Lead Time, Reorder Points and Safety Stock

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When to reorder and how much extra inventory to keep: reorder points, service levels using the z-score, and safety stock when demand and lead times are uncertain.

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Introduction

In the lesson on economic order quantity, we looked at how much to buy at a time to minimize ordering and holding costs. But that model assumes materials are available the moment they are ordered. In practice, that never happens. Days, sometimes weeks, pass between the decision to buy and the point when the materials are ready for use.

That wait is called lead time, also known as replenishment time or delivery time. It covers every stage, not just shipping:

  • placing and approving the order (in public agencies, the entire procurement process);
  • the supplier picking or manufacturing the materials;
  • transport to the company;
  • receiving, checking and inspecting the materials until they are cleared for use.

Lead time brings uncertainty: the order has been placed, but we do not know exactly how much stock will be used before it arrives, or whether it will arrive on the promised day. This lesson answers two questions: when to order (the reorder point) and how much extra stock to hold to manage that uncertainty (safety stock).

Uncertainty

Many factors can change demand for a product unpredictably: weather, the economy, government policies, marketing campaigns (ours and our competitors'), fashion and people's incomes. Supply has its own surprises, too.

During lead time, two things may differ from expectations:

  • Demand. If it slows, the new batch arrives while stock is still on hand, tying up more capital and taking up more space. If it picks up, stock runs out before the batch arrives.
  • Lead time itself. If delivery comes early, stock rises above the planned level. If it is delayed (because of transport problems, materials rejected during inspection or a manufacturing failure at the supplier), the wait grows longer and stock may run out first.

Both kinds of surplus cost money, but both kinds of shortage usually cost much more: lost sales, halted production and unhappy customers. That is why companies hold additional stock to cushion against uncertainty: safety stock. It is a reserve to draw on while waiting for replenishment. But holding that reserve also costs money, so the challenge is deciding how much to keep.

Reorder point: when to order

When delivery takes time, the order must be placed before stock runs out: when the stock remaining is enough to cover lead time. That level is the reorder point (PP), also called the replenishment point:

PP = d × LT + ES

where d is average demand per unit of time (per day, for example), LT is lead time in the same unit, and ES is safety stock. The d × LT term is expected consumption before the order arrives; ES provides a buffer against the unexpected.

Example. Mr. Manoel sells 20 savory snacks a day. His supplier, Ms. Maria, takes 2 days to make and deliver them. He orders a new batch of 80 snacks whenever stock falls to 60. Expected consumption during lead time is 20 × 2 = 40 snacks; by ordering at 60, he holds ES = 60 − 40 = 20 snacks as safety stock, equivalent to one day's sales.

Stock falls to the reorder point, an order is placed, and the batch arrives one lead time later, when stock is at the safety-stock level
Stock falls to the reorder point, an order is placed, and the batch arrives one lead time later, when stock is at the safety-stock level

In the chart, safety stock is the "floor" that, on average, is never used: when everything goes as expected, the batch arrives just as stock reaches that level. It is drawn on only when demand rises or delivery is delayed. How this system works in detail, with the same quantity ordered whenever stock reaches PP, is covered in the lesson on continuous and periodic review.

A store sells an average of 50 units of a product per day, its supplier delivers in 6 days, and the store holds 60 units of safety stock. At what stock level should it place an order?

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Answer: 360 units — PP = 50 × 6 + 60 = 300 + 60 = 360. At 300, the reorder point would offer no protection: stock would run out in half the cycles when demand was above average.

Safety stock and service level

How much safety stock should a company hold? That depends on two things: the degree of uncertainty and the risk of a stockout the company is willing to accept.

