Which costs more: a forecast 10 per cent low or 10 per cent high?
A biased forecast does not add noise to the stock. It moves the reorder point, which is the same thing as changing the service level, and on AF-7 the two directions are nowhere near symmetric.
On part AF-7, a forecast that runs 10 per cent low costs Aldermere Filtration 8,952.25 EUR a year. A forecast that runs 10 per cent high does not cost it anything: it saves 2,168.93 EUR a year. Same part, same supplier, same error in size, and the low one is worse by 11,121.18 EUR. The reason is not that over-forecasting is a virtue. It is that a 10 per cent over-forecast lifts the board’s 98 per cent service level to within 0.0004 of a standard deviation of the economic optimum, and quietly collects the whole of the gap the board left on the table.
The part and the error
AF-7 sells 1,840 units a week with a weekly standard deviation of 462, and the supplier delivers in six weeks with a standard deviation of 1.9. Demand during lead time averages 11,040 units, with a standard deviation of 3,674.60, and the board has voted a cycle service level of 98 per cent. That is a z of 2.0537 and a safety stock of 7,546.70 units, and at the economic batch of 4,384.50 units it leaves 588.83 units short a year.
A bias is not scatter. Safety stock is sized against the scatter of demand around its forecast; a bias moves the forecast itself, and the safety stock is then measured from the wrong centre. A forecast 10 per cent low puts the reorder point 1,104 units too low (10 per cent of 1,840 a week for six weeks). The stock carried falls by 1,104 units and the protection against a stock-out falls with it. A forecast 10 per cent high does the opposite. In units of the lead-time standard deviation, 1,104 units is 0.3004, so the effective z moves from 2.0537 to 1.7533 in one direction and to 2.3542 in the other.
Each unit held costs 3.8324 EUR a year (28.60 EUR of landed cost at a holding rate of 13.4 per cent) and each unit short loses the margin of 18.90 EUR. Those two prices are all the Forecast bias sheet needs.
Bias
Shift, units
Effective z
Holding
Shortage
Net a year
−10%
−1,104
1.7533
−4,230.97
+13,183.22
+8,952.25
−5%
−552
1.9035
−2,115.48
+5,471.58
+3,356.10
+5%
+552
2.2040
+2,115.48
−3,805.65
−1,690.17
+10%
+1,104
2.3542
+4,230.97
−6,399.90
−2,168.93
Workbook 4, Forecast bias sheet, rows 5 to 8. EUR a year against the board’s 98 per cent; a plus is a cost.
The holding column is symmetric, as it must be: 1,104 units at 3.8324 EUR is 4,230.97 EUR whichever way they move. The shortage column is not. Losing 1,104 units of cover adds 13,183.22 EUR of lost margin; gaining the same 1,104 units removes only 6,399.90. The ratio of the net effects is 4.13 at 10 per cent and 1.99 at 5 per cent, so the asymmetry grows with the size of the error.
Why the two directions differ
The first reason is the shape of the tail. Expected shortage falls away steeply as the reorder point rises and climbs steeply as it drops. At the board’s z the part is short 588.83 units a year. Move the reorder point down by 1,104 units and the shortfall more than doubles, to 1,286.36; move it up by the same amount and it falls to 250.22. The first move adds 697.52 units of lost sales, the second removes 338.62. A unit short costs 4.93 times what a unit held costs, so the shortage side dominates both rows.
The second reason is the more useful one. The board’s 98 per cent is not the optimum. Workbook 2 sets the economic optimum on its Three policies sheet: a cycle service level of 99.0708 per cent, a z of 2.3538, and a total cost of 54,685.21 EUR a year against the board’s 56,854.14. The gap is 2,168.94 EUR. A 10 per cent over-forecast takes the effective z to 2.3542. It lands on the optimum, and the 2,168.93 it saves is the gap to three tenths of a cent.
An over-forecast of 10 per cent is not a forecasting error that happens to pay. It is a service level of 99.07 per cent that nobody voted on, and it pays because the level the board did vote on is below the optimum. The bias that lands exactly on the optimum is 9.99 per cent, a shift of 1,102.47 units. Fix the board’s figure at the optimum and every bias, in either direction, becomes a cost.
