### Service levels

Several definitions of service levels are used in the literature as well as in practice. These may differ not only with respect to their scope and to the number of considered products but also with respect to the time interval they are related to. These performance measures are the *Key Performance Indicators* (KPI) of an inventory node which must be regularly monitored. If the controlling of the performance of an inventory node is neglected, the decision maker will not be able to optimize the processes within a supply chain.

$\alpha$ Service Level

The $\alpha$ service level is an **event-oriented** performance criterion. It measures the probability that *all* customer orders arriving within a given time interval will be completely delivered from stock on hand, i.e. without delay.

Two versions are discussed in the literature differing with respect to the time interval within which the customers arrive. With reference to a *demand period*, α denotes the probability that an arbitrarily arriving customer order will be completely served from stock on hand, i.e. without an inventory-related waiting time (period $\alpha_p$ service level):

$\alpha_p = P\{\mathrm{Period Demand} \leq \mathrm{Inventory on hand at the beginning of a period}\}$

In order to determine the safety stock that guarantees a target $\alpha_p$ service level, the stationary probability distribution of the inventory on hand must be known. This version of $\alpha$ is also called *ready rate*.

If an *order cycle* is considered as the standard period of reference, then $\alpha$ denotes the probability of no stock-out within an order cycle which is equal to the proportion of all order cycles with no stock-outs (cycle $\alpha_c$ service level):

$\alpha_p = P\{\mathrm{Demand during replenishment lead time} \leq \mathrm{Inventory on hand at the beginning of the lead time}\}$

This second definition, which is often used in operations management textbooks, is based on the idea of not running out of stock during the time between the re-ordering and the order arrival (the lead time).

$\beta$ Service Level

The β service level (fill rate) is a **quantity-oriented** performance measure describing the proportion of total demand within a reference period which is delivered without delay from stock on hand:

$\beta=1-\frac{\mathrm{Expected backorders per time period}}{\mathrm{Expected demand per time period}}$

This is equal to the probability that an arbitrary demand unit is delivered without delay.

Because, contrary to the variations of the $\alpha$-service level, the $\beta$-service level does not only reflect the stock-out *event* but also the *amount backordered*, it is widely used in industrial practice.

$\gamma$ Service Level

The $\gamma$ service level, a time- and quantity-related performance criterion, serves to reflect not only the amount of backorders but also the waiting times of the demands backordered. The $\gamm$ service level is defined as follows:

$\gamma=1-\frac{\mathrm{Expected backorder level per time period}}{\mathrm{Expected demand per time period}}$

The $\gamma$ service level is rarely used in industrial practice.

Further performance indicators

- Duration of a stockout situation

- Customer order waiting time

The customer witing time plays a key role in supply chain optimization. In a supply chain, the replenishment lead time of a customer node (e.g. a retailer) is the customer waiting time observed at the supplier (e.g. a central warehouse). In this situation, high low upstream safety stock can be compensated by low downstream safety stock, and vice versa. The interrelating mechanism is the waiting time.