The newsvendor model

Some decisions get one shot. A café bakes croissants each morning, and whatever is left at night is nearly worthless. A clothing brand orders a winter collection once. A sweet shop stocks up for Diwali. You must choose a quantity before you know the demand, and you can be wrong in two directions:

• Too many: you paid for something you can’t sell. That is the overage cost, $C_o$ = cost − salvage value.
• Too few: a customer wanted it and you had none. That is the underage cost, $C_u$ = price − cost, the margin you missed.

For the café, a croissant sells for ₹100, costs ₹40 to make and fetches ₹10 at night from a juice stall. So $C_o$ = ₹30 and $C_u$ = ₹60: running short hurts twice as much as over-baking.

The model says to stock up to the critical ratio:

That means: bake enough to cover demand on about two days out of three. If daily demand averages 80 with a standard deviation of 20, the 67th percentile is about $80 + 0.43 \times 20 \approx 89$. So the café should bake around 89 croissants, not the 80 it sells on an average day.

The marginal argument

Think about the last croissant you might bake. If demand turns out higher than your quantity $q$, that croissant earns its margin $C_u$. If demand is lower, it is wasted and costs $C_o$. Bake it if the expected gain beats the expected loss:

Keep adding croissants until the two sides balance, which happens where:

So the critical ratio is the probability of not running out that you should aim for. For normal demand, $q^{*} = \mu + z\,\sigma$, with $z$ read from the normal table at the critical ratio.

When demand isn’t normal

With real data you rarely need the normal curve. Sort past demand (or the scenarios of a probabilistic forecast) and pick the smallest quantity whose cumulative share reaches the critical ratio. With 30 days of history and a ratio of 0.67, sort the days from lowest to highest demand and stock the 20th value. For slow sellers, a Poisson or empirical distribution is more honest than a normal one.

Getting the costs right

• Underage $C_u$: the margin lost per unit short, plus any goodwill or penalty. If customers switch to another of your products, the true loss is smaller.
• Overage $C_o$: what you paid minus whatever you recover (salvage, markdown price or return credit), plus disposal cost.
• Use marginal costs only. Rent and salaries are paid whatever you bake.

Sales are not demand

When the café sells out at 2 p.m., the till records 89 croissants, not the 110 people who wanted one. Sales data is censored at the stock level. Plan from raw sales and you underestimate demand, bake less, sell out earlier, record even lower “demand” and spiral down. The fixes: record stockout times, estimate lost demand from the sales rate before the stockout, or use statistical methods built for censored data.

People order too close to the average

In experiments, people facing newsvendor problems consistently order between the average demand and the optimal quantity: too little when the margin is high, too much when it is low. This pull-to-centre effect (Schweitzer and Cachon, 2000) persists even with experience, which is a strong argument for computing the quantity instead of trusting gut feel.

Contracts: who carries the leftover risk?

In a supply chain, the retailer orders and carries the leftovers, while the manufacturer earns on every unit ordered. The retailer’s critical ratio uses its own margin, which is thinner than the chain’s total margin, so it orders less than would be best for the chain as a whole. This is double marginalization. Contracts fix it by sharing the risk:

• Buy-back: the manufacturer takes back unsold units at a partial refund, lowering the retailer’s $C_o$. Common in publishing.
• Revenue sharing: the retailer pays a low wholesale price and shares its revenue. Video rental chains used this with film studios in the late 1990s.
• Options and quantity flexibility: the retailer reserves capacity and adjusts the final quantity later.

Each one raises the retailer’s critical ratio towards the chain’s, so more is stocked and total profit grows.

Where else it appears

The same logic sets airline overbooking (an empty seat is perishable), hotel room blocks, fashion and seasonal buys, capacity reservations, and the “last-time buy” of spare parts before a component is discontinued. It even sets service targets: in a periodic-review system with backorder cost $b$ and holding cost $h$ per period, the best cycle service level is $b/(b+h)$.