Newsvendor model: order once, and the average is the wrong number

The formula, in words

• Two prices, and neither of them is the unit price. What matters is what changes when you order one more: it either sits there afterwards, costing you what you paid net of whatever you recover, or it is not there when it is wanted, costing you what that absence costs. A cost you pay whichever quantity you choose does not belong in either figure, because it cannot move the choice.
• A quantity's bill is two expectations added together. For each row of your table, work out how many units would be left over and how many would be missing, weight each by the chance of that row, price the two at your two prices, and add. Revenue is nowhere in it. This compares quantities against each other and deliberately does not rebuild an income statement.
• Only the rows can win. Between two adjacent rows the bill changes by the same amount for every unit, so it is a straight line with no bend inside the gap. A straight line has its lowest point at one end. The calculation therefore lands on rows, and nothing is rounded away.
• The step and the ratio are the same fact. Moving up one row costs you the leftover price on every outcome at or below where you are, and saves you the shortfall price on every outcome above it. Setting those equal gives one number, the shortfall price as a share of the two together, and the step turns from saving to costing exactly where the cumulative chance passes it. The step chart is the staircase chart, measured from that number instead of from zero.
• It lands on a tread, so it lands in jumps. The ratio is a smooth number and the staircase is not. Two quite different shortfall prices often land on the same row, and a small change can move the landing by a whole row. Both behaviours are the arithmetic being honest about a table with seven rows in it.

A worked example

The table as shipped. Seven rows 100 units apart, from 200 to 800, with the chances typed against them.

The chances come to 100%, so no normalising is needed on the shipped table. The average demand is $10 + 30 + 80 + 125 + 120 + 84 + 64 = 513$ units, and 513 is not one of the seven outcomes. Demand will be 200 or 300 or 400 or 500 or 600 or 700 or 800. It will never be 513.

The two prices. A unit costs $180 and a leftover recovers $60, so a leftover costs $180 - 60 = 120$ dollars. A unit short is expedited and waited for, at $480. The ratio is $480 \div (480 + 120) = 0.80$.

The staircase, all seven rows.

Where the ratio lands, and the tie it produces. The first cumulative chance at or above 0.80 is the one against 600 units, and it is 0.80 exactly. That equality is why 600 and 700 cost the same: the step between them is zero. On these numbers the arithmetic has no preference between two quantities 100 units apart, and the difference between them is 80 more units of stock on average against 20 fewer units of shortfall. That is not a defect of the model. It is the model finishing its work and handing the decision back.

What the average costs. The row nearest the average demand is 500 units, and it costs $39,240 against $27,240 at the two rows the ratio reaches. That is $12,000 more, or 44% more, for picking the number a forecast hands you. The forecast itself, read as a certainty, predicts a bill of nothing at all: buy 513, sell 513. No quantity on the table costs nothing, and the lowest bill available is $27,240.

How far the prices can move before the row does. With a leftover at $120, the row the ratio lands on stays at 600 units for every shortfall price above $120 \times 0.60 \div 0.40 = 180$ dollars and up to $120 \times 0.80 \div 0.20 = 480$ dollars. A shortfall price anywhere between $181 and $480 lands on the same row, a range of more than two and a half times, which is a useful thing to know before a meeting spends an hour on whether it is $300 or $400.

Service, which is the output rather than the input. The chance of covering demand entirely is the cumulative chance itself: 35% at 400 units, 60% at 500, 80% at 600. The share of demand met from stock is one minus the expected shortfall over the average: $1 - \frac{133}{513} = 74.1\%$ at 400, $1 - \frac{68}{513} = 86.7\%$ at 500, $1 - \frac{28}{513} = 94.5\%$ at 600. Nobody typed a service level and two of them came out anyway, which is the sentence that connects this page to both of its neighbours.

The same average, three shapes.

Same average, same two prices, three different bills and two different rows. Under Tighter the cumulative runs 2, 8, 28, 66, 87, 96, 100, so 0.80 still lands on 600 and the bill there is $120 \times 104 + 480 \times 17 = 12{,}480 + 8{,}160 = 20{,}640$ dollars: $6,600 less for knowing more, with nothing else changed. Under A longer tail the cumulative runs 5, 15, 37, 63, 79, 88, 100, so 0.80 clears 600 by a single point and lands on 700, where the bill is $120 \times 199 + 480 \times 12 = 23{,}880 + 5{,}760 = 29{,}640$ dollars.

What this does not model

Anything that recovers value from a leftover other than at the price typed, including a second season, a transfer to another site or a warranty pool. A shortfall that costs differently depending on how short you are or how long it lasts. Substitution, where a customer takes something else and the shortfall costs less than typed, and its opposite, where a shortfall loses the account. Any chance to react partway through with a second, smaller commitment, which changes the answer and generally lowers the first order. Quantity breaks, minimum order quantities, pallet and container rounding, and the supplier's own lot size. Where the seven chances came from, which is the input carrying the most weight and the least evidence: a short history misses the tail, a trend makes old periods incomparable, and a judgement folded into a rounded forecast cannot be argued with. And the largest omission of all: a cost taken from an allocation rather than from cash moves the arithmetic for reasons that have nothing to do with the decision.

Check it yourself

The average demand is 513 units in all three tables, the two prices never move, and the row the ratio lands on goes 600, 600, 700 while the bill at that row goes $27,240, $20,640, $29,640. Nothing about the average changed and the answer did. Units sold plus units left over are $485 + 115$, and units sold plus units short are $485 + 28$. At 600 units the table says 485 units sold, 115 left over and 28 short, and 485 plus 115 is the 600 you ordered while 485 plus 28 is the 513 you expected to want.

Basis

Each bill is the average across the seven outcomes in your table, weighted by the chances you typed.

Assumptions

One order. No reordering inside the window. Leftovers cost what you typed, shortfalls cost what you typed.

Model version

1.0