Nobody upstream is irrational. The order-up-to rule itself turns a steady market into lumpy orders, and the size of the swing can be written down before a single week is simulated.
The bullwhip ratio is the variance of the orders a stocking point places divided by the variance of the demand it sees. For an order-up-to policy that forecasts with a p-week moving average and holds L weeks of cover, it is 1 + 2L/p + 2L²/p². On a part selling 1,840 units a week with a standard deviation of 462, seven weeks of cover and a four-week average give a ratio of 10.62: the supplier sees orders with a standard deviation of 1,506 units, 3.26 times the swing in real demand.
Worked in full in Supply Chain Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The bullwhip effect is usually explained with stories about panic buying and promotions. Those make it worse, but they are not needed. A perfectly disciplined planner running a textbook reorder rule amplifies demand variability on the way upstream, and the amount follows from two numbers the planner chose: how many weeks the order-up-to level covers, and how many weeks the forecast averages.
| Input | Value |
|---|---|
| Mean weekly demand | 1,840 units |
| Standard deviation of weekly demand | 462 units |
| Supplier lead time | 6 weeks |
| Review period | 1 week |
| Cover the order-up-to level protects (L) | 7 weeks |
| Forecast: moving average of the last p weeks | p = 4 |
Each week the planner sets the order-up-to level at L times the forecast plus a safety stock, and orders whatever brings the inventory position back up to it. The order is therefore last week's demand, to replace what was sold, plus the change in the order-up-to level.
Order-up-to level: St = L × Mt + safety stock, where Mt is the average of the last p weeks.
The average moves by (Dt−1 − Dt−1−p) ÷ p each week, so the level moves by L/p times that.
Order: qt = Dt−1 + (L/p)(Dt−1 − Dt−1−p) = (1 + L/p) Dt−1 − (L/p) Dt−1−p
With L = 7 and p = 4, L/p = 1.75: each order is 2.75 times one week's demand minus 1.75 times the demand of a week that has just left the average.
That one line is the whole mechanism. A week of high sales is counted once as replenishment and again, multiplied by seven weeks of cover, as evidence that the level should rise. Four weeks later the same week drops out of the average and the level falls back, so the order is cut.
With independent weeks, the variance of a weighted sum is the sum of the squared weights times the demand variance:
Var(q) ÷ Var(D) = (1 + L/p)² + (L/p)² = 1 + 2L/p + 2L²/p²
= 1 + 3.50 + 6.125 = 10.62
Order standard deviation: 462 × √10.625 = 462 × 3.26 = 1,506 units.
In Excel, with cover in B1, the averaging window in B2 and the demand standard deviation in B3: =1+2*B1/B2+2*B1^2/B2^2 for the ratio and =B3*SQRT(1+2*B1/B2+2*B1^2/B2^2) for the order standard deviation.
The variance of weekly demand is 213,444 units squared; the variance of weekly orders is 2,267,843. Average orders still equal average demand, 1,840 units, because nothing is lost or created. Only the spread changes.
| Week | Demand | Order placed after it |
|---|---|---|
| Normal week | 1,840 | 1,840 |
| Demand one standard deviation high | 2,302 | 3,110 |
| Following three weeks, demand normal | 1,840 | 1,840 |
| The high week leaves the four-week average | 1,840 | 1,032 |
Demand moved by 462 units once. The supplier saw one order 1,270 units above normal and, a month later, one 808 units below. Its planners, running the same rule on their own forecasts, amplify that again.
| Weeks averaged (p) | Ratio, 2-week lead time | Order SD | Ratio, 6-week lead time | Order SD |
|---|---|---|---|---|
| 4 | 3.62 | 880 | 10.62 | 1,506 |
| 8 | 2.03 | 658 | 4.28 | 956 |
| 13 | 1.57 | 579 | 2.66 | 753 |
| 26 | 1.26 | 518 | 1.68 | 599 |
| 52 | 1.12 | 489 | 1.31 | 528 |
Two levers, and both work through the ratio L/p. Cutting the lead time from six weeks to two at a four-week average takes the ratio from 10.62 to 3.62 and the order standard deviation down 42 per cent, which is a benefit that rarely appears in the business case for a nearer supplier; the stock side of that case is worked in how much inventory a shorter lead time frees up. Smoothing the forecast does the same job for free, at the price of reacting more slowly to a genuine shift in demand. With seven weeks of cover, the average has to span 20 weeks before the ratio falls below 2.
The supplier sizes its own safety stock on the orders it receives, not on the market. Holding a 98 per cent cycle service level (z = 2.054) over its own two-week lead time:
On end demand: 2.054 × 462 × √2 = 1,342 units, 0.7 weeks of sales.
On the orders it actually sees: 2.054 × 1,506 × √2 = 4,374 units, 2.4 weeks of sales.
Extra stock created by the ordering rule alone: 3,032 units, for a buffer 3.26 times as large.
That stock is paid for somewhere in the chain, and usually comes back in the price. It is also why sharing point-of-sale data with a supplier is worth more than it looks: the supplier can then forecast on the 462, not the 1,506.
The ratio is defined on variances. Reporting 10.62 as if it described the swing in orders, rather than 3.26, overstates the problem threefold and leads to oversized buffers upstream. The opposite error is as common: measuring orders against demand over a period that includes a promotion or a price change, then blaming the ordering rule for a ratio that the event created. Measure the ratio on clean weeks first; the formula above tells you what the rule alone should produce. The same separation of forecast error from lead-time effects sits behind safety stock from forecast error.
Check: if the measured ratio is close to 1 + 2L/p + 2L²/p², the amplification is mechanical and the fix is the forecast window or the lead time. If it is much higher, look for batching, rationing games or promotions.
Write the order as a function of past demand and the bullwhip ratio drops out in one line. With seven weeks of cover and a four-week moving average, a stocking point multiplies the variance it passes upstream by 10.62. The variance split behind the same part number, and how much of it comes from the supplier's lead time rather than the market, is laid out in the free workbooks for this case.
A ratio of 1.0 means orders vary exactly as much as sales, which is what a pure pass-through policy achieves. Anything above that is amplification created by forecasting and buffering. In the illustrative case, 7 weeks of cover with a 4-week moving average gives 10.62; lengthening the average to 20 weeks brings it to 1.94, and to 52 weeks 1.31. Below about 2 the amplification usually matters less than the forecast error itself.
Each forecast revision is multiplied by the weeks of cover the order-up-to level must protect. With 7 weeks of cover and a 4-week average, a one-week rise in demand of 462 units lifts the next order by 1,270 units, and the drop when that week leaves the average cuts a later order by 808. Cutting the lead time from 6 to 2 weeks takes the ratio from 10.62 to 3.62 and the order standard deviation down 42 per cent.
The usual definition is on variance: variance of orders over variance of demand. The standard deviation ratio is its square root. Here a variance ratio of 10.62 is a standard deviation ratio of 3.26. Quoting the first as if it were the second overstates how much wider the supplier's safety stock must be by a factor of more than three.
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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