Service level

Forecast errors often follow an approximately normal distribution: most periods fall close to the average, and few fall far from it. On that assumption, we choose a service level: the probability of avoiding a stockout during a replenishment cycle. Each service level corresponds to a number of standard deviations, the z factor, taken from the normal distribution table:

Service levelz (number of standard deviations)
50%0
60%0,25
80%0,84
85%1,04
90%1,28
95%1,65
97,5%1,96
98%2,05
99%2,33
99,9%3,09

(Some books round the z value for 98% to 2,06 and the value for 95% to 1,64; the differences are small. In a spreadsheet, the INV.NORMP.N function, or INV.NORMP in LibreOffice, gives the z value for any service level: INV.NORMP.N(0,95) = 1,645.)

Choosing 95% means accepting that demand during lead time will exceed available stock in 5% of replenishment cycles. This is not the same service-level measure used in the lesson on inventory policy, which counts requests fulfilled: here we measure the chance of getting through a cycle without a stockout. Note the 50% figure, too: with z = 0, meaning no safety stock, stock runs out in half of cycles because demand exceeds the average half the time.

Example. Average demand over a period is 1.200 units, with a standard deviation of 200. At a 95% service level (z = 1,65), the highest demand we want to cover is:

Maximum demand = Average demand + z × σ = 1.200 + 1,65 × 200 = 1.530 units

The 330 units above the average are safety stock:

ES = z × σ

Adjusting the standard deviation for lead time

We often know the standard deviation of daily demand, but delivery takes several days. If demand varies independently from day to day, the standard deviation of demand during lead time is:

σ(LT) = σd × √LT

and safety stock is:

ES = z × σd × √LT

where σd is the standard deviation of daily demand and LT is lead time in days (the units must match: if the standard deviation is weekly, LT must be in weeks). Why the square root rather than simple multiplication? Over several days, high- and low-demand days partly offset each other. Total variation grows, but more slowly than the number of days.

Worked examples

a) A supermarket sells an average of 514 cases a day, with a standard deviation of 73 cases. The supplier takes 1 day to deliver. For a 90% service level (z = 1,28):

ES = 1,28 × 73 × √1 = 93,4 → 94 cases

(Always round safety stock up: rounding down would put the service level slightly below the target.)

b) The supplier's distribution center has moved, increasing lead time from 1 to 9 days:

ES = 1,28 × 73 × √9 = 1,28 × 73 × 3 = 280,3 → 281 cases

Lead time increased ninefold, but safety stock only tripled (√9 = 3).

c) With lead time at 9 days, the company decides to raise its service level to 95% (z = 1,65):

ES = 1,65 × 73 × 3 = 361,4 → 362 cases

Safety stock in cases (9-day lead time) at each service level: the curve climbs sharply near 100%
Safety stock in cases (9-day lead time) at each service level: the curve climbs sharply near 100%

The chart illustrates the most important lesson here: each additional percentage point of service level requires more stock than the last. Raising it from 90% to 95% adds 81 cases; going from 95% to 99% adds another 149 (2,33 × 219 = 510,3 → 511); and 100% would require infinite stock. That is why no company guarantees 100%. Instead, companies commonly set higher levels for critical or higher-margin items and lower levels for others (the ABC and XYZ classifications covered in another lesson help guide that choice).

An item's daily demand has a standard deviation of 40 units, and lead time is 4 days. How much safety stock is needed for a 95% service level (z = 1,65)?

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Answer: 132 units — ES = 1,65 × 40 × √4 = 1,65 × 40 × 2 = 132. The result of 264 comes from multiplying by lead time (4) instead of its square root (2).

A company wants to raise an item's service level from 95% (z = 1,65) to 99% (z = 2,33), without changing demand or lead time. How much does safety stock increase?

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Answer: About 41% — ES is proportional to z, and 2,33 ÷ 1,65 ≈ 1,41. Four percentage points of service level require 41% more safety stock, showing how the cost rises near 100%.

