How do you calculate safety stock from forecast error?

When stock is planned against a forecast, the buffer covers what the forecast misses, and how much a better forecast saves depends on how reliable the supplier is.

Julian R. Sterling · from Supply Chain Management

When stock is planned against a forecast, safety stock should cover the forecast error, not the swings in demand: SS = z × √(L × σ_e² + d² × σ_L²), with σ_e the standard deviation of weekly forecast errors. For part AF-7 in Supply Chain Management a forecast that cuts the weekly spread from 462 units to an error of 193.8 cuts the 98 per cent safety stock from 2,324 to 975 units if lead time is fixed, but only from 7,547 to 7,246 units once the supplier's lead-time variability is counted, a 4.0 per cent saving.

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

Why the forecast error is the right sigma

The textbook safety stock formula uses the standard deviation of demand around its average. That is right when the plan is the average: every week the system expects 1,840 units and holds stock against anything else. Most companies plan against a forecast that moves with the season, the promotion and the order book. What the buffer has to absorb is then the part of demand the forecast did not see, which is the forecast error. Measured that way, a good forecast is worth stock. How much depends on what else the buffer is covering.

The data: thirteen weeks of forecast against actual

AF-7, one-week-ahead forecast, illustrative history; error = actual minus forecast
Week	Actual	Forecast	Error
1	1,390	1,640	−250
2	2,480	2,260	220
3	1,730	1,820	−90
4	2,030	2,070	−40
5	1,240	1,530	−290
6	2,640	2,360	280
7	1,910	1,990	−80
8	1,580	1,720	−140
9	2,220	2,010	210
10	1,140	1,460	−320
11	2,140	2,100	40
12	1,650	1,800	−150
13	1,770	1,870	−100
Total	23,920	24,630	−710

The actuals average 1,840 a week with a standard deviation of 456, close to the 462 the book measures over a longer history and uses below. The other inputs are the book's: a 6-week lead time with a standard deviation of 1.9 weeks, a 98 per cent cycle service level (z = 2.0537), 28.60 EUR a unit and a 13.4 per cent holding rate.

The calculation, step by step

1. Bias = mean error = −710 / 13 = −54.6 a week: the forecast runs 2.97 per cent high

2. σ_e = standard deviation of the errors = 193.8 (RMSE 194.1; 1.25 × MAD = 1.25 × 170.0 = 212.5)

3. Lead-time demand variance from the forecast: 6 × 193.8² = 225,462 (against 6 × 462² = 1,280,664 from raw demand)

4. Lead-time variance from the supplier: 1,840² × 1.9² = 12,222,016

5. SS = 2.0537 × √(225,462 + 12,222,016) = 7,246 units

Excel: =NORM.S.INV(0.98)*SQRT(L*STDEV.S(Errors)^2+d^2*sL^2)

The result

AF-7 safety stock at 98 per cent, by the sigma used
Sigma used	Sigma	SS, fixed lead time	SS, variable lead time	Value, variable
Demand around its mean	462.0	2,324	7,547	215,836
Forecast error, standard deviation	193.8	975	7,246	207,231
Forecast error, 1.25 × MAD	212.5	1,069	7,259	207,609

If the lead time were fixed, the forecast would release 1,349 units, 58.0 per cent of the buffer and 38,581 EUR of stock. With the supplier as it is, it releases 301 units, 8,605 EUR, and 1,153 EUR a year of holding cost. The forecast removed four-fifths of the demand-side variance, but demand was never the problem on this part: with the forecast in place, 98.2 per cent of the remaining variance is the supplier's lead time.

What if: the supplier became more regular

Safety stock saved by the forecast, by lead-time standard deviation
σ_L, weeks	SS on demand sigma	SS on forecast error	Units saved	Saving
1.9	7,547	7,246	301	4.0%
1.0	4,436	3,903	534	12.0%
0.5	2,995	2,126	869	29.0%

The two levers are complements. A better forecast is worth almost nothing while the supplier is erratic, and nearly three times as many units once the supplier is brought to half a week. The order of work on AF-7 is supplier first, forecast second, which is the argument the book makes with the variance split.

The common mistakes

Using the RMSE without removing bias. Here the bias is small and the RMSE of 194.1 is close to the 193.8 standard deviation. A biased forecast should be corrected, not buffered: AF-7's forecast runs 54.6 units a week high, so over a 6-week lead time the reorder point already carries 328 units the plan does not know it has.
Scaling a one-week error by √L when errors are correlated. The formula assumes each week's error is independent. If one week's miss tends to repeat the next, with a correlation of 0.3 the 6-week error has a standard deviation of 611, not 475. Measure the error of the forecast over the lead time itself whenever the history allows.
Taking 1.25 × MAD as exact. It is the normal-distribution shortcut. On this history it gives 212.5 against 193.8, which matters little here and more on a part where demand dominates.

Takeaway

Plan to a forecast, buffer the forecast error: SS = z × √(Lσ_e² + d²σ_L²).
For AF-7, 7,246 units instead of 7,547: a 4.0 per cent saving, because the supplier dominates.
Correct bias in the forecast; use the standard deviation of errors in the buffer.
Fix lead-time variability first; the forecast's value grows as it falls.

The forecast-bias sweep from minus 10 to plus 10 per cent and the lead-time table are in the free workbooks for this case; what a biased forecast costs in euros is worked in which costs more: a forecast 10 per cent low or 10 per cent high.

Questions readers ask

Should safety stock use forecast error or demand standard deviation?

Forecast error, whenever replenishment is planned against a forecast, because the buffer only has to absorb what the forecast did not anticipate. On part AF-7 the forecast error is 193.8 a week against a demand spread of 462, which with a fixed lead time cuts the 98 per cent safety stock from 2,324 to 975 units.

How do you convert MAD to a standard deviation for safety stock?

Multiply the mean absolute deviation of forecast errors by about 1.25, the ratio that holds for normally distributed errors. On thirteen illustrative weeks for AF-7 the MAD is 170.0, giving 212.5, against a measured standard deviation of 193.8. Use the measured figure when you have enough history; the shortcut slightly overstates here.

Why did a better forecast barely reduce safety stock?

Because most of the uncertainty over the lead time came from the supplier. For AF-7 the lead-time term is 12,222,016 against 225,462 from forecast error, 98.2 per cent of the variance. The forecast saves 301 units at a 1.9-week lead-time spread, but 869 units, 29.0 per cent, once the spread is cut to 0.5 weeks.

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 calculate safety stock when lead time varies?z times the root of L x demand variance plus d squared x lead-time variance: 7,547 units for AF-7 at 98 per cent, against 2,324 if lead time is treated as fixed.
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.
How do you calculate a reorder point with safety stock?Reorder point = demand over the lead time + safety stock: 11,040 + 7,547 = 18,587 units for part AF-7, compared with stock on hand plus stock on order.
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.
How do you calculate the economic order quantity (EOQ)?EOQ is the square root of 2 x annual demand x order cost / annual holding cost per unit: 4,384.5 units for 95,680 a year, 385 an order and 3.8324 to hold.
How do you calculate EOQ with quantity discounts?Price each break at its EOQ or minimum quantity, add purchase cost, take the lowest total. For AF-7, 0.5% off at 8,000 saves 10,628 a year; 1% at 20,000 only 4,385.

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