Intermittent demand: Croston, SBA, TSB

Some items sell like this: 0, 0, 8, 0, 0, 0, 7, 0, 9, 0. Spare parts, slow-moving SKUs and specialist chemicals behave this way. Ordinary exponential smoothing struggles because it blends the zeros and the bursts into a small average that is wrong in every single period: too high when nothing sells, far too low when a burst arrives.

Croston’s method (1972) asks two separate questions:

1. When demand happens, how big is it? (smoothed demand size, $z$)
2. How many periods usually pass between demands? (smoothed interval, $p$)

The forecast per period is the size divided by the gap: $z / p$. If bursts average 8 units and come every 3 periods, the forecast is about 2.7 units a period. That isn’t a prediction that 2.7 units sell each week; it is the average rate, which is what replenishment needs.

Two improvements matter in interviews:

• SBA (Syntetos-Boylan approximation) fixes a known upward bias in Croston by multiplying the forecast by $(1 - \alpha/2)$.
• TSB (Teunter-Syntetos-Babai) tracks the probability that demand occurs and updates it every period, zeros included. Croston only updates when a sale happens, so when a part stops selling it keeps forecasting the old rate for ever. TSB’s forecast fades as the part dies out, so you stop buying stock nobody will need.

Accuracy also needs different measures here. MAPE breaks on zero-demand periods, so use scaled errors or service-based measures instead.

The update rules

Croston updates only in periods with demand. When a demand of size $y$ arrives $q$ periods after the previous one:

In periods with zero demand, nothing changes.

SBA multiplies the result by $(1 - \alpha/2)$ to remove Croston’s upward bias.

TSB replaces the interval with a demand probability $d$, updated every period with its own constant $\beta$: towards 1 when demand occurs and towards 0 when it doesn’t. The size $z$ updates only when demand occurs, and the forecast is $d \times z$.

That last comparison is why TSB matters for parts at risk of becoming obsolete.

From forecast to stock

An intermittent forecast is a rate, but stocking decisions need the distribution of demand over the lead time. Two common approaches:

• A compound distribution: assume demand occurrences follow a Poisson process and sizes follow another distribution, then compute the stock that meets the service target.
• Bootstrapping: resample past periods, zeros included, to simulate many possible lead-time totals, and read off the quantile you need.

The policy, often a periodic review with an order-up-to level, is then set from that distribution, not from a normal curve built around the average rate.

Where Croston’s bias comes from

Croston divides one smoothed estimate by another. Even if each is unbiased, the average of a ratio is not the ratio of the averages: random variation in the interval estimate pushes the expected forecast above the true rate. Syntetos and Boylan (2001) showed this bias analytically, and SBA’s factor of $(1 - \alpha/2)$ is a simple approximate correction. The bias grows with $\alpha$, one reason small smoothing constants (0.05 to 0.2) are usual for intermittent items.

Why the usual error measures mislead

• MAPE divides by actual demand, so zero periods make it undefined.
• MAE has a hidden flaw: it is minimized by the median. For an item that sells nothing in most periods the median is zero, so a forecast of zero every period can score the best MAE while being useless for stocking.
• Better choices are errors scaled by a benchmark, such as MASE or a scaled squared error, which reward getting the average rate right, or measures of the inventory outcome itself: the service achieved and the stock needed to achieve it.

Simpler and smarter tricks

• Aggregate over time: forecasting quarterly totals and splitting them down can beat forecasting weekly zeros directly (the ADIDA idea).
• Aggregate over items: forecast a family of similar parts and allocate by share.
• Use the installed base: demand for spare parts depends on how many machines are in service, how old they are and when they are maintained. Forecasting from the installed base usually beats any pure time-series method, especially near end of life.
• Question stocking altogether: for very lumpy items, it may be better to hold no local stock, keep one central buffer, or agree lead times with customers.

The bigger picture

Intermittent items are typically a large share of SKUs, a small share of revenue and a large share of slow and obsolete stock. Getting them right is less about a clever forecast than about the combination: a sensible rate, a realistic distribution, a service target that reflects how critical the item is, and a review that catches decline early.