Question: 10. A product has annual demand forecasted to be $8000$ units with a standard deviation of $1500$ units, a cost of goods sold of $900$ and a fixed cost of $3500$ when each order is placed. The holding cost is expected to be $18\%$ of the variable costs. The order lead time is $12$ weeks and a service level of $92\%$ is desired. What is the optimal order quantity and the order frequency?

1. **State the problem:** We need to find the optimal order quantity and order frequency for a product with given demand, costs, and service level. 2. **Given data:** - Annual demand, $D = 8000$ units - Standard deviation of demand, $\sigma = 1500$ units - Cost of goods sold per unit, $C = 900$ - Fixed ordering cost, $S = 3500$ - Holding cost rate, $h = 18\%$ of variable cost - Lead time, $L = 12$ weeks - Service level, $SL = 92\%$ 3. **Calculate holding cost per unit per year:** $$H = h \times C = 0.18 \times 900 = 162$$ 4. **Calculate the Economic Order Quantity (EOQ):** The EOQ formula is: $$EOQ = \sqrt{\frac{2DS}{H}}$$ Substitute values: $$EOQ = \sqrt{\frac{2 \times 8000 \times 3500}{162}} = \sqrt{\frac{56000000}{162}} = \sqrt{345679.01} \approx 588.01$$ 5. **Calculate order frequency:** Order frequency is how many orders per year: $$\text{Order frequency} = \frac{D}{EOQ} = \frac{8000}{588.01} \approx 13.61$$ 6. **Calculate cycle time (time between orders):** There are 52 weeks in a year, so: $$\text{Cycle time} = \frac{52}{13.61} \approx 3.82 \text{ weeks}$$ 7. **Calculate safety stock for service level:** Service level $92\%$ corresponds to a z-score $z \approx 1.41$ (from standard normal tables). Demand during lead time $L$ weeks: $$D_L = D \times \frac{L}{52} = 8000 \times \frac{12}{52} \approx 1846.15$$ Standard deviation during lead time: $$\sigma_L = \sigma \times \sqrt{\frac{L}{52}} = 1500 \times \sqrt{\frac{12}{52}} \approx 1500 \times 0.4811 = 721.65$$ Safety stock: $$SS = z \times \sigma_L = 1.41 \times 721.65 \approx 1017.53$$ 8. **Final optimal order quantity including safety stock:** $$Q^* = EOQ + SS = 588.01 + 1017.53 = 1605.54$$ **Answer:** - Optimal order quantity $\approx 1606$ units - Order frequency $\approx 13.6$ orders per year (or every $3.82$ weeks)