How far over-forecasting keeps paying
If a high forecast saves money at 10 per cent, the obvious next question is where it stops. The sheet prints four biases; the same formulas run on further rows give the rest of the curve.
Bias
Effective cycle service level
Units short a year
Net a year
−20%
92.69%
2,614.37
+29,820.65
−15%
94.55%
1,850.21
+17,493.48
−10%
96.02%
1,286.36
+8,952.25
0%
98.00%
588.83
0.00
+10%
99.07%
250.22
−2,168.93
+15%
99.39%
158.53
−1,786.39
+20%
99.60%
98.52
−805.06
+25%
99.75%
60.04
+583.27
The net figures at −10 and +10 per cent are the sheet’s own; the other rows run the same formulas on further biases, and the service level is NORM.S.DIST of the effective z.
An over-forecast stays cheaper than an unbiased forecast up to a bias of 23.04 per cent, and turns into a cost only beyond it. An under-forecast is a cost from the first unit, and a steep one: 20 per cent low costs 29,820.65 EUR a year, 3.33 times the cost of 10 per cent low for twice the error. The curve is flat above the board’s level and steep below it, and a forecasting process whose errors are equal in size on both sides will still lose money on balance.
Set against the argument the board had
The book spends its middle chapters on the service level, and Workbook 2 prices the whole dispute: taking the cycle service level anywhere from 97 to 99.5 per cent moves the total cost by 6,197.72 EUR a year at most. A forecast that runs 10 per cent low costs 8,952.25, which is 1.44 times the entire range the board was arguing over. The service level is the number that gets voted on; the bias is the number that decides what service level the part actually receives.
That is why the effective service level column matters more than the net column. A 10 per cent under-forecast delivers 96.02 per cent against a vote of 98. Nobody decided that, and nobody sees it in the safety-stock calculation, because the calculation is correct: it is measured from a centre that is wrong.
The optimum used here is the one Workbook 2 computes, with the batch held at the economic order quantity. Case two in the companion files sets the batch and the service level together and takes a further 683.37 EUR a year off the cost; it moves the optimum to 98.7622 per cent with a larger batch. The Forecast bias sheet keeps the batch fixed and moves only the reorder point, which is what a biased forecast of lead-time demand does.
What to do with it
Measure the bias before the scatter. A mean error over the last twelve months of lead-time demand, with its sign, is the first control figure. The standard deviation of the error is the second.
Translate every bias into the service level it delivers. “Forecast 10 per cent low” means nothing to a board; “AF-7 is running at 96.02 per cent against a vote of 98” is a decision it has already taken, now being broken.
Do not leave a known over-forecast in place because it pays. If it pays, the voted service level is wrong. Move the vote to the optimum and remove the bias; the saving is the same 2,168.94 EUR, and it no longer depends on the forecast being wrong by the right amount.
Treat an under-forecast as the expensive direction. On these inputs a low bias costs 4.13 times what the same high bias saves, and 20 per cent low costs more than three times what 10 per cent low does.
A bias is a stock policy that nobody voted on. The only question is whether the policy it sets is better or worse than the one the board chose, and on AF-7 the answer depends entirely on the sign.
Reproducing it in the workbook
Open Workbook 4, Pooling, Bias and the Holding Rate. The inputs are on the Assumptions sheet: demand in C5 and C6, lead time in C8 and C9, landed cost in C11, the four holding components in C12 to C15, the board’s service level in C17. The lead-time standard deviation of 3,674.60 is C22, the board’s z of 2.0537 is C26 and the 588.83 units short a year is C28. The Forecast bias sheet holds the first table in rows 5 to 8: the bias in column B, the shift in C, the effective z in D, holding in E, shortage in F and the net effect in G. Overtype B8 with 0.2 or 0.25 to reproduce the extended rows; the Checks sheet will then flag rows 31 to 34, which is the file computing, not breaking. NORM.S.DIST of column D gives the effective service level.
The optimum is in Workbook 2, Service Levels and the Money, on the Three policies sheet: the service level in C7, its z in D7, the total cost in I7, the gap to the board in I10 and the 6,197.72 EUR dispute in I11. The blank twins in the blank set run the same sheets on your own part number.
The workbooks behind this article
Every figure above is a live formula in the free companion files for
Supply Chain Management. Each workbook ends with a Checks sheet
setting the printed figure beside the computed one. No account and no email address.
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