Fill Rate

409. Fill Rate

409.1. Fill Rate (Type II, $𝛽$ )

The expected fraction of demand satisfied directly from on-hand inventory. Counts units, not events.

Fill rate = 𝛽 = \frac{𝐸 [units served]}{𝐸 [units demanded]} = 1 - \frac{𝐸 [units short]}{𝐸 [units demanded]}

409.1.1. Why fill rate, not CSL?

Cycle service level (CSL) treats every stockout — small or large — as a single failure. Fill rate is unit-weighted: a cycle that’s 1 unit short is treated very differently from a cycle that’s 100 units short. Most operational service-level promises to customers (e.g., “we fulfill 99% of unit demand”) refer to fill rate.

CSL → frequency of stockout incidents. Fill rate → fraction of demand met.

409.1.2. Computing fill rate (continuous demand)

For a (Q, r) policy with normal lead-time demand:

Expected shortage per cycle = 𝐸 [{(𝐷_{𝐿} - 𝑟)}^{+}] = 𝜎_{𝐿} \cdot 𝐿 (𝑧)

where $𝑧 = (𝑟 - 𝜇_{𝐿}) / 𝜎_{𝐿}$ and $𝐿 (𝑧) = 𝜑 (𝑧) - 𝑧 (1 - Φ (𝑧))$ is the standard normal loss function (see [newsvendor expected_profit](../newsvendor/expected_profit.typ) for the derivation).

Each cycle covers $𝑄$ units of demand on average. So:

𝛽 = 1 - \frac{𝐸 [shortage per cycle]}{𝐸 [demand per cycle]} = 1 - \frac{𝜎_{𝐿} 𝐿 (𝑧)}{𝑄}

Equivalently:

Required 𝜎_{𝐿} 𝐿 (𝑧) = (1 - 𝛽) 𝑄

To hit fill rate $𝛽$ , choose $𝑟$ such that $𝐿 (𝑧) = (1 - 𝛽) 𝑄 / 𝜎_{𝐿}$ .

409.1.3. Why fill rate is usually higher than CSL

For the same $𝑟$ :

CSL = $𝑃 (𝐷_{𝐿} \leq 𝑟) = 1 - 𝛼$
Fill rate = $1 - 𝜎_{𝐿} 𝐿 (𝑧) / 𝑄$

Whenever $𝑄$ is much larger than $𝜎_{𝐿}$ , the fill rate exceeds the CSL. Reason: most cycles run no stockout at all (contributing 100% fill); the few that do stock out are usually short by only a few units (each contributing > 99% fill on its own cycle).

Practical ranges:

Lean inventory ( $𝑄 / 𝜎_{𝐿} \approx 5$ ): fill rate $\approx$ CSL.
Standard ( $𝑄 / 𝜎_{𝐿} \approx 20$ ): fill rate $\approx$ CSL + 5%.
Bulk ( $𝑄 / 𝜎_{𝐿} \approx 50$ ): fill rate can be 99% even at CSL = 80%.

409.1.4. Setting $𝑟$ for target fill rate

Iterative or table-based. Algorithm:

Given target $𝛽$ , solve $𝐿 (𝑧) = (1 - 𝛽) 𝑄 / 𝜎_{𝐿}$ for $𝑧$ .
Use a normal loss function table (or numerically solve $𝑧$ ).
Set $𝑟 = 𝜇_{𝐿} + 𝑧 𝜎_{𝐿}$ .

Most software does this automatically.

Example

Given (same params as CSL example):

$𝑄 = 775$ , $𝜇_{𝐿} = 462$ , $𝜎_{𝐿} = 18.7$
Target fill rate: $𝛽 = 99 %$ (more demanding than CSL = 95%)

Step 1 — required loss-function value

𝐿 (𝑧) = (1 - 𝛽) \frac{𝑄}{𝜎_{𝐿}} = (1 - 0.99) \cdot \frac{775}{18.7} = 0.414

Step 2 — invert the loss function

Looking up: $𝐿 (0.10) \approx 0.351$ , $𝐿 (0.05) \approx 0.378$ , $𝐿 (0.00) \approx 0.399$ , $𝐿 (- 0.05) \approx 0.421$ .

So $𝑧 \approx - 0.04$ gives $𝐿 (𝑧) \approx 0.414$ . (For high fill rate, we’d usually expect $𝑧 > 0$ — but since $𝑄$ is much larger than $𝜎_{𝐿}$ , even a small safety stock yields high fill rate. In some cases $𝑧$ even goes negative for moderate fill rates.)

Step 3 — reorder point

𝑟 = 𝜇_{𝐿} + 𝑧 𝜎_{𝐿} = 462 + (- 0.04) \cdot 18.7 \approx 461

Almost no safety stock! Setting $𝑟 \approx 461$ achieves 99% fill rate because the order quantity 775 is so large compared to lead-time variability 18.7 that stockouts are rare and small.

Step 4 — same setup, target CSL = 95%

Compare: CSL 95% would set $𝑟 = 493$ (safety stock 31). Fill rate at $𝑟 = 493$ :

𝑧 = \frac{493 - 462}{18.7} \approx 1.66, 𝐿 (1.66) \approx 0.0207

𝛽 = 1 - 18.7 \cdot \frac{0.0207}{775} \approx 1 - 0.0005 \approx 99.95 %

Implication: setting CSL to 95% gives a fill rate of essentially 100% in this regime. The CSL target is much more demanding than the fill-rate target for moderate-Q regimes.

This explains why retailers usually quote fill rate (99%) rather than CSL (which would have to be e.g. 70% to give the equivalent operational meaning).