Safety stock and reorder level with lead time

(a) Definitions

• Lead time: the time between placing an order and receiving the goods in stock.
• Reorder level (point): the stock level at which a new order is placed; $ROP = \text{demand during lead time} + \text{safety stock}$.
• Safety stock (buffer stock): extra stock kept to protect against variation in demand and lead time. It is kept because demand and supply are uncertain, and a stock-out causes lost sales or production stoppage.

(b) Constant lead time

Demand during lead time $= d \times L = 50 \times 9 = 450$ units. Standard deviation of demand during lead time:

$$\sigma_L = \sigma_d \sqrt{L} = 8\sqrt{9} = 24\ \text{units}$$ $$SS = z\,\sigma_L = 1.645 \times 24 = 39.5 \approx 40\ \text{units}$$ $$ROP = 450 + 39.5 = 489.5 \approx 490\ \text{units}$$

(c) Variable lead time ($\sigma_{LT} = 2$ days)

$$\sigma_{dL} = \sqrt{L\sigma_d^2 + d^2\sigma_{LT}^2} = \sqrt{9(64) + 2500(4)} = \sqrt{10576} = 102.8\ \text{units}$$ $$SS = 1.645 \times 102.8 = 169.2 \approx 169\ \text{units}$$ $$ROP = 450 + 169.2 = 619.2 \approx 619\ \text{units}$$

Variation in lead time raises the required safety stock more than four times, showing the value of reliable suppliers.

Answer: (b) Safety stock = 40 units, ROP = 490 units; (c) safety stock = 169 units, ROP = 619 units.