Suppose that in the inventory​ problem, the storage cost depends on the maximum inventory​ size, rather than the average. In this​ case, the number of units that should be ordered or manufactured to minimize the total cost is qequalsStartRoot StartFraction fM Over k EndFraction EndRoot
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A book publisher wants to know how many times a year a print run should be scheduled. Suppose it costs ​$1200 to set up the printing​ process, and the subsequent cost per book is so low it can be ignored. Suppose further that the annual warehouse cost is ​$4 times the maximum number of books stored. Assuming 5600 copies of the book are needed per​ year, how many books should be printed in each print​ run?

We are dealing with an **inventory problem** where the storage cost depends on the maximum inventory size, rather than the average. The goal is to determine the **optimal quantity $q$** to print per run that minimizes total costs.

The formula for the optimal order quantity $q^*$ in this setting is:  
\[
q^* = \sqrt{\frac{2 \cdot f \cdot M}{k}}
\]
where:  
- $f$ = Fixed setup cost per run (\$1200)  
- $M$ = Annual demand (5600 copies)  
- $k$ = Storage cost per unit (\$4 per book stored)  

### Step 1: Substituting known values into the formula
Given:  
- $f = 1200$  
- $M = 5600$  
- $k = 4$  

\[
q^* = \sqrt{\frac{2 \cdot 1200 \cdot 5600}{4}}
\]

### Step 2: Simplifying the expression inside the square root
\[
2 \cdot 1200 \cdot 5600 = 13,440,000
\]
\[
\frac{13,440,000}{4} = 3,360,000
\]

### Step 3: Calculating the square root
\[
q^* = \sqrt{3,360,000} = 1833.33
\]

Since the number of books must be a whole number, we round it to 1833 books per run.

### Step 4: Conclusion
The optimal number of books to print per print run is approximately **1833 books**.

---

Would you like further clarifications or additional insights?  

Here are 5 related questions you might find useful:  
1. How does this solution change if the storage cost depends on the **average** inventory?  
2. What if the fixed setup cost increases? How would that impact the optimal print run size?  
3. How do we calculate the **total annual cost** with this order quantity?  
4. What happens if the demand increases to 10,000 books per year?  
5. How would we handle non-negligible **per-unit production costs** in this calculation?

**Tip:** The formula used here is a variation of the **Economic Order Quantity (EOQ)** model, with modifications for maximum inventory cost.

A book publisher wants to know how many times a year a print run should be scheduled. Suppose it costs ​$1200 to set up the printing​ process, and the subsequent cost per book is so low it can be ignored. Suppose further that the annual warehouse cost is ​$4 times the maximum number of books stored. Assuming 5600 copies of the book are needed per​ year, how many books should be printed in each print​ run?

Learn how to calculate the optimal print run size for a book publisher using the EOQ model. With a setup cost of $1200 per print run and an annual demand of 5600 books, this problem involves finding the quantity that minimizes storage costs. Ideal for college-level or advanced high school students.

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