is $31 per order. It take about 5 days for an order to arrive, and the demand for 1 week is 120 units. (This is a corporate operation, and the 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 inventory?F) What is the total annual inventory cost, including cost of the 6,100 units?
Mathematics · College · Thu Feb 04 2021
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To solve these problems, we use the Economic Order Quantity (EOQ) formula and its related concepts.
A) The EOQ formula is:
\[ EOQ = \sqrt{\frac{2DS}{H}} \]
Where: D = Demand (annually) S = Ordering cost per order H = Holding (carrying) cost per unit per year
Plugging in the values from the problem statement:
\[ EOQ = \sqrt{\frac{2 * 6100 * 31}{8}} \] \[ = \sqrt{379,400} \] \[ = \sqrt{379,400} \] \[ ≈ 616 units \]
Therefore, the EOQ is approximately 616 units.
B) The average inventory if the EOQ is used is: \[ Average \ Inventory = \frac{EOQ}{2} \] \[ = \frac{616}{2} \] \[ = 308 units \]
C) The optimal number of orders per year is:
\[ Number \ of \ Orders = \frac{D}{EOQ} \] \[ = \frac{6100}{616} \] \[ ≈ 9.9 \ orders \]
Rounded up, Thomas will make approximately 10 orders per year.
D) The optimal number of days between any two orders is:
\[ Days \ per \ Order = \frac{Number \ of \ working \ days \ per \ year}{Number \ of \ Orders} \] \[ = \frac{250}{9.9} \] \[ ≈ 25.25 \ days \]
E) The annual cost of ordering and holding inventory is the sum of the annual ordering costs and the annual holding costs. The annual ordering cost is:
\[ Annual \ Ordering \ Cost = Number \ of \ Orders * S \] \[ = 9.9 * 31 \] \[ ≈ 306.9 \ dollars \]
The annual holding cost is:
\[ Annual \ Holding \ Cost = Average \ Inventory * H \] \[ = 308 * 8 \] \[ = 2464 \ dollars \]
Thus, the annual cost of ordering and holding inventory is:
\[ Total \ Inventory \ Costs = Annual \ Ordering \ Cost + Annual \ Holding \ Cost \] \[ = 306.9 + 2464 \] \[ ≈ 2770.9 \ dollars \]
F) The total annual inventory cost, including the cost of the 6,100 units, includes the purchase cost and the inventory costs. The purchase cost is:
\[ Purchase \ Cost = D * Cost \ per \ Unit \] \[ = 6100 * 101 \] \[ = 616,100 \ dollars \]
Adding the total annual inventory costs from E:
\[ Total \ Annual \ Inventory \ Cost = Purchase \ Cost + Total \ Inventory \ Costs \] \[ = 616,100 + 2770.9 \] \[ ≈ 618,870.9 \ dollars