Two measures both called service level, 1,709 units of safety stock apart, and the batch size that links them.
To convert a cycle service level into a fill rate, compute the expected units short per replenishment cycle, σ × G(z), where G is the standard normal loss function, and divide by the order quantity: fill rate = 1 − σG(z) ÷ Q. For part AF-7, a 98 per cent cycle service level is a fill rate of 99.38 per cent. Run the other way, a 98 per cent fill rate needs only a 94.39 per cent cycle service level and 1,709 fewer units of safety stock.
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The two measures answer different questions. Cycle service level is the share of replenishment cycles with no stock-out at all. Fill rate is the share of demand served from stock. A board that votes for "98 per cent service" without saying which has not yet decided how much stock to hold.
The case is Aldermere Filtration's part AF-7, from the companion files of Supply Chain Management.
| Input | Value |
|---|---|
| Annual demand (1,840 a week, 52 weeks) | 95,680 units |
| Standard deviation of demand during lead time, σ | 3,674.60 units |
| Order quantity, Q (economic order quantity) | 4,385 units |
| Orders a year | 21.82 |
| Cycle service level voted by the board | 98% |
| Purchase cost / holding rate | 28.60 EUR / 13.4% |
The order quantity is the economic order quantity on an order cost of 385 EUR and a holding cost of 3.83 EUR a unit a year. σ combines demand and lead-time variability; how it is built is shown in how to calculate safety stock when lead time varies.
G(z) = φ(z) − z × (1 − Φ(z))
Expected units short per cycle = σ × G(z)
Fill rate = 1 − σG(z) ÷ Q
=NORM.S.DIST(B1,FALSE)-B1*(1-NORM.S.DIST(B1,TRUE)) gives G(z).The reverse needs a search. For a 98 per cent fill rate the required G(z) is 0.02 × 4,385 ÷ 3,674.60 = 0.0239, which is reached at z = 1.5888, a cycle service level of 94.39 per cent. Goal Seek on the cell above does it in one step.
| Cycle service level | z | Safety stock | Fill rate | Units missing a year |
|---|---|---|---|---|
| 90.0% | 1.28 | 4,709 | 96.03% | 3,796 |
| 95.0% | 1.64 | 6,044 | 98.25% | 1,675 |
| 97.0% | 1.88 | 6,911 | 99.03% | 932 |
| 98.0% | 2.05 | 7,547 | 99.38% | 589 |
| 99.0% | 2.33 | 8,548 | 99.72% | 272 |
| 99.5% | 2.58 | 9,465 | 99.87% | 127 |
| 99.9% | 3.09 | 11,355 | 99.98% | 22 |
A 98 per cent cycle service level means a stock-out in about 0.44 of the 21.82 cycles a year. A 98 per cent fill rate means 1.22 stock-out cycles a year, each one short on average by well under half an order. Reading the board's 98 per cent as a fill rate would hold 5,838 units of safety stock instead of 7,547, releasing 1,709 units and 48,865.25 EUR of inventory.
Releasing that stock is not free. The fill-rate policy loses 1,325 more units of sales a year. At a unit margin of 18.90 EUR that is 25,038.06 EUR of margin against 6,547.94 EUR of holding cost saved, so on AF-7 the cheaper-looking policy is 18,490.12 EUR a year worse. Convert the measure first; then price both.
Fill rate depends on Q, because a longer cycle has more units to dilute the same shortfall. Cycle service level does not.
| Order quantity | Fill rate at 98% CSL | Units missing a year | CSL for a 98% fill rate | Safety stock needed |
|---|---|---|---|---|
| 2,192 | 98.77% | 1,178 | 96.93% | 6,873 |
| 4,385 | 99.38% | 589 | 94.39% | 5,838 |
| 8,769 | 99.69% | 294 | 89.93% | 4,695 |
Halve the batch and the same safety stock serves fewer units from stock. Teams that cut order quantities to reduce cycle stock, without touching the service target, quietly lower the fill rate their customers see.
The service level table and the fill-rate solver are in the free workbooks on the Supply Chain Management companion page. What a biased forecast does to the same service level is in which costs more: a forecast 10 per cent low or 10 per cent high.
Cycle service level is the probability that a replenishment cycle has no stock-out. Fill rate is the share of demand served immediately from stock. Because a stock-out usually misses only part of a cycle's demand, fill rate is higher: on AF-7 a 98 per cent cycle service level gives a 99.38 per cent fill rate and 589 units missing a year.
It depends on the order quantity and the spread of lead-time demand. For AF-7, with an order quantity of 4,385 and a standard deviation of 3,674.60, the required loss function value is 0.0239, reached at z of 1.5888: a cycle service level of 94.39 per cent and 5,838 units of safety stock.
Yes, at the same safety stock. The expected shortage per cycle is unchanged, but it is spread over more units. On AF-7, at a 98 per cent cycle service level, a batch of 2,192 gives a 98.77 per cent fill rate and a batch of 8,769 gives 99.69 per cent.
The service level table and the fill-rate solver are in the second workbook on the companion page, for chapters 6 to 11 of Supply Chain Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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