When lead time varies too

So far, we have assumed the supplier always delivers on time. If lead time also varies, safety stock must cover both uncertainties. Each additional day of delay uses an average of d more units, so variation in delivery time is multiplied by daily demand. Assuming demand and lead time are independent, the combined formula is:

ES = z × √(σd² × LT + d² × σLT²)

where d is average daily demand, σd is the standard deviation of daily demand, LT is average lead time, and σLT is the standard deviation of lead time (in days). Consider two special cases:

  • if lead time is fixed (σLT = 0), the formula reduces to ES = z × σd × √LT;
  • if demand is constant (σd = 0), it reduces to ES = z × d × σLT.

Example. An item has average demand of 100 units a day, with a standard deviation of 20. Average lead time is 4 days, with a standard deviation of 1 day. For a 95% service level:

ES = 1,65 × √(20² × 4 + 100² × 1²) = 1,65 × √(1.600 + 10.000) = 1,65 × √11.600 = 1,65 × 107,7 = 177,7 → 178 units

Demand uncertainty alone would call for 1,65 × 20 × 2 = 66 units. Uncertainty about delivery time nearly triples the safety stock needed. In practice, that means a reliable supplier saves money: reducing variation in delivery times often frees up more stock than improving demand forecasts.

In the example above (d = 100, σd = 20, LT = 4 days, z = 1,65), the supplier now always delivers in exactly 4 days. What is the new safety-stock level?

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Answer: 66 units — with σLT = 0, only demand variation remains: 1,65 × 20 × √4 = 66. The supplier's consistent delivery reduced safety stock from 178 to 66 units.

Where the standard deviation comes from

To use the formulas, we need to estimate σ. There are two approaches:

  • Using historical demand: Calculate the standard deviation of daily (or weekly) sales over a representative period. In a spreadsheet, the DESVPAD.A function does this. This approach works well when demand has no strong trend or seasonality.
  • Using forecast errors: What matters most is how far actual demand differs from what was forecast. Calculate the standard deviation of the errors, or use the mean absolute deviation (MAD) covered in the demand forecasting lesson: for normally distributed errors, σ ≈ 1,25 × MAD. Better forecasts mean smaller errors and less safety stock.

Lead time must be measured, not taken from the supplier’s promise. Recording the date of each order and delivery reveals the actual lead time and how much it varies.

Exercises

Easy

1) Daily demand for sliced cheese at a supermarket is normally distributed, with a mean of 50 kg and a standard deviation of 10 kg. The supplier always takes 3 days to deliver. The supermarket wants a 95% service level. How much safety stock should it hold?

2) Item X has average demand of 500 units per day and a daily demand standard deviation of 20 units. Replenishment lead time is fixed at 4 days. What safety stock is needed for a 98% service level?

3) (Logistics exam, CEFET/RJ, 2018) Item Y has average demand of 1.500 units per day and a daily demand standard deviation of 30 units. Replenishment lead time is fixed at 4 days. What safety stock is needed for 95%?

4) Item Z has average demand of 5.200 units per day and a daily demand standard deviation of 50 units. Replenishment lead time is fixed at 4 days. What safety stock is needed for 90%?

5) A company has not calculated the standard deviation of demand, but it knows that the mean absolute deviation (MAD) of its daily forecast errors is 24 units. Lead time is 4 days. Estimate the daily standard deviation and the safety stock needed for 90%.

Hard

6) (Logistics exam, CEFET/RJ, 2018) A candy distributor has found that daily demand for peppermint candy is normally distributed, with a mean of 5.000 kg and a standard deviation of 350 kg. The manufacturer’s delivery time is also normally distributed, with a mean of 5 days and a standard deviation of 2 days. How much safety stock should the distributor hold for a 95% service level?

7) For items X, Y and Z in exercises 2 to 4, calculate the reorder point and compare the safety stock with demand during lead time.

8) A store operates 5 days a week. Demand for a product is 400 units per week, with a weekly standard deviation of 50 units. The supplier delivers in 10 business days. For a 95% service level, calculate the safety stock and reorder point.

