Chapter 2 of 4

Safety Stock and the Newsvendor

3 sections · ~6 min read · 3 practice questions · part of Demand & Inventory Planning

Learning objectives

  • Compute safety stock from demand variability, lead time, and target service level
  • Use the newsvendor model to set a one-time order quantity under uncertainty
  • Explain how service level choice, lead time variability, and unit margin trade off against each other

Safety stock fundamentals

Safety stock is inventory held to absorb the random difference between expected and actual demand during the lead time. Under a normal approximation, Safety Stock = k x sigma_LT, where sigma_LT is the standard deviation of demand over the lead time and k is a safety factor tied to the desired cycle service level (the probability of not stocking out during a replenishment cycle). For independent weekly demand, sigma_LT = sigma_d x sqrt(L) when lead time L is in the same units as demand variance. When both demand and lead time are variable, the two sources of variance add: sigma_LT^2 = L x sigma_d^2 + d^2 x sigma_L^2, where d is mean demand per period and sigma_L is the standard deviation of the lead time itself, expressed in the same periods. Note the asymmetry: the demand term grows with L, so its standard deviation grows only with sqrt(L), while the lead-time term grows with the square of mean demand. Compare the two variance terms to see whether demand error or lead-time variation contributes more. This calculation assumes independent demand periods and lead time; check those assumptions before using it to set a buffer.

Service level types

Cycle service level (CSL) is the fraction of replenishment cycles that do not stock out before the next order arrives. Fill rate is the fraction of unit demand that is satisfied from stock, which measures units rather than cycles and therefore cannot be read as the same service probability. Ready rate is the fraction of orders shipped complete on first attempt. The three metrics answer different questions and lead to different safety stock levels for the same target. Raising CSL from 95 percent to 99 percent lifts k from 1.65 to 2.33, and because safety stock scales linearly with k, safety stock and its carrying cost rise by about 41 percent. Going further, from 99 percent to 99.9 percent, lifts k from 2.33 to 3.09, another 33 percent, for a service gain of less than one percentage point. That is the shape of the curve: each additional increment of service costs more than the last, because the normal distribution's tail thins out faster than k grows. The lesson is to buy the cheapest unit of service improvement, which is often fill rate on a handful of critical items rather than a uniform CSL target across the whole catalogue.

Newsvendor for one-shot decisions

When the firm must commit to a single order before knowing demand (a seasonal item, a one-off purchase, a perishables order), the newsvendor model gives the right quantity. Compute the critical ratio = (Cu) / (Cu + Co), where Cu is the per-unit cost of under-stocking (lost margin, lost goodwill, expedite) and Co is the per-unit cost of over-stocking (holding, markdown, disposal). Order the quantity Q such that P(Demand <= Q) = critical ratio. If Cu = $40, Co = $10, critical ratio = 0.80, so order up to the 80th percentile of the demand distribution. The model is simple, transparent, and works well for any demand distribution for which you can estimate percentiles.

Worked example

Problem

Mean weekly demand = 200 units with weekly sigma = 40 units. Lead time = 3 weeks and is reliable (no lead-time variance). Target cycle service level = 97.5 percent. The firm orders in lots of 600 units. Compute (a) sigma over lead time, (b) safety stock, (c) reorder point, and (d) implied fill rate assuming normally distributed lead-time demand.

Step by step

  1. k for 97.5 percent CSL: z = 1.96 (approximately).
  2. sigma_LT = sigma_d x sqrt(L) = 40 x sqrt(3) = 40 x 1.7321 = 69.28 units.
  3. Safety stock = k x sigma_LT = 1.96 x 69.28 = 135.79 units.
  4. Mean demand over lead time = 200 x 3 = 600 units. Reorder point = 600 + 135.79 = 735.79 units, round to 736 units.
  5. Fill rate needs the expected shortage per cycle: ESC = sigma_LT x L(z), where L(z) is the standard normal unit loss function, L(z) = phi(z) - z x (1 - Phi(z)).
  6. For z = 1.96: phi(1.96) = 0.0584 and 1 - Phi(1.96) = 0.0250, so L(1.96) = 0.0584 - 1.96 x 0.0250 = 0.0584 - 0.0490 = 0.0094.
  7. ESC = 69.28 x 0.0094 = 0.651 units short per replenishment cycle.
  8. Fill rate = 1 - ESC / Q, where Q is the order quantity of 600 units. Fill rate = 1 - 0.651 / 600 = 1 - 0.00109 = 99.89 percent.

Answer. sigma over lead time = 69.28 units, safety stock = 135.79 units, reorder point = 736 units, implied fill rate about 99.89 percent. Fill rate sits well above the 97.5 percent cycle service level because a stockout event rarely means losing an entire cycle's demand: it usually means missing a few units at the very end of the cycle. Note that fill rate depends on the order quantity, while cycle service level does not, so a firm that halves its lot size holds the same safety stock but delivers a lower fill rate.

Practice

Work each question before opening the solution.

  1. Lead time variance grows because a key lane is unreliable. If the LT standard deviation becomes 1.5 weeks while other parameters are unchanged, by how much does safety stock change?

    Show solution for question 1

    sigma_LT^2 = L x sigma_d^2 + d^2 x sigma_L^2 = 3 x 40^2 + 200^2 x 1.5^2 = 4,800 + 90,000 = 94,800. sigma_LT = 307.90 units. Safety stock at k = 1.96 is 1.96 x 307.90 = 603.48 units, about 4.4 times the original 135.79 units. Note that the lead-time term ($90,000) is nearly nineteen times the demand term ($4,800), so essentially all of the increase comes from lane unreliability, not from demand noise. Stabilizing the lane is worth far more here than improving the forecast.

  2. A perishable item has selling price $12, unit cost $5, salvage $1 if unsold. Estimate the critical ratio ignoring goodwill costs and state which percentile of the demand distribution to order up to.

    Show solution for question 2

    Cu = lost margin = 12 - 5 = $7. Co = over-stock cost = 5 - 1 = $4. Critical ratio = 7 / (7 + 4) = 7 / 11 = 0.6364. Order up to the 63.6 percentile of the demand distribution.

  3. Compute the percentage increase in safety stock when a planner moves a SKU from 95 percent CSL to 99 percent, and then from 99 percent to 99.9 percent. What does the comparison say about where to spend a service budget?

    Show solution for question 3

    Safety stock is k x sigma_LT, so it scales with k alone. From 95 to 99 percent, k goes 1.65 to 2.33, an increase of 0.68 / 1.65 = 41 percent for 4 points of service, or about 10 percent of stock per point. From 99 to 99.9 percent, k goes 2.33 to 3.09, an increase of 33 percent for 0.9 points, or about 37 percent of stock per point. Service gets nearly four times more expensive per point in that upper range, so a fixed service budget buys far more by lifting several mid-range items from 95 to 99 than by pushing one item from 99 to 99.9.