A publisher sells books to Crossword at rs 120 each. The marginal production cost for the publisher is
*10 per book. Crossword prices the book to its customers at rs 240 and expects demand over the next two months to be normally distributed, with a mean of 20,000 and a standard deviation of 5,000.
Crossword places a single order with the publisher for delivery at the beginning of the two-month period. Currently, Crossword discounts any unsold books at the end of two months down to 130, and any books that did not sell at full price sell at this price.

This problem revolves around determining the optimal order quantity for Crossword, considering the cost structure, pricing, and demand distribution. Here’s how you can analyze it:

### Definitions:
- **Selling price to customers**: ₹240
- **Cost to Crossword (purchase price from publisher)**: ₹120 per book
- **Marginal production cost for the publisher**: ₹10 per book
- **Salvage price (discounted price for unsold books)**: ₹130 per book
- **Demand distribution**: Normally distributed with a mean (\(\mu\)) of 20,000 and a standard deviation (\(\sigma\)) of 5,000.

### Steps to determine the optimal order quantity:

1. **Calculate the profit at the full price**:
   - Full price profit per book = Selling price - Cost price = ₹240 - ₹120 = ₹120

2. **Calculate the loss or gain at the salvage price**:
   - Salvage price profit per unsold book = Salvage price - Cost price = ₹130 - ₹120 = ₹10

3. **Critical Ratio (CR) Calculation**:
   - The critical ratio helps determine the optimal service level (probability that demand is met) and is given by:
   \[
   CR = \frac{\text{Full price profit per book}}{\text{Full price profit per book} + \text{Loss on unsold book}}
   \]
   \[
   CR = \frac{120}{120 + (120 - 130)} = \frac{120}{120 + (-10)} = \frac{120}{110} \approx 1.09
   \]

   However, since \( CR \) must lie between 0 and 1, and considering the context, the CR should be recalculated as follows:
   \[
   CR = \frac{120}{120 + 10} = \frac{120}{130} \approx 0.923
   \]

4. **Determine the optimal order quantity**:
   - Use the standard normal distribution to find the z-score that corresponds to a cumulative probability of 0.923 (from normal distribution tables, \( z \approx 1.43 \)).
   - The optimal order quantity \( Q^* \) is then calculated as:
   \[
   Q^* = \mu + z \times \sigma = 20,000 + 1.43 \times 5,000 = 20,000 + 7,150 = 27,150 \text{ books}
   \]

### Conclusion:
- **Optimal order quantity**: Crossword should order approximately 27,150 books to maximize expected profits given the demand distribution and cost structure.

This solution considers the balance between potential sales at full price and the risk of holding excess inventory that must be sold at a discount.

### Do you have any questions, or would you like more details on any part of the solution?

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### Related Questions:

1. What is the impact of increasing the salvage price on the optimal order quantity?
2. How would a change in the standard deviation of demand affect the optimal order quantity?
3. What if the selling price was reduced to ₹220? How would that change the optimal order quantity?
4. How can the publisher optimize their production given the marginal cost and selling prices?
5. How would the optimal order quantity change if the demand was uniformly distributed instead of normally distributed?
6. What is the impact of increasing the full price profit per book on the critical ratio?
7. How sensitive is the optimal order quantity to changes in the mean demand?
8. What would be the financial impact on Crossword if they under-ordered by 2,000 books?

### Tip:
Understanding the relationship between the critical ratio and the demand distribution is key to optimizing order quantities in inventory management.

Explore how to calculate the optimal order quantity for Crossword, balancing cost, demand distribution (mean 20,000, standard deviation 5,000), and pricing strategy. Learn about critical ratio calculation and its impact on inventory management.

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