Safety stock

Why is a shop that never once runs out of an item almost certainly ordering too much of it?

A shopkeeper says, with some pride, that in four years no customer has ever been told an item was out. That sounds like the definition of running a shop well, and yet an inventory planner would wince, because the sentence is not really a claim about service — it is a claim about a tail, and tails are expensive. So what is being bought with that record, and why is it suspicious rather than admirable?

Start with why any stock is held beyond what sells. Replenishment is not instant: between placing an order and receiving it — the lead time — the shelf is on its own, and whatever demand arrives must be met from what was already there. So the only question that matters is how much demand over the lead time could exceed what you expected.

Split the shelf in two. One part covers expected lead-time demand — cycle stock, and not optional. The other covers the amount by which reality might overshoot, and unlike the first it is a choice. If lead-time demand has an average and a spread, you order when the shelf falls to the average plus some number of spreads, and that multiplier is the whole decision. Pick it from the fraction of cycles you are willing to get through without running dry and the normal distribution hands you the number: about 1.28 spreads for 90 per cent of cycles, 1.64 for 95, 2.33 for 99, 3.09 for 99.9.

Look at what those numbers do. Going from 95 per cent to 99 costs 2.33 over 1.64, about 41 per cent more stock, to buy four percentage points; going from 99 to 99.9 costs a further third of that larger pile for nine-tenths of one point. The tail thins faster than the buffer grows, so each additional nine is bought at a worse price than the last.

Now put the shopkeeper's boast through that arithmetic, because "never once" is a statement about many draws, not one. A weekly reorder means fifty-two independent chances a year to be caught short. For even a coin-flip chance of getting through a single year untouched, each cycle must survive with probability about 98.7 per cent — a multiplier of 2.22, some 35 per cent above a 95 per cent policy. A genuinely clean four-year record puts the multiplier near 3 and above. That is the arithmetic behind the wince.

Is the target really the cause, though? Test it the other way. Suppose the shop sells to one contracted canteen that orders the same quantity every Tuesday. The spread is near zero, so a perfect record costs nothing at all, and no amount of it is evidence of over-ordering. Raise the target instead on a genuinely variable item and the excess appears immediately with no change whatever in forecasting skill. So neither variability nor the target alone is the culprit: the buffer is their product, and better forecasting is the honest way to shrink one of them.

Which exposes the sharper point. The spread has to be estimated, and a buffer's edge is only ever observed by crossing it. A shop that never stocks out has censored its own data: it cannot tell a well-sized buffer from a wildly excessive one, because the only cheap instrument for locating that edge is the occasional mild stockout it has spent money to eliminate.

Three refinements hold this together.

First, "ordering too much" does not mean the whole shelf is excessive — only the tail portion of it. Cycle stock is dictated by the order frequency; safety stock is the discretionary part, and it is the only part the service target moves.

Second, "service level" hides two measurements shops routinely conflate: cycle service level is the fraction of replenishment cycles with no stockout, while fill rate is the fraction of units demanded supplied on time. The same policy scores very differently on the two, and the multipliers above belong to the first.

Third, the honest limits. The normal assumption is reasonable for a fast-moving item and poor for a slow one — three units a month is lumpy, not bell-shaped, and needs a different method. Lead-time variability often matters more than demand variability, and when both vary the buffer must cover the combination. And for a genuine few — an intensive-care drug, the part grounding an aircraft — a stockout is a catastrophe rather than a lost sale, and something near a perfect record really is right. The claim here is narrower than it sounds: for ordinary goods, a perfect record is a purchase and deserves to be priced like one.

This is the same calculation an airline runs when overbooking, in the opposite direction: both size a buffer against the spread of a total, but for a perishable seat the asymmetry points the other way, so the buffer is set below capacity. And a shop driving its buffers toward zero is choosing the just-in-time end of this same curve, with the exposure that implies.

• How intermittent, lumpy demand is forecast when the normal assumption plainly fails.
• Why one central buffer for several shops needs less stock than the same shops holding their own.

Every term the collection defines is gathered in the glossary.