Lead time and demand are independent.

The formula, in words

• The reorder point covers one interval and no other. Not a year, not a cycle, not a month. It covers the stretch between deciding to order and the order being usable, because that is the only stretch in which nothing you decide can change what arrives. Under continuous review that stretch is the lead time. If you look at the stock once a week instead, add the week: an order missed on Monday cannot become a receipt any sooner for having been noticed on Tuesday.
  $$P=L+R$$
  where $P$ is the protection period, $L$ the lead time, and $R$ the review interval, in days.
• Two sources of uncertainty, and one of them is usually the larger. Demand varies over the interval and the interval itself varies. Combine them as the demand variance accumulated over the average lead time plus the lead time variance multiplied by the square of the daily rate. The second term is the one people leave out, and on the numbers shipped here it is 98% of the total.
  $$\sigma_P=\sqrt{P\sigma_D^2+\mu_D^2\sigma_L^2}$$
  where $\sigma_P$ is the protection-demand spread in units, $\sigma_D$ the daily demand spread, $\mu_D$ the daily demand mean, and $\sigma_L$ the lead-time spread in days.
• The buffer is that figure times a factor you choose, and the factor is the whole decision. Everything else on this page is arithmetic. The factor is a judgement about consequences wearing a decimal point.
  $$B=k\sigma_P$$
  where $B$ is the safety stock in units and $k$ the safety factor.
• Naming the number is not the same as defining it. A cycle service level is read straight off the normal distribution: the factor is the percentile. A fill rate cannot be, because it depends on how much demand a cycle carries, so a bigger lot meets the same fill rate with less buffer, and the same spoken percentage lands somewhere else entirely.
  $$\alpha=\Phi(k),\qquad\beta=1-\frac{\sigma_P G(k)}{Q}$$$$\begin{gathered}\alpha=\Phi(k), \\ \beta=1-\frac{\sigma_P G(k)}{Q}\end{gathered}$$
  where $\alpha$ is cycle service, $\Phi$ the standard normal distribution function, $\beta$ the fill rate, $G$ the standard normal loss function, and $Q$ the lot size in units.
• Round the buffer up, never down. Rounding down leaves the cover below the number you just named, and the whole point of naming it was to decide how much cover you wanted. The reorder point shown is the protection demand plus that rounded buffer, so the two figures on the page always add up. The annual cost of carrying it is worked out from the unrounded buffer, which is why it agrees to the cent with the service level lesson running the same inputs.
  $$r=\mu_D P+\lceil B\rceil,\qquad H=Bch$$$$\begin{gathered}r=\mu_D P+\lceil B\rceil, \\ H=Bch\end{gathered}$$
  where $r$ is the reorder point in units, $H$ the annual holding cost in dollars, $c$ the unit cost in dollars, and $h$ the annual holding rate.
• The trigger gets crossed, not touched. Every formula on this page assumes you order at the instant the position reaches the reorder point. Real transactions arrive in lumps and land below it. The distance below is the undershoot, and it comes straight off your cover, so a policy sized for one number behaves like a smaller one.
  $$r_e=r-u$$
  where $r_e$ is effective cover and $u$ the undershoot in units.

What this does not model

The lot size, which is typed rather than worked out, so nothing here balances ordering cost against carrying cost. Any correlation between demand and lead time, which is exactly backwards from reality: a congested supplier is slow in the same weeks everybody wants more. Orders crossing, which the variance formula quietly assumes away while also assuming the lead times are independent, and those two assumptions are not comfortable together once more than one order is open. The difference between demand variation and forecast error, which matters more than anything else on this list: the buffer should absorb what your forecast gets wrong, and this page sizes it against how much demand moves, which double counts whatever the forecast already predicts and buys more stock than the numbers deserve. Stock committed but not yet issued. Lost sales as opposed to backorders, which changes both the accounting and the arithmetic. Partial deliveries, supplier allocation in a shortage, minimum order quantities, and the shipping quantity a supplier will actually accept. Review effort, which is real and is charged nowhere here.

A worked example

The inputs. 100 units a day, demand described as normal, a 30 day lead time whose spread is 12 days, a lot of 2,000 units, a unit cost of $40 and a holding rate of 22% a year. The number named is 95%, and it is named as a cycle service level: the chance a replenishment cycle ends without running out.

The safety factor, the buffer and the reorder point. At 95% as a cycle service level the factor is the 95th percentile of the standard normal, 1.64485. The buffer is $1.64485 \times 605.60 = 996.12$ units, rounded up to 997, and the reorder point is $3{,}000 + 997 = 3{,}997$ units. Holding that buffer costs $996.12 \times 40 \times 0.22 = 8{,}765.87$ dollars a year, worked out from the unrounded figure, which is the same figure the service level lesson prints for the same inputs, to the cent.

So the two readings of one sentence. 3,997 units against 3,372. 625 units of buffer, and $5,499 a year, decided by which of two definitions the person who wrote the policy had in mind.

