A business hires you as a cost consultant. They want to purchases 650 whizits per year from a distributer to sell to their customers. Each whizit costs $3.85 to purchase, it costs the business $1.46 to store each whizit and each shipment of whizits costs $18.  

Find the number of whizits  that should be ordered in each shipment to minimize the cost for the business. Whta is the final minimal cost?

SHOW ALL OF YOUR WORK IN A NEAT AND ORGANZIED MANNER. INCLUDE SKETCHES OR PICTURES OF YOUR GRAPH.

The given problem is a typical Economic Order Quantity (EOQ) problem, which aims to minimize the total cost for a business by determining the optimal number of items to order per shipment. Let's work through this step by step.

### Step 1: Identifying Variables
We need to minimize the total cost $ C $, which includes:
1. **Ordering cost**: The cost associated with ordering the product.
2. **Holding (or carrying) cost**: The cost of storing the product.
3. **Purchase cost**: This is fixed since it doesn't depend on the order quantity. Thus, it will not affect the optimal order size.

The formula for the Economic Order Quantity (EOQ) is:

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

Where:
- $ D $ = Demand per year (650 whizits per year),
- $ S $ = Ordering cost per order ($18 per shipment),
- $ H $ = Holding cost per unit per year ($1.46 to store each whizit).

### Step 2: Calculating the EOQ
Plug in the given values into the EOQ formula:

\[
EOQ = \sqrt{\frac{2 \times 650 \times 18}{1.46}}
\]

Now, calculate the value inside the square root.

\[
EOQ = \sqrt{\frac{23,400}{1.46}} = \sqrt{16,027.40}
\]

\[
EOQ \approx 126.57
\]

Since we can't order a fraction of a whizit, we round this to 127 whizits. Therefore, the optimal order size is **127 whizits**.

### Step 3: Total Cost Calculation
The total cost includes:
1. **Ordering cost**: This is calculated as the number of orders per year multiplied by the cost per order. The number of orders per year is $ \frac{D}{EOQ} $, and the cost per order is $ S $.
   
   Ordering cost:
   \[
   \text{Ordering cost} = \frac{D}{EOQ} \times S = \frac{650}{127} \times 18 \approx 5.12 \times 18 = 92.16
   \]

2. **Holding cost**: This is calculated as the average inventory level multiplied by the holding cost per unit. The average inventory level is $ \frac{EOQ}{2} $, and the holding cost per unit is $ H $.
   
   Holding cost:
   \[
   \text{Holding cost} = \frac{EOQ}{2} \times H = \frac{127}{2} \times 1.46 = 63.5 \times 1.46 = 92.71
   \]

3. **Purchase cost**: The purchase cost is simply the total number of whizits needed per year multiplied by the purchase price per unit. However, since the purchase cost doesn't depend on the order quantity, we do not need to calculate this to find the EOQ.

### Step 4: Total Minimal Cost
The total minimal cost is the sum of the ordering cost and the holding cost:

\[
\text{Total minimal cost} = \text{Ordering cost} + \text{Holding cost} = 92.16 + 92.71 = 184.87
\]

Thus, the total minimal cost for the business is approximately **$184.87**.

### Summary:
- The optimal number of whizits that should be ordered in each shipment is **127 whizits**.
- The minimal total cost is **$184.87**.

#### Graphical Representation
The graph of total cost vs. order quantity would typically show a U-shaped curve. The total cost initially decreases as the order size increases (due to lower ordering costs), but after a certain point, the total cost starts to increase again (due to higher holding costs). The EOQ represents the point where this curve reaches its minimum.

Would you like to see more details or have any questions about the steps?

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Here are 5 questions related to this topic:
1. How does an increase in the ordering cost affect the EOQ?
2. What happens to the EOQ if the holding cost per unit increases?
3. How would the EOQ change if the demand for whizits doubled?
4. If the purchase cost per whizit changes, will it affect the EOQ?
5. Can the EOQ model be used for products that are perishable? Why or why not?

**Tip**: The EOQ formula assumes constant demand and uniform holding costs. For fluctuating demand, you might need more advanced models like the dynamic lot-sizing model.

A business hires you as a cost consultant. They want to purchase 650 whizits per year from a distributor to sell to their customers. Each whizit costs $3.85 to purchase, it costs the business $1.46 to store each whizit, and each shipment of whizits costs $18. Find the number of whizits that should be ordered in each shipment to minimize the cost for the business. What is the final minimal cost?

Find out how to calculate the optimal order size for a business that purchases 650 whizits per year, with storage costs of $1.46 per unit and shipping costs of $18 per order. This detailed solution covers EOQ calculations and the minimal total cost, providing key insights for business operations and inventory management.

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