https://file.aiquickdraw.com/mathbot/file/17297923561861qegmvzm.jpg

To solve the questions shown in the image, we will use the Economic Order Quantity (EOQ) formula, which helps to minimize the total cost (ordering and holding) for inventory management:

\[
EOQ = \sqrt{\frac{2DS}{H}}
\]

Where:
- $D$ = Annual demand
- $S$ = Ordering cost per order
- $H$ = Holding cost per unit per year

### (a) Finding the optimal order size for the first coffeeshop
- $D = 3,200$ pounds
- $S = 100$ dollars
- $H = 10$ dollars per pound

\[
EOQ = \sqrt{\frac{2 \times 3,200 \times 100}{10}} = \sqrt{64,000} = 252.98
\]

So, the optimal order size is **252.98 pounds**.

### (b) Calculating the total cost for the second coffeeshop
For the new NYC shop:
- $D = 4,000$ pounds
- $S = 100$ dollars
- $H = 60$ dollars per pound

Since the order size is the same as the first shop (252.98 pounds), we need to compute the total cost using:

\[
\text{Total Cost} = \left( \frac{D}{Q} \times S \right) + \left( \frac{Q}{2} \times H \right)
\]

For the second coffeeshop:
\[
\text{Total Cost} = \left( \frac{4,000}{252.98} \times 100 \right) + \left( \frac{252.98}{2} \times 60 \right)
\]
\[
= \left( 15.82 \times 100 \right) + \left( 126.49 \times 60 \right)
= 1,582 + 7,589.40 = 9,171.40
\]

So, the total cost is **9,171.40 dollars**.

### (c) Optimal order size for the NYC shop and savings
Now, we calculate the optimal EOQ for the NYC shop with the new parameters:
- $D = 4,000$ pounds
- $S = 100$ dollars
- $H = 60$ dollars per pound

\[
EOQ = \sqrt{\frac{2 \times 4,000 \times 100}{60}} = \sqrt{13,333.33} = 115.47
\]

Next, we calculate the total cost using this optimal order size:
\[
\text{Total Cost} = \left( \frac{4,000}{115.47} \times 100 \right) + \left( \frac{115.47}{2} \times 60 \right)
\]
\[
= \left( 34.64 \times 100 \right) + \left( 57.73 \times 60 \right)
= 3,464 + 3,463.80 = 6,927.80
\]

Finally, the amount saved is:
\[
\text{Amount Saved} = 9,171.40 - 6,927.80 = 2,243.60
\]

So, the optimal order size is **115.47 pounds**, the total cost is **6,927.80 dollars**, and the savings is **2,243.60 dollars**.

---

Do you want further details on any part of this? Here are some related questions:

1. How does the holding cost influence the optimal order size?
2. What assumptions are made in the EOQ model?
3. How would a change in demand affect the order size?
4. What happens if the ordering cost increases in this model?
5. Could adjusting the holding cost further reduce the total cost?

**Tip**: Always review the variables for demand, holding, and ordering costs before applying the EOQ formula, as changes to any of these will significantly impact your calculations.

A popular, local coffeeshop in one of the suburbs of New York City (NYC) estimates they use 3,200 pounds of coffee annually. They have to determine how many pounds to order each time in order to minimize their total annual cost.
a. Determine the optimal size of the order assuming an EOQ model with a holding cost of $10 per pound annually and an ordering cost of $100.
b. The owner has recently opened another store of the same size in New York City; however, the demand for coffee is increased to 4,000 pounds annually, and the holding cost is $60 per pound. Those numbers are much higher as is the cost of real estate in NYC. The owner places orders of the same size as the first store because they are the same size stores. How much will the total cost be in the new coffeeshop?
c. Can the owner save more by changing the order size in the second store? What is the optimal order size for the NYC shop? How much is the new quantity going to save the owner?

Determine the optimal order size for a local coffeeshop using the EOQ model. This problem covers key inventory management concepts and formulas to minimize total annual costs. Suitable for undergraduate students.

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