Safety stock

You sell 1,000 units a month, give or take 250, and the supplier delivers in 21 days, give or take 5. Ask for 95% and most calculators give 436 units of safety stock. That 95% is the cycle service level: in 95 of 100 order cycles you never run out.

A customer who hears "95% service" usually means something else: 95 of every 100 units they order arrive on time. That is the fill rate, and it is much cheaper. A cycle that runs out usually runs out in its last days and misses a few units, not the whole order. Ordering 1,008 at a time, a 98% fill rate needs 277 units, which would score only 85.2% as a cycle service level. Decide which one you promise before you size the buffer.

The other lesson is where the buffer comes from. Of the 436 units, 94 are there only because the lead time varies by ± 5 days. A more reliable supplier is worth that much stock.

Uncertainty over the lead time

Monthly demand d and its standard deviation are turned into days (a month is 365 ÷ 12 days; the spread scales with the square root of time, assuming days vary independently). Over the protection interval T, the lead time L, or L plus the days between checks when you order on a fixed day, the standard deviation of demand is σ = √(T × σ_d² + d² × σ_L²), with σ_L the spread of the lead time. The first term is uneven demand, the second the uncertain delivery date (Chopra, Reinhardt and Dada, 2004, after Silver and Peterson). The tool shows both. Textbooks write σ_L for the spread of lead-time demand; here σ_L is the spread of the lead time itself, and σ the spread of demand over it.

Lead time from a quote

Few buyers know the standard deviation of their supplier's lead time, but most know the range. Given the shortest a, the usual m and the longest b, the tool reads the quote as a triangular distribution: mean (a + m + b) ÷ 3, variance (a² + b² + m² − ab − am − bm) ÷ 18. Without a usual value it uses a uniform one: mean (a + b) ÷ 2, standard deviation (b − a) ÷ √12. For 14 / 21 / 35days that is 23.3 ± 4.4 days, or 24.5 ± 6.1 from the range alone. Both treat the range as the edge of every delivery. A delivery that came later than the "longest" means the spread is larger than this; widen the range, or measure the spread from past orders.

Cycle service level

Safety stock = z × σ, where z is the point of the normal distribution below which the target share falls: 1.282 for 90%, 1.645 for 95%, 2.326 for 99%. The reorder point is d × L + safety stock; on a fixed-day cycle, the order-up-to level is d × (L + R) + safety stock.

Fill rate

The units you expect to be short in one cycle are σ × G(k), where G is the standard normal loss function, G(k) = φ(k) − k × (1 − Φ(k)). The fill rate is 1 − σ × G(k) ÷ Q, with Q the order quantity (on a fixed-day cycle, the demand between checks). The tool solves for k and sets safety stock = k × σ. If k comes out below zero the order size alone already gives that fill rate, and the buffer is zero.

When not to trust it

Everything above assumes demand over the lead time follows a normal curve. That fails for slow movers and for very uneven demand, and the tool warns instead of pretending precision: when demand over the lead time varies by more than half its mean (a common rule of thumb), when fewer than about 10 units are sold over it (Axsäter, 2013), and when a pasted history has demand in fewer than one period of every 1.32 (the intermittent-demand cut-off of Syntetos, Boylan and Croston, 2005). Seasonal or trending demand needs a forecast first: the spread should be the forecast error, not the swing of the season. The fill-rate formula is the standard backorder approximation (Silver, Pyke and Peterson's G(k)); it slightly understates the true fill rate, so it errs on the side of more stock.

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What is the safety stock formula?

Safety stock = z × σ, where σ is the standard deviation of demand over the lead time and z the factor for the service level (1.645 for 95%). With both demand and lead time varying, σ = √(L × σ_d² + d² × σ_L²): L the lead time, d and σ_d the average and spread of daily demand, σ_L the spread of the lead time.

How do I calculate the reorder point?

Reorder point = average demand during the lead time + safety stock. With 1,000 units a month and a 21-day lead time that is 690 + 436 = 1,126 units for a 95% cycle service level.

What is the difference between service level and fill rate?

The cycle service level is the share of replenishment cycles with no stockout at all; the fill rate is the share of demand shipped from stock. With orders as large as in the example here, the same buffer scores much higher on fill rate (with very small orders it can be the other way round): 95% cycle service level needs 436 units of safety stock, a 98% fill rate only 277, which is a 85.2% cycle service level.

How do I estimate the lead time spread if I only know the shortest and longest delivery?

Read the range as a distribution. With a usual lead time as well, a triangular one: a quote of 14 / 21 / 35 days gives a mean of 23.3 days and a standard deviation of 4.4. With only the range, a uniform one: (longest − shortest) ÷ √12 = 6.1 days. Both assume no delivery falls outside the range. The tool does this when you pick "Shortest, usual, longest".

How do I find the standard deviation of demand?

Take at least 6, better 12 or more, past periods of demand and compute the sample standard deviation (STDEV.S in Excel). Paste them into the tool and it does this for you, in your browser. Weekly figures are converted to a month assuming the weeks vary independently.