One order, no second chance: how the margin and the loss on leftovers set how far above or below the forecast to buy.
The newsvendor order quantity is the demand level you have a probability of not exceeding equal to the critical ratio: underage cost divided by the sum of underage and overage costs. For an illustrative seasonal product with a margin of €27.00 and a €12.00 loss on each unsold unit, the ratio is 69.2 per cent, so with demand of 8,000 ± 2,000 units you order 9,005, not the forecast of 8,000. That earns €3,689 more in expectation than ordering the forecast and €13,681 more than ordering for a 95 per cent in-stock probability.
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 newsvendor model is for one buying decision with no second chance: a seasonal line from an overseas supplier whose lead time is longer than the season, a promotional run, a last production batch. Order too few and the margin on the missing units is lost; order too many and the leftovers go to a liquidator for less than they cost. The question is not what demand will be but how far above or below the forecast to place a bet, and the answer comes from the two costs.
| Input | Value |
|---|---|
| Forecast demand for the season, mean | 8,000 units |
| Standard deviation of demand (forecast error, 25% of mean) | 2,000 units |
| Selling price | 45.00 |
| Landed unit cost | 18.00 |
| Salvage value of a unit left at the end of the season | 6.00 |
Demand is taken as normally distributed around the forecast, with the standard deviation measured from past forecast errors on comparable seasons, not from the spread of sales across weeks. Unmet demand is lost, with no further penalty.
Underage cost, the margin lost on a unit you could have sold: Cu = 45.00 − 18.00 = 27.00.
Overage cost, the loss on a unit you could not sell: Co = 18.00 − 6.00 = 12.00.
Critical ratio = Cu ÷ (Cu + Co) = 27.00 ÷ 39.00 = 0.6923.
z = Φ−1(0.6923) = 0.502. Order quantity Q* = μ + zσ = 8,000 + 0.502 × 2,000 = 8,000 + 1,005 = 9,005.
In Excel: =NORM.INV(Cu/(Cu+Co),Mean,SD), or with a discrete demand table, the smallest quantity whose cumulative probability reaches the ratio.
Why the ratio works: the last unit ordered sells with probability 1 − F(Q) and earns 27.00, and is left over with probability F(Q) and loses 12.00. At the optimum the two are equal: 27.00 × 0.3077 = 12.00 × 0.6923 = 8.31. One unit more and the expected loss is larger than the expected gain.
Expected sales use the standard normal loss function L(z) = φ(z) − z(1 − Φ(z)), equal to 0.1971 at the optimum. Expected lost sales = σ × L(z); expected sales = μ minus that; leftovers = Q minus expected sales.
| Policy | Order | Expected sales | Expected leftovers | Expected lost sales | In-stock probability | Expected profit |
|---|---|---|---|---|---|---|
| Order the forecast | 8,000 | 7,202 | 798 | 798 | 50.0% | 184,883 |
| Critical ratio | 9,005 | 7,606 | 1,399 | 394 | 69.2% | 188,572 |
| 95% in-stock target | 11,290 | 7,958 | 3,332 | 42 | 95.0% | 174,891 |
Expected profit = 45.00 × sales + 6.00 × leftovers − 18.00 × order. At 9,005 that is €188,572. Ordering the forecast leaves €3,689 on the table, because with a margin more than twice the overage loss the bet should lean towards having stock. Ordering for 95 per cent in-stock, z = 1.645, overshoots by 2,285 units and costs €13,681: each extra unit near the top is almost always left over.
With perfect foresight the season would earn 27.00 × 8,000 = €216,000. The €27,428 gap to the optimum is the cost of the forecast error, and it is the number that justifies spending on a better forecast or a shorter lead time.
| Case | Overage cost | Critical ratio | Order | Expected leftovers | Expected profit |
|---|---|---|---|---|---|
| Salvage 0.00 | 18.00 | 60.0% | 8,507 | 1,077 | 181,229 |
| Salvage 6.00, base | 12.00 | 69.2% | 9,005 | 1,399 | 188,572 |
| Salvage 12.00 | 6.00 | 81.8% | 9,817 | 2,015 | 198,572 |
| Standard deviation 1,000 | 12.00 | 69.2% | 8,502 | 699 | 202,286 |
| Standard deviation 3,000 | 12.00 | 69.2% | 9,507 | 2,098 | 174,858 |
A better liquidation channel raises both the order and the profit: when leftovers lose less, it pays to carry more. At the same critical ratio, uncertainty cuts profit: halving the forecast error to 1,000 adds €13,714, which is the case for buying closer to the season even at a higher unit cost.
Two mistakes are common. The first is ordering the forecast, which implicitly sets a 50 per cent critical ratio and is right only when the margin equals the overage loss. The second is borrowing a service level target from replenished stock, 95 or 98 per cent, and applying it to a one-shot buy. In a reorder-point system the stock left at the end of a cycle is not wasted: it serves the next cycle, so the trade-off is a holding cost against a shortage. That is a different rule, worked in how to calculate the optimal service level for a stocked part, and the newsvendor ratio gives the wrong answer there just as a 95 per cent target gives the wrong answer here.
Check the costs, not the formula. If a stock-out also loses the customer's next order, add that to Cu. If unsold stock has to be stored for a year before it can be sold off, add the holding cost to Co. The ratio is only as good as the two numbers in it.
Order quantity = forecast + z × standard deviation, with z set by Cu ÷ (Cu + Co): 69.2 per cent and 9,005 units here, €3,689 better than ordering the forecast. For stock you replenish, the book's service-level and safety stock models for one part number are in the free workbook for this case.
It is the underage cost divided by the sum of underage and overage costs, and it is the optimal probability that demand does not exceed the order. With a margin of 27.00 lost per missed sale and 12.00 lost per unsold unit, the critical ratio is 27.00 divided by 39.00, or 69.2 per cent, which puts the order 0.502 standard deviations above the forecast.
Only if the margin on a sale equals the loss on a leftover, because ordering the forecast implies a 50 per cent critical ratio. In the illustrative case ordering 8,000 instead of 9,005 lowers expected profit from 188,572 to 184,883. When leftovers are expensive relative to the margin, the right order is below the forecast.
A higher salvage value reduces the overage cost, which raises the critical ratio and the order. Moving salvage from 0.00 to 12.00 takes the overage cost from 18.00 to 6.00, the ratio from 60.0 to 81.8 per cent and the optimal order from 8,507 to 9,817 units, with expected profit rising from 181,229 to 198,572.
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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