The reorder point (ROP) tells you exactly when to place the next replenishment order so inventory does not run out during lead time. It combines expected demand during lead time with a buffer for uncertainty. This page explains the core formulas used in supply chain analytics and includes multiple numerical examples for both stable and variable environments.

1. What the Reorder Point Represents

The reorder point is the inventory level at which a new order must be placed. It ensures that:

• Demand during lead time is covered.
• Variability in demand or lead time is absorbed by safety stock.
• Service-level targets are met (probability of avoiding stockouts).

The basic structure is always:

\[ \text{Reorder Point} = \text{Demand During Lead Time} + \text{Safety Stock} \]

2. Reorder Point with Constant Demand and Constant Lead Time

This is the simplest case, used when demand is stable and predictable:

\[ \text{ROP} = D \cdot L \]

• \(D\) = average demand per period
• \(L\) = lead time in periods

Numerical Example

• Daily demand = 40 units
• Lead time = 6 days

\[ \text{ROP} = 40 \cdot 6 = 240 \text{ units} \]

This version does not include safety stock. It is only appropriate when variability is negligible.

3. Reorder Point with Safety Stock (Variable Demand)

When demand varies but lead time is constant, use:

\[ \text{ROP} = D \cdot L + Z \cdot \sigma_d \cdot \sqrt{L} \]

• \(Z\) = service-level factor (e.g., 1.65 for 95%)
• \(\sigma_d\) = standard deviation of demand per period

Numerical Example

• Average daily demand = 100 units
• Std dev of daily demand = 20 units
• Lead time = 5 days
• Service level = 95% → \(Z = 1.65\)

Demand during lead time: \[ 100 \cdot 5 = 500 \]

Safety stock: \[ 1.65 \cdot 20 \cdot \sqrt{5} = 74 \]

\[ \text{ROP} = 500 + 74 = 574 \text{ units} \]

4. Reorder Point with Variable Demand and Variable Lead Time

When both demand and lead time fluctuate, use the full variability formula:

\[ \text{ROP} = D \cdot L + Z \cdot \sqrt{L\sigma_d^2 + D^2\sigma_L^2} \]

• \(\sigma_L\) = standard deviation of lead time

Numerical Example

• Average daily demand \(D = 80\)
• Std dev of demand \(\sigma_d = 15\)
• Average lead time \(L = 7\) days
• Std dev of lead time \(\sigma_L = 2\)
• Service level = 90% → \(Z = 1.28\)

Demand during lead time: \[ 80 \cdot 7 = 560 \]

Safety stock: \[ \sqrt{7(15^2) + 80^2(2^2)} = \sqrt{1575 + 25600} = 164.8 \] \[ \text{Safety Stock} = 1.28 \cdot 164.8 = 211 \]

\[ \text{ROP} = 560 + 211 = 771 \text{ units} \]

5. Reorder Point Using Fill Rate (β‑Service Level)

Fill rate focuses on the percentage of demand filled, not the probability of avoiding stockouts. A common approximation:

\[ \text{Safety Stock} = \sigma_{DL} \cdot \sqrt{2\ln\left(\frac{1}{1-\beta}\right)} \]

Then: \[ \text{ROP} = D \cdot L + \text{Safety Stock} \]

Numerical Example

• Demand during lead time std dev = 50 units
• Fill rate target = 95%
• Average demand during lead time = 300 units

\[ \text{Safety Stock} = 50 \cdot \sqrt{2\ln(20)} = 122 \]

\[ \text{ROP} = 300 + 122 = 422 \text{ units} \]

6. Reorder Points for Intermittent or Lumpy Demand

When demand is sporadic (e.g., spare parts), normal assumptions break down. Two alternatives are common:

• Poisson‑based reorder points — for low, random demand.
• Croston‑based reorder points — uses forecast error from intermittent demand forecasting.

Poisson Example

• Average weekly demand = 4 units
• Lead time = 3 weeks
• Service factor \(Z = 1.5\)

\[ \text{Safety Stock} = \lambda L (Z – 1) = 4 \cdot 3 \cdot 0.5 = 6 \]

\[ \text{ROP} = 12 + 6 = 18 \text{ units} \]

7. Summary Table of Reorder Point Formulas

8. Practical Tips for Using Reorder Points

• Recalculate ROP whenever lead times change.
• Use higher service levels for critical or high‑margin items.
• Use lower service levels for low‑value or non‑critical items.
• Review safety stock and ROP quarterly to avoid overstocking.
• Ensure demand and lead‑time variability are based on actual historical data.