And each delivers the other, which is the check. At the cycle service reorder point, $G(1.64485) = 0.020893$, so the expected shortfall is $605.60 \times 0.020893 = 12.65$ units a cycle. Over $36{,}500 \div 2{,}000 = 18.25$ cycles a year that is 230.9 units, out of 36,500, so 95% of cycles ending clean is 99.4% of units shipped from stock. At the fill rate reorder point the shortfall is $605.60 \times 0.165127 = 100.00$ units a cycle by construction, 1,825 a year, which is exactly 5% of 36,500, while the chance of a clean cycle is $\Phi(0.61304) = 73.0\%$. 95% of units shipped from stock is 73% of cycles ending clean. Both are true. Neither is the other.

One thing worth reading twice in that table: 99% named as a cycle service level and 99.9% named as a fill rate are 4,409 and 4,415, six units apart. They are the same policy under two very different sounding promises.

The replay, transaction by transaction. Opening stock 4,700 units, nothing on order, nothing owed. The reorder point is 3,997, the lot is 2,000, the lead time is 30 days, and six issues total 6,000 units across 60 days, which is the 100 a day typed above.

Inventory position is on hand plus on order less backorders on every single line, and the whole table closes: $4{,}700 + 4{,}000 - 6{,}000 = 2{,}700$, with $2{,}700 + 2{,}000 = 4{,}700$ still in position.

The three moments the table exists for. On day 13 the position drops to 3,500 and an order goes in: the position jumps by 2,000 and on hand does not move by a unit. On day 37 on hand hits its floor at zero while net stock keeps going to −800, because on hand cannot record what you owe and net stock can. And on day 43 two thousand units physically arrive: on hand rises by 1,200 after 800 go straight out to the backorder, on order falls by 2,000, and inventory position does not move at all, because those units had been counted since day 13.

Where the two conventions split. On day 37, deducting the backorder gives a position of 3,200, and not deducting it gives 4,000. The reorder point is 3,997, which sits between them. So the broader convention orders on day 37 and the narrower one waits until day 43, when the receipt takes its reading to $1{,}200 + 2{,}000 = 3{,}200$ and it orders then. Three days of difference, from a definition. And there is nothing accidental about how close 4,000 sits to 3,997: with two lots of 2,000 already open and on hand at zero, the narrower reading cannot fall below 4,000 at all, so under that convention no further order is possible until something arrives.

What comparing against the shelf would have done. Run the identical ledger with one line changed, so the reorder point is tested against on hand instead of inventory position. It orders on days 13, 21, 29, 37, 43, 46 and 51: seven orders and 14,000 units against three orders and 6,000 units, and it ends day 60 holding 4,700 on hand with 8,000 more on the water. Every order after the first was placed because the previous one had not arrived yet.

Undershoot, and what it does to the number you named. The three placements happened at positions of 3,500, 3,800 and 3,200 against a trigger of 3,997, so the trigger was undershot by 497, 197 and 797 units, an average of 497 units. Cover is measured from where you actually order, so the effective cover is $3{,}997 - 497 = 3{,}500$, which is 500 above the protection demand, a factor of $500 \div 605.60 = 0.826$, and a cycle service level of 79.5% rather than the 95% named. Two honest caveats, both of which push the same way. That figure treats the undershoot as a fixed shift when it is itself a spread, and allowing for its spread would take the number lower. And one of the six issues, 2,600 units, is larger than the 2,000 unit lot, which is precisely the case where a standard undershoot correction is least reliable and also precisely the case where ignoring undershoot is most wrong.

Check it yourself

Move the lead-time spread from 12 days to 0, holding the rest still. The buffer falls from 997 units to 136 and the cost of holding it falls from $8,766 a year to $1,189. That gap, $7,577 a year, is the most a perfectly steady 30 day lead time could be worth to you in carrying cost, and it is the figure to take into a conversation about a more reliable route or a more reliable supplier, because nothing you pay above it can be justified by this saving alone.

The buffer goes to −410 units and the reorder point to 2,590, below the 3,000 the lead time consumes. That is not a fault: at that lot size the order itself covers 95% of units, and the arithmetic is saying that more than half the cycles may dip into backorder while 95 of every 100 units still ship from stock.

Sources

Silver, Pyke and Thomas, Inventory and Production Management in Supply Chains, 4th edition. The source for the four state measures, the protection interval, the two term lead time demand variance, cycle service against fill rate, undershoot and the fixed lot policy.

Nahmias and Olsen, Production and Operations Analysis, 7th edition. The source for separating the lot from the trigger, the narrower inventory position, expected lead time demand, negative safety stock, the two service definitions and the implied shortage price.

Cachon and Terwiesch, Matching Supply with Demand, 4th edition. An independent statement of the state accounting, protection interval, service definitions and implied shortage price, and of why a waiting promise is incomplete until its measured wait is named.

Chopra, Supply Chain Management, 7th global edition, chapter 12. The source for item fill rate against order fill rate and why a complete order is a harder promise than a filled line.

None of the material consulted writes the reorder point or safety stock as an algebraic expression. The algebra on this page is standard and no formula is attributed to an author.

Equation trace

1. Calculate protection period demand and its spread.
2. Translate the named service definition into its own safety factor.
3. Add protection demand and the shown safety stock, then round to the displayed reorder point.
4. Replay each issue, order placement, and receipt in date order.
5. Compare the trigger with inventory position, not on hand stock alone.

Source anchors

• Inventory and Production Management in Supply Chains. Provides published inventory control context for reorder points and service measures.
• Foundations of Inventory Management. Provides published context for continuous review policies and inventory state definitions.

Calculation version

1.0