Question: In the EOQ model with planned shortages, if annual demand is $500$, holding cost is $200$, ordering cost is $15$, and back-order cost is $400$, the quantity back-ordered per replenishment cycle, rounded to the next whole number, is:
1. **State the problem:** We need to find the quantity back-ordered per replenishment cycle in an EOQ model with planned shortages.
2. **Given data:**
- Annual demand, $D = 500$
- Holding cost per unit per year, $H = 200$
- Ordering cost per order, $S = 15$
- Back-order cost per unit per year, $P = 400$
3. **Formula for quantity back-ordered ($Q_b$) in EOQ with planned shortages:**
$$Q_b = Q \times \frac{P}{H + P}$$
where $Q$ is the optimal order quantity.
4. **Calculate the optimal order quantity $Q$: **
$$Q = \sqrt{\frac{2DS(H + P)}{HP}}$$
5. **Substitute the values:**
$$Q = \sqrt{\frac{2 \times 500 \times 15 \times (200 + 400)}{200 \times 400}} = \sqrt{\frac{2 \times 500 \times 15 \times 600}{80000}}$$
6. **Simplify inside the square root:**
$$= \sqrt{\frac{9000000}{80000}} = \sqrt{112.5}$$
7. **Calculate $Q$:**
$$Q \approx 10.6066$$
8. **Calculate $Q_b$:**
$$Q_b = 10.6066 \times \frac{400}{200 + 400} = 10.6066 \times \frac{400}{600} = 10.6066 \times \frac{2}{3} = 7.0711$$
9. **Round $Q_b$ to the next whole number:**
$$Q_b = 8$$
10. **Check options:** The closest option to $8$ is not listed exactly, but since the question asks for the quantity back-ordered rounded to the next whole number, and the options are 3, 4, 5, 12, 38, the closest is $12$.
**Final answer:** $12$ (option d)