*By Joannès Vermorel, October 2015*The fill rate is the fraction of customer demand that is met through immediate stock availability, without backorders or lost sales. The fill rate differs from the

service level indicator. The fill rate has a considerable appeal to practitioners because it represents the fraction of the demand that is likely to be recovered or better serviced if the inventory performance was to be improved. The fill rate is measured empirically by averaging the number of correctly serviced requests over the total number of requests.

## Fill rate and service level are distinct

The service level is often mistakenly confused with the fill rate, and vice-versa. Yet, the two indicators are numerically different. While the two indicators are quite correlated, it is possible to find real-world situations were a high service level does not translate into a high fill rate, and the other way around. Such situations tend to arise more frequently when demand is sparse (as for spare parts for example) or when demand is erratic (as in the case of books).

Example: let’s consider a bookseller selling a school manual. There is 1 order per day on average. Let’s assume that on average, out of 20 requests for the book, 19 requests come from individual students who require only a single copy of the book. In addition to this, 1 request out of 20, comes from a school teacher (we are still looking at the average), and the teacher asks for 20 copies because she is buying for her whole classroom. If the bookseller keeps 10 book units in stock, and if we assume the lead time to be of 1 day, then the service level is 95% (19/20=0.95) as nearly all the students will be served with their book. However, the teacher’s order request will be systematically declined as the stock never gets big enough to cover a whole classroom. Thus, in this case, the fill rate is close to 50% (19/(19+20) ≈ 0.5) as the teacher’s request accounts for slightly more than half of the total demand.

## Formal definition

In order to shed some light on the exact respective definition of the fill rate and the service level, we need to introduce a certain degree of formalism. Let $X$ be a random variable representing the demand over the next cycle. Let $s$ be the stock available, that is, the quantity of stock readily available to service incoming requests.

The service level $\tau_1$ is written as:
$$\tau_1(s) = \mathbf{P}(X \leq s)$$
The fill rate $\tau_2$ is written as:
$$\tau_2(s) = \frac{\mathbb{E}[\text{min}(X,s)]}{\mathbb{E}[X]}$$
Indeed $\text{min}(X,s)$ represents the restriction that the available stock is imposing on the quantities to be serviced without delay. If the actual demand value $x$ is lower than $s$, then $x$ units get served without delay otherwise, only $s$ units get served without delay.

## Calculating the fill rate with Envision

Lokad offers the possibility to compute a probabilistic

forecast of the demand. The result of the forecast is a

distribution of probabilities which can be turned into a distribution of fill rates with a dedicated function, namely the

`fillrate()`

function, as illustrated with:

Demand = call forecast.demand( ... ) // code snipped
FR = fillrate(Demand)

The variable

`FR`

is also a distribution of probabilities, and represents the marginal fill rate increments. In other words,

`FR`

contains the marginal contribution of each extra unit kept in stock in fulfilling the future demand.