Safety Stock with a Variable Lead Time
In one sentence
When the lead time varies, inventory protection must account for the uncertainty of both daily consumption and the expected time.
By the end of this lesson
- Calculate the two components of total demand variance under independent demand and replenishment.
- Find the raw safety, the rounded protection and the order threshold from the given data.
- Compare, numerically, the options of reducing replenishment variability and of reducing daily demand variability.
- Stop using the independence formula when demand and replenishment are correlated.
Why it matters
The safety calculation with a fixed time assumes that the day on which the order will be ready for use does not change. In reality a delay in replenishment can lead to more consumption while the material is awaited. Even if demand stays the same, a longer time has to be covered. With a variable lead time, safety stock also covers this additional uncertainty. Combining only the average delivery day with the daily demand deviation leaves the volatility in the time invisible. But calculating two different uncertainties separately and adding the safety quantities directly is generally not correct either. Under independent conditions the quantity that combines is the variance; the standard deviation and the protection are found afterwards.
The model of this lesson requires that the daily demands be independent of each other and come from the same distribution, and also that the lead time be independent of these demands. The variance of the total quantity consumed during the time replenishment takes consists of two parts. The first part is the daily volatility within the average number of days. The second part is the volatility arising over the average consumption because the number of days changes. If the daily mean is high, even a small deviation in time can have a large effect. Showing the two terms separately makes it possible to judge, apart from raising inventory, which process improvement will be more effective.
The calculation of the variance under these conditions and the use of the service factor are separate assessments. The total variance formula is valid when its conditions are met; choosing protection with the factor z requires an approximately normal representation of total replenishment demand. If replenishment is from time to time cut very long, distributions with the same mean and standard deviation can carry different tail risks. The appearance of 1.65 in the calculation therefore does not give a firm guarantee of a service percentage in real life. The shape of the distribution, the data coverage and rare events are examined. Repeated heavy delays are not left as unexplained events to be covered just by a higher factor.
In a relationship where the supplier is also loaded when demand rises, the independence assumption is especially doubtful. If high consumption falls exactly in long replenishment periods, the conditions in which the most protection is needed arise together. Calculating the total deviation with the independent model can miss this pairing. Demand and replenishment records are therefore stored with their time order and pairings preserved. Having only two averages in separate files is not enough to examine the relationship. In correlated data, demand and time must be assessed together according to the condition; adding an arbitrary extra percentage to the independent formula does not turn a wrong assumption into a verified model.
Formula
Mean replenishment demand under the independent model (units) = E(L) × E(d)
Replenishment demand variance (units²) = E(L) × σd² + E(d)² × σL²
Replenishment demand standard deviation (units) = √[E(L) × σd² + E(d)² × σL²]
Safety stock (units) = z × Replenishment demand standard deviation
Reorder point (units) = Average replenishment demand + Safety stock
Time base: σd is the deviation of one day's quantity in units
L is used as the number of one-day periods in the first variance term and as days in the time term
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- Step-by-step instructions
- Worked example with real numbers
- Where the method misleads
- Common mistakes