How do you calculate safety stock when lead time varies?

Add the variances, not the standard deviations, and find out who really owns the uncertainty: the market or the supplier.

Julian R. Sterling · from Supply Chain Management

When both demand and lead time vary, safety stock is z × √(L × σ_d² + d² × σ_L²): the two variances add, not the two standard deviations. For a part selling 1,840 units a week with a standard deviation of 462, on a 6-week lead time that varies by 1.9 weeks, the 98 per cent cycle service level needs 7,547 units of safety stock. The demand-only formula gives 2,324, short by 69.2 per cent, because 90.52 per cent of the variance comes from the supplier.

Worked in full in Supply Chain Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →

Most planning systems still default to the textbook formula that treats lead time as fixed. On a part with a reliable supplier that is harmless. On a part with an erratic one it is the single largest error in the stocking policy, and it hides because stock-outs are blamed on demand.

The assumptions

The case is Aldermere Filtration's part AF-7, the fictional example that runs through the companion files of Supply Chain Management.

Part AF-7. Illustrative case.
Input	Value
Mean weekly demand, d	1,840 units
Standard deviation of weekly demand, σ_d	462 units
Mean lead time, L	6 weeks
Standard deviation of lead time, σ_L	1.9 weeks
Cycle service level (z)	98% (2.0537)
Purchase cost delivered	28.60 EUR
Holding rate	13.4% a year

Demand and lead time must be in the same time unit. Weekly demand with a lead time in days is the most common way this formula produces nonsense.

The calculation, step by step

Variance of demand during lead time = L × σ_d² + d² × σ_L²
Safety stock = z × √variance Reorder point = d × L + safety stock

From demand: 6 × 462² = 1,280,664.
From lead time: 1,840² × 1.9² = 12,222,016.
Total variance 13,502,680; standard deviation 3,674.60 units.
Safety stock = 2.0537 × 3,674.60 = 7,547 units. Reorder point = 11,040 + 7,547 = 18,587.
In Excel: =NORM.S.INV(B6)*SQRT(B3*B2^2+B1^2*B4^2) with d in B1, σ_d in B2, L in B3, σ_L in B4 and the service level in B6.

The result

Where the variance comes from.
Source	Variance	Share
Demand (market)	1,280,664	9.48%
Lead time (supplier)	12,222,016	90.52%
Total	13,502,680	100%

Nine tenths of the uncertainty on AF-7 is the supplier's. The safety stock of 7,547 units is worth 215,836 EUR and costs 28,922 EUR a year to hold at 13.4 per cent. Any conversation about forecast accuracy on this part is a conversation about the smaller tenth.

What if: the supplier became more reliable

Safety stock at a 98% cycle service level, demand unchanged.
σ_L, weeks	Lead-time share of variance	Safety stock, units	Value, EUR	Holding cost a year, EUR
0.0	0.0%	2,324	66,471	8,907
0.5	39.8%	2,995	85,665	11,479
1.0	72.6%	4,436	126,881	17,002
1.5	85.6%	6,126	175,213	23,479
1.9	90.5%	7,547	215,836	28,922
2.5	94.3%	9,729	278,247	37,285
3.0	96.0%	11,572	330,973	44,350

Taking the supplier's spread from 1.9 weeks to 0.5 releases 4,551 units, 130,171 EUR of stock and 17,443 EUR a year of holding cost on safety stock alone. By contrast, doubling the demand standard deviation to 924 with the supplier unchanged lifts safety stock only to 8,553. The lever on this part is the supplier, not the forecast. The book's reliability workbook values the same move, 1.9 weeks to 0.5, at 24,154.77 EUR a year on its full policy cost, against the 17,443 EUR of safety-stock holding counted here.

The common mistakes

Ignoring lead-time variability. z × σ_d × √L gives 2,324 units. Held against the true spread of 3,674.60, that stock delivers a cycle service level of about 73.6 per cent, not the 98 per cent the board voted for.
Adding the standard deviations. z × (σ_d√L + dσ_L) gives 9,504 units, 25.9 per cent too many: 55,980 EUR of stock and 7,501 EUR a year for no service gain. Independent sources of variation add in variance.
Estimating σ_L from the supplier's promise. Measure it from goods receipts, order date to receipt date. A quoted lead time has no spread; a delivered one does.

Takeaway

Safety stock = z × √(Lσ_d² + d²σ_L²): 7,547 units for AF-7.
Split the variance before arguing about the forecast: here 90.52 per cent is lead time.
The supplier's regularity is worth money, and it can be priced into the supply contract.

The variance split and the lead-time table from 0.0 to 3.0 weeks are in the free workbooks for this case on the Supply Chain Management companion page. The same part's service level is read as a fill rate in how to convert a cycle service level into a fill rate, and the cost of a biased forecast in which costs more: a forecast 10 per cent low or 10 per cent high.

Questions readers ask

What is the safety stock formula with variable demand and lead time?

Safety stock equals z times the square root of (mean lead time times demand variance plus mean demand squared times lead-time variance). For AF-7, with 1,840 units a week, a 462 standard deviation, a 6-week lead time and a 1.9-week spread, that is 2.0537 times 3,674.60, or 7,547 units at a 98 per cent cycle service level.

Why not add the demand and lead-time standard deviations?

Because independent sources of variation add in variance, not in standard deviation. Adding the two spreads gives 4,628 units of standard deviation instead of 3,674.60, and a safety stock of 9,504 units, 25.9 per cent too much. On AF-7 the excess ties up 55,980 EUR of stock for no service gain.

How much does lead-time variability add to safety stock?

On AF-7, almost everything. With a perfectly regular supplier the 98 per cent safety stock would be 2,324 units; at a 1.9-week spread it is 7,547. Cutting the spread to 0.5 weeks releases 4,551 units and 17,443 EUR a year of holding cost on the safety stock alone.

Read the whole case

This article is one calculation from Supply Chain Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.

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Also on this site

How do you convert a cycle service level into a fill rate?Fill rate is 1 minus sigma times G(z) over the order quantity. On part AF-7 a 98 per cent cycle service level is a 99.38 per cent fill rate.
Which costs more: a forecast 10 per cent low or 10 per cent high?On part AF-7 a forecast 10 per cent low costs 8,952.25 EUR a year; one 10 per cent high saves 2,168.93, because it lifts the board's 98 per cent to the optimum.
Is a longer warranty worth more than a better remedy?Paying the repair instead of the part is worth 200,336.62; the extra warranty year is worth 196,996.34, and only 23,019.80 if the remedy stays at the part.
When does an indexed price overtake a dearer firm bid?A bid at 104.00 indexed after 12 months crosses a 109.50 bid firm for 24 in month 19. A 878,350.00 headline saving falls to 87,304.04 over the term.
How do you calculate activity-based costing per unit?Pool over driver count gives the rate; counts times rates over units gives overhead a unit. A bracket costing 287.69 on labour hours costs 436.91 on activities.
How do you calculate construction cost escalation to the midpoint?Compound the inflation rate from the base date to mid-construction: 24,000,000 escalated 21 months at 4.0 per cent adds 1,705,117, or 7.10 per cent.

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