---
title: "Safety stock and reorder level with lead time"
subject: "Industrial Engineering and Management"
chapter: 4
key: "semester_6/industrial_engineering_and_management:safety-stock-and-reorder-level"
source: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/safety-stock-and-reorder-level
site: BCT Engineering
---

# Safety stock and reorder level with lead time

Industrial Engineering and Management · Chapter 4: Inventory Control

- Asked in: year not given, 3+5 marks
- Video: [Safety Stock Calculation Explained | 5 Methods Of Calculating Safety Stock | Safety Stock Inventory](https://www.youtube.com/watch?v=cz_t4a8Uxhs) — 10 Min For Supply Chain
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/safety-stock-and-reorder-level
- Whole chapter: https://www.bctengineer.com/md/semester_6/industrial_engineering_and_management/chapter-4

## Question

(a) Define safety stock, reorder level and lead time, and state why safety stock is kept. (b) The daily demand of an item is normally distributed with mean 50 units and standard deviation 8 units. The lead time is constant at 9 days. Find the safety stock and the reorder level for a 95% service level (z = 1.645). (c) If the lead time itself varies with a standard deviation of 2 days, find the new safety stock and reorder level.

## Short answer

With constant lead time, safety stock = 40 units, ROP = 490 units for 95% service. With variable lead time (σ=2 days), safety stock = 169 units, ROP = 619 units.

## Answer

#### (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.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is the reorder level (ROP) defined in the answer?
   A. Demand during lead time minus safety stock
   B. Demand during lead time plus safety stock
   C. Safety stock multiplied by lead time
   D. Average demand per day times z
   Answer: B. ROP = demand during lead time + safety stock, so a new order is placed while enough stock remains to cover lead-time demand and the buffer. (Key point: Reorder level)

2. Daily demand has sigma_d = 8 units and the lead time is a constant 9 days. What is the standard deviation of demand during lead time?
   A. 72 units
   B. 8 units
   C. 24 units
   D. 2.67 units
   Answer: C. sigma_L = sigma_d x sqrt(L) = 8 x 3 = 24 units, because variances add over days, not standard deviations. (Key point: Std deviation over lead time)

3. For a 95% service level (z = 1.645) with constant lead time, what are safety stock and ROP?
   A. SS = 40, ROP = 490
   B. SS = 40, ROP = 450
   C. SS = 169, ROP = 619
   D. SS = 72, ROP = 522
   Answer: A. SS = 1.645 x 24 = 39.5 (about 40); ROP = 450 + 39.5 = about 490 units. (Key point: Safety stock and ROP, constant lead time)

4. When lead time varies with sigma_LT = 2 days, which expression gives the standard deviation of demand during lead time?
   A. sqrt(L x sigma_d^2 + d^2 x sigma_LT^2)
   B. sigma_d x sqrt(L) + d x sigma_LT
   C. sigma_d x sigma_LT
   D. sqrt(L) x sigma_LT
   Answer: A. Both demand variation and lead-time variation contribute: sqrt(9x64 + 2500x4) = 102.8 units, giving SS of about 169 and ROP of about 619. (Key point: Variable lead time)

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