9) A company sells an average of 200 units of an item per day, with a daily standard deviation of 30 units, and wants a 95% service level. It can buy from two suppliers at the same price: A always delivers in exactly 6 days; B delivers in an average of 4 days, but with a standard deviation of 1,5 days. Holding one unit in stock costs R$ 6 per year.
a) Calculate the safety stock and reorder point for each supplier.
b) Which supplier has the lower safety-stock holding cost, and by how much per year?

10) A company uses 18.000 units of an item per year and operates 300 days a year. Each order costs R$ 100, and holding one unit costs R$ 4 per year. The daily demand standard deviation is 15 units, and lead time is fixed at 4 days. For a 95% service level, calculate:
a) the economic order quantity (see the lesson on economic order quantity);
b) the safety stock;
c) the reorder point;
d) the average inventory, assuming it equals the safety stock plus half the order quantity;
e) the annual cost of holding the safety stock.

Answer Key

1) ES = z × σd × √LT = 1,65 × 10 × √3 = 1,65 × 10 × 1,732 = 28,6 → 29 kg (rounded up).

2) ES = 2,05 × 20 × √4 = 2,05 × 20 × 2 = 82 units (using z = 2,06, as some textbooks do, gives 82,4 → 83).

3) ES = 1,65 × 30 × √4 = 1,65 × 30 × 2 = 99 units.

4) ES = 1,28 × 50 × √4 = 1,28 × 50 × 2 = 128 units.

5) σ ≈ 1,25 × MAD = 1,25 × 24 = 30 units per day. ES = 1,28 × 30 × √4 = 76,8 → 77 units.

6) Because lead time also varies, use the combined formula:
ES = 1,65 × √(σd² × LT + d² × σLT²) = 1,65 × √(350² × 5 + 5.000² × 2²)
= 1,65 × √(612.500 + 100.000.000) = 1,65 × √100.612.500 = 1,65 × 10.030,6 = 16.550,5 → 16.551 kg.
Nearly all of the safety stock accounts for lead-time uncertainty: a 2-day standard deviation at demand of 5.000 kg per day has a much greater effect than daily demand variability of 350 kg.

7) PP = d × LT + ES.
X: 500 × 4 + 82 = 2.082 units (ES = 4% of lead-time demand of 2.000);
Y: 1.500 × 4 + 99 = 6.099 units (ES = 1,7% of 6.000);
Z: 5.200 × 4 + 128 = 20.928 units (ES = 0,6% of 20.800).
When demand is stable (the standard deviation is small relative to the mean), safety stock is small relative to lead-time demand.

8) The units must match: the standard deviation is weekly, so express lead time in weeks: 10 business days ÷ 5 = 2 weeks.
ES = 1,65 × 50 × √2 = 1,65 × 50 × 1,414 = 116,7 → 117 units.
PP = 400 × 2 + 117 = 917 units.
(Using LT = 10 with a weekly standard deviation gives 1,65 × 50 × 3,16 = 261, more than twice as much.)

9) a) Supplier A (fixed lead time): ES = 1,65 × 30 × √6 = 1,65 × 30 × 2,449 = 121,2 → 122; PP = 200 × 6 + 122 = 1.322.
Supplier B (variable lead time): ES = 1,65 × √(30² × 4 + 200² × 1,5²) = 1,65 × √(3.600 + 90.000) = 1,65 × 305,9 = 504,8 → 505; PP = 200 × 4 + 505 = 1.305.
b) Supplier A is cheaper: it requires 505 − 122 = 383 fewer units of safety stock, saving 383 × 6 = R$ 2.298 per year. The supplier with the shorter average lead time costs more because its delivery time is less predictable.

10) a) LEC = √(2 × 18.000 × 100 ÷ 4) = √900.000 ≈ 949 units.
b) ES = 1,65 × 15 × √4 = 49,5 → 50 units.
c) Daily demand: 18.000 ÷ 300 = 60. PP = 60 × 4 + 50 = 290 units.
d) EM = 949 ÷ 2 + 50 = 474,5 + 50 = 524,5 units.
e) 50 × 4 = R$ 200 per year.

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