The academic name is off-putting and the maths is usually shown with integrals. Neither is necessary. The model answers one question every buyer already faces: you order once, demand is uncertain, and both mistakes cost money.
Where the name comes from
A newsagent buys papers each morning. Unsold copies are worthless by evening; copies they run out of are sales walking out the door. They must commit before knowing demand, and they cannot restock mid-day. Formalised in the 1950s, the problem turns out to describe far more than newspapers: seasonal apparel, event catering, vaccine batches, and, the case that concerns us, the first order for a product with no sales history.
The two costs
Underage cost (Cu) is what one unit of unmet demand costs you. The textbook answer is the gross margin you did not earn. In distribution the honest answer is higher: fill rate is a top-three criterion when customers choose a supplier, and customers who hit a stock-out measurably shrink their follow-up orders (Conexiom, vendor research). A contractor who calls for a new item and hears "two weeks" has learned something about you, not just about that item.
Worth being exact about what this tool does with that, because the honest answer and the charged answer are not the same. It charges the lost gross margin and nothing else. The relationship cost is real, unmeasured, and yours to add: if a stock-out on this item would cost you more than the margin on it, raise the underage figure yourself and watch the quantity move. Every number on the result comes from inputs you can see, and this is one of them.
Overage cost (Co) is what one unsold unit costs. Not the purchase price, the purchase price minus whatever you eventually recover. That is the figure for a buy you will not repeat; for a catalogue item it is much smaller, and the section below says why. For a distributor that means the unit cost, plus roughly a quarter of its value per year in carrying charges while it sits (Industrial Supply Magazine, US trade press), minus a liquidation value that typically lands between ten and forty percent of cost.
The result, without calculus
Think about adding one unit to your order. It earns Cu if demand turns out high enough to sell it, and costs Co if it does not. If the probability of selling that marginal unit is p, its expected value is p × Cu − (1 − p) × Co. Keep adding units while that is positive; stop when it hits zero. Rearranged, you stop when
p = Cu / (Cu + Co)
That fraction is the critical ratio, and the rule it produces is this: order enough to cover demand up to its critical-ratio quantile. If the ratio is 0.7, order the quantity demand has a 70% chance of falling short of, so you plan to run out in only 30% of scenarios, and to have stock left in the other 70%. The optimum is a point in the demand distribution, not a point estimate of demand, and the two coincide only in the special case where both costs are equal.
A worked example
A valve costs €88, sells at €129, and unsold units clear at 45% of cost, the default this site uses, and the optimistic end of what dead stock actually fetches. Cu = €129 − €88 = €41. Co = €88 − €40 = €48. The critical ratio is 41.00 ÷ (41.00 + 48.40) = 0.46. So you cover demand only up to its 46th percentile, meaning you deliberately accept running out in 54% of scenarios, because leftovers hurt more than shortages here.
Now suppose you negotiate returns with the supplier at 80% of cost. Co falls to €18, the ratio jumps to 0.70, and the correct order rises sharply. The demand forecast never changed. This is the model's most practical lesson: the terms of the deal move the right quantity as much as the demand estimate does, and buyers spend far more time refining the forecast than the terms.
Where it stops being enough
The model assumes one order, one selling season, and a known demand distribution. For a first buy, the first two hold well, the honest planning horizon is lead time plus the time until your next order, because after that you can reorder. The third is the hard part: for a new item you do not have a demand distribution, and pretending otherwise is where most applications of the model go wrong.
The distribution has to come from somewhere defensible: comparable launches you have run before, corrected for the stock-outs that truncated their sales curves, see why your sales history understates demand. Simulating from those comparables gives you a distribution with an audit trail behind it, rather than a normal curve with a standard deviation somebody guessed.
When the leftover is not actually dead
Everything above assumes one selling window and no reorder, which is what makes it a single-period model. That fits a seasonal buy, a promotion, or an item you will not restock. It does not fit a catalogue item you replenish every month, and most of what a distributor buys is a catalogue item.
The difference is what happens to stock left at the end of the cover period. On a one-time buy it is dead and you liquidate it. On a continuing item it is not dead, it is next month's stock: it sells in the following weeks, and all your early order really cost you was the money tied up in it, plus the risk the item gets dropped before it clears. That is carrying charges over a few weeks rather than a write-down of half the unit cost, often ten times less. Use the single-period Co on a continuing item and you will systematically under-buy, and reject minimums that were fine.
The trade-off does not change: Cu is the same, and you still stop where one more unit stops paying for itself. The arithmetic does change. Once the cost of being long depends on how long the stock sits, Co is no longer one number, so there is no fixed ratio to take a quantile at and the tool searches for the quantity that maximises the expected outcome directly. The critical ratio still describes the answer it lands on, read at the overhang that answer produces, but it is a reading of the optimum rather than the formula that found it. The tool asks which kind of buy this is and says on the result which of the two it applied.
One more caveat worth having: the expected-profit curve is flat near its peak. Small errors in the critical ratio or the distribution barely matter; large errors in either direction matter enormously. Use the model to get into the right neighbourhood, not to defend a precise number.
Try it
The Newsvendor Calculator takes both costs and a demand estimate and returns the optimum with the full trade-off curve. If you would rather start from a product than from costs, the Initial Order Quantity Calculator applies the same maths to a launch, deriving the demand distribution for you.