Safety stock

A commonly used approach calculates[6][7] the safety stock based on the following factors:

• Demand is the number of items consumed by customers, usually a succession of independent random variables.
• Lead time is the delay between the time the reorder point (inventory level which initiates an order[8]) is reached and renewed availability.
• Service level is the desired probability of meeting demand during the lead time without a stockout. If the service level is increased, the required safety stock increases, as well.
• Forecast error is an estimate of how far actual demand may be from forecast demand.

Assuming that demand during successive unit time periods are independent and identically distributed random variables drawn from a normal distribution, the safety stock can be calculated as:[9]

${\displaystyle SS=z_{\alpha }\times {\sqrt {E(L)\sigma _{D}^{2}+(E(D))^{2}\sigma _{L}^{2}}}}$

where,

• ${\displaystyle \alpha }$ is the service level, and ${\displaystyle z_{\alpha }}$ is the inverse distribution function of a standard normal distribution with cumulative probability ${\displaystyle \alpha }$; for example, ${\displaystyle z_{\alpha }}$=1.65 for 95% service level. The service level can be easily calculated in Excel by typing in the formula =normsinv(probability%). For eg entering =normsinv(95%) will return 1.65 as the answer.[10]
• ${\displaystyle E(L)}$ and ${\displaystyle \sigma _{L}}$ are the mean and standard deviation of lead time.
• ${\displaystyle E(D)}$ and ${\displaystyle \sigma _{D}}$ are the mean and standard deviation of demand in each unit time period.[11]

The reorder point can then be calculated as:

${\displaystyle ROP=E(L)\cdot E(D)+SS}$

The first term in the ROP formula ${\displaystyle E(L)E(D)}$ is the average demand during the lead time. The second term ${\displaystyle ss}$ is the safety stock. If the lead time is deterministic, i.e. ${\displaystyle \sigma _{L}=0}$, then the ROP formula is simplified as ${\displaystyle ROP=L\cdot E(D)+z_{\alpha }\sigma _{D}{\sqrt {L}}}$.

No universal formula exists for safety stock, and application of the one above can cause serious damage. [12] [13] It makes several implicit assumptions:

• The assumption that demand is a succession of independent normal random variables: First, real demand cannot be negative. If the ratio of standard deviation to mean is quite high, this will skew the distribution (compared to the normal distribution), leading to consistent overestimation of safety stock by this formula. Second, and more importantly, demand is often influenced by external random factors which persist for more than one time period, so that the successive demands are not independent. With a very large number of sources (for example, consumers of a central retail warehouse), that may not be an issue, but otherwise it is (for example, for a manufacturer that supplies these retail warehouses)
• The use of average and standard demand assumes it is constant. For seasonal demand (for example high in summer, low in winter), the formula will consistently produce stock outs in summer and waste in winter. Similar errors apply for demand that grows or declines. That does not invalidate the formula, but it influences the parameters to be input into the formula in each time period.
• Lead time is extremely hard to quantify in complex manufacturing and/or purchase environment, which has become the norm in global supply chains that span many independent partners. In practice, lead time is estimated by a rule of thumb that hardly improves on estimating safety stock with a rule of thumb. Even when lead time is correctly quantified, the formula assumes supply (production and purchase) is statistically constant, which is not always the case.

Another popular approach described by Nahmias[14] uses the standardized unit loss integral L(z), given by:

${\displaystyle L(z)=\int _{z}^{+\infty }\left(t-z\right)\phi (t)dt}$

Where ${\displaystyle \phi (t)}$ is cumulative distribution function for the standard normal. Let β be the proportion of demands met from stock (service level), Q the order quantity and σ the standard deviation of demand, then the following relationship holds:

${\displaystyle L(z)=(1-\beta )Q/\sigma }$

In this case, the safety stock is given by:

${\displaystyle SS=z_{\beta }\sigma }$

and the expected number of units out of stock during an order cycle is given by σL(z).[15]