Safety stock when both daily demand and supplier lead time vary

If a supplier usually takes four days but sometimes takes longer, multiplying daily demand by four gives only an expected requirement. It does not describe the stock needed when extra demand days occur. Daily demand variability and lead-time variability create different sources of uncertainty.

Peter King's safety-stock article, hosted by MIT, discusses combining these sources under independence assumptions and distinguishes cycle service level from fill rate. The calculation below makes the underlying random-sum model explicit so the variance formula can be checked rather than copied without context.

The aim is to estimate demand accumulated while a replenishment is outstanding. It is not to choose the order quantity, measure every possible supply disruption, or guarantee that a particular stock level will meet all customer requests.

Assume daily demand is independent from day to day, with the same mean of 20 units and standard deviation of 5 units. Lead time is a random whole number of days with mean 4 and standard deviation 1. Assume lead time is independent of the demand sequence.

These are hypothetical inputs, not a claim about any supplier. The standard deviations imply daily demand variance of 25 units squared and lead-time variance of 1 day squared. All observations use the same day definition, such as operating days rather than a mixture of operating and calendar days.

Let L denote lead time. Total demand is the sum of the demands on each of those L days. That random number of daily observations is important. It is a different model from drawing one uncertain daily rate and multiplying that same rate by every day of a random lead time.

For a known lead time of L days, expected total demand is 20 × L and its variance is 25 × L. Independent daily variances add. Averaging the conditional variance over possible lead times gives 25 × 4, or 100.

There is also variation in the conditional mean because L itself changes. The variance of 20 × L is 20 squared multiplied by the variance of L, or 400. Adding the two pieces gives total lead-time demand variance of 500.

More generally, the random-sum result is mean lead time × daily demand variance + daily mean demand squared × lead-time variance. The expected total demand is mean lead time × daily mean demand. No additional product-of-variances term belongs in this particular independent random-sum model.

Suppose an analyst uses a normal approximation and a one-sided 95% probability target. Using a z value of 1.645, the illustrative safety stock is 1.645 multiplied by 22.36, or approximately 36.78 units. Adding expected demand of 80 gives 116.78 units.

Rounding that threshold upward produces 117 units. If lead time were instead fixed at four days, the standard deviation would be the square root of 100, or 10. The corresponding approximate threshold would be 96.45, rounded to 97 units.

The 20-unit difference between rounded thresholds comes from acknowledging lead-time uncertainty in this example. It is not a universal adjustment. Different means, variances or dependence patterns can make the lead-time contribution smaller or much larger.

The mean and variance result follows from the stated random-sum assumptions. The normal approximation adds a separate distributional assumption about the accumulated demand. Knowing only means and standard deviations does not establish that approximation or prove an exact 95% service level.

A skewed lead-time distribution, a rare long delay or lumpy demand can make the relevant upper tail differ from the normal model. Use the observed or otherwise justified distributions to evaluate the threshold when those features matter.

A cycle service target concerns the chance of avoiding a stockout over the specified replenishment exposure. It is not automatically the percentage of units supplied immediately. The amount of unmet demand during a stockout affects fill rate, so the two measures should not be used interchangeably.

Demand can cluster across consecutive days. In that case, variances of daily demands do not simply add without covariance terms. The four-day demand variance may exceed four times the daily variance even when the daily summary statistics look stable.

Lead time may also lengthen during periods of high demand, creating dependence between the number of days and what happens during them. The independent random-sum calculation then omits part of the relationship. Preserve that relationship in a suitable joint model or use evidence from actual replenishment episodes.

Distinguish random variation from a known seasonal change. A forecast of next month's demand may have a different level from the historical daily average. Adding safety stock around an obsolete mean does not repair the underlying forecast.

A continuously reviewed reorder point is normally compared with the defined inventory position, taking relevant outstanding orders and commitments into account. Looking only at units physically on the shelf can trigger repeated orders for stock already on its way.

If inventory is reviewed only periodically, the exposure includes the time until the next review as well as replenishment time. The example's lead-time-only threshold should not be copied into a weekly review system without adapting the protection period.

Use demand data that represent requests or consumption opportunities, not merely shipments capped by a stockout. A history that records zero sales while shelves were empty can understate both expected demand and its uncertainty. Lead times likewise need consistent start and availability timestamps.

In this illustration, changing lead-time length contributes 400 of the 500 variance units. That points toward investigating delivery reliability as well as demand forecasting. It does not establish which improvement is cheapest or feasible.

Present the two variance components, the chosen probability approximation and the observed service results together. Management can then see why the buffer exists and test whether a smaller buffer follows from better reliability, a different service objective or merely a more optimistic assumption.