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The problem provided involves calculating the optimal lot size, annual cost, and cycle inventory for a furniture manufacturer based on various pricing tiers, shipping costs, and holding costs. Here's a breakdown of the problem:

### Given Data:
- **Monthly demand**: 20,000 sq.ft.
- **Shipping cost**: $400 per shipment (independent of quantity).
- **Pricing scheme**:
  - $1.00 per sq.ft. for orders under 20,000 sq.ft.
  - $0.98 per sq.ft. for orders between 20,000 and 40,000 sq.ft.
  - $0.96 per sq.ft. for orders larger than 40,000 sq.ft.
- **Holding cost rate**: 18% of the purchase price.
  
### Required:
1. **Optimal lot size** using Economic Order Quantity (EOQ).
2. **Annual cost** of the policy.
3. **Cycle inventory** of plywood.
4. Comparison of the cycle inventory with the scenario where plywood is sold at $0.96/sq.ft without quantity discounts.

---

### To Solve:
We'll use the **EOQ formula** for finding the optimal lot size:

\[
Q^* = \sqrt{\frac{2DS}{H}}
\]
Where:
- $D$ is the annual demand (which is $20,000 \times 12 = 240,000 \, \text{sq.ft.}$ per year),
- $S$ is the order (shipping) cost ($400),
- $H$ is the annual holding cost per unit (which is 18% of the unit cost).

We will calculate the EOQ for each price tier and find the total annual cost using the formula:

\[
\text{Total Annual Cost} = \frac{D}{Q}S + \frac{Q}{2}H + D \times \text{Unit Price}
\]

Then, we will compare the cycle inventory in both scenarios.

I'll calculate the optimal lot size and other values for each price tier.### Calculated EOQ for Each Price Tier:
- For **$1.00 per sq.ft.**: Optimal order size $ Q^* = 32,660 \, \text{sq.ft.} $
- For **$0.98 per sq.ft.**: Optimal order size $ Q^* = 32,991 \, \text{sq.ft.} $
- For **$0.96 per sq.ft.**: Optimal order size $ Q^* = 33,333 \, \text{sq.ft.} $

### Holding Costs:
- For $1.00 per sq.ft., the holding cost per unit is **$0.18**.
- For $0.98 per sq.ft., the holding cost per unit is **$0.1764**.
- For $0.96 per sq.ft., the holding cost per unit is **$0.1728**.

Next, I will calculate the total annual cost for each price tier.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you need further assistance, feel free to let me know.

Would you like details on the formulas used or have any specific questions? Here are five related questions you could explore:
1. How do holding costs impact inventory decisions in EOQ?
2. What would the EOQ be if the annual demand changed?
3. How does the order quantity affect cycle inventory?
4. What is the impact of changing shipping costs on total cost?
5. How do pricing tiers affect the EOQ model?

**Tip**: When calculating EOQ for multiple price tiers, always compare the total costs, not just the EOQ, to determine the best option.

A furniture manufacturer uses 20,000 sq.ft. of plywood per month. Their trucking company charges $400 per shipment independent of the quantity purchased. The manufacturer offers an all-unit quantity discount with a price of $1.00 per sq.ft. for orders under 20,000 sq.ft., $0.98 per sq.ft. for orders between 20,000 sq.ft. and 40,000 sq.ft., and $0.96 per sq.ft. for orders larger than 40,000 sq.ft. The producer incurs a holding cost of 18%. What is the optimal lot size? What is the annual cost of such a policy? What is the cycle inventory of plywood? How does it compare with the cycle inventory if the manufacturer did not offer a quantity discount but sold all plywood at $0.96 per sq.ft.?

Learn how to calculate the optimal lot size, annual cost, and cycle inventory for a furniture manufacturer that uses 20,000 sq.ft. of plywood per month. This example applies the Economic Order Quantity (EOQ) model with quantity discounts, analyzing pricing tiers and shipping costs.

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