The cost-minimising cycle service level for a reorder-point system, worked on one part, and why the newsvendor ratio gives the wrong answer here.
In a reorder-point system the cost-minimising probability of a stock-out in each replenishment cycle is Q × H ÷ (p × D): order quantity times the annual holding cost of a unit, over the shortage cost of a unit times annual demand. For part AF-7 that is 0.929 per cent, so the optimal cycle service level is 99.07 per cent. The 98 per cent the board voted costs 2,169 EUR a year more.
Worked in full in Supply Chain Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Service levels are usually set by habit or by a board vote, then defended as policy. They can be computed. The trade-off is simple: one more unit of safety stock is held all year and costs its holding cost every year; it only earns its keep in the cycles where demand during the lead time reaches it. The optimum is the level at which those two are equal at the margin.
| Input | Value |
|---|---|
| Mean weekly demand, standard deviation | 1,840 / 462 units |
| Lead time, standard deviation | 6 / 1.9 weeks |
| Standard deviation of demand over the lead time σ | 3,674.6 units |
| Annual demand D | 95,680 units |
| Order quantity Q (the EOQ) | 4,384.5 units |
| Holding cost H (13.4% of 28.60) | 3.8324 |
| Shortage cost p (lost margin: 47.50 less 28.60) | 18.90 |
A stock-out on AF-7 is treated as a lost sale, so each missing unit costs its margin. If your customers wait instead, use what a backorder actually costs: expediting, penalties, the share of orders that cancel.
Strictly, the rule below is the backorder form. When every shortage is a lost sale, the exact rule adds Q × H to the denominator, Q × H ÷ (p × D + Q × H), which gives 0.921 per cent and 99.08 per cent. The difference is a hundredth of a point, so the simpler form is used throughout, as in the case workbook.
=NORM.S.INV(1-Q*H/(p*D)) gives the z.The numerator Q × H is worth a second look: at the EOQ it equals the whole annual ordering plus holding cost of the cycle stock, 16,803.18. The denominator is the margin on a year's sales. The optimum is set by how expensive stock is relative to the margin it protects, scaled by how often the part is exposed.
| Cycle service level | Safety stock | Holding | Shortage | Total | Above optimum | Fill rate |
|---|---|---|---|---|---|---|
| 95.00% | 6,044 | 23,164 | 31,664 | 54,828 | 16,946 | 98.25% |
| 97.00% | 6,911 | 26,486 | 17,608 | 44,094 | 6,212 | 99.03% |
| 98.00% | 7,547 | 28,922 | 11,129 | 40,051 | 2,169 | 99.38% |
| 99.00% | 8,548 | 32,761 | 5,136 | 37,897 | 15 | 99.72% |
| 99.07% | 8,649 | 33,147 | 4,735 | 37,882 | 0 | 99.74% |
| 99.50% | 9,465 | 36,274 | 2,395 | 38,670 | 788 | 99.87% |
| 99.90% | 11,355 | 43,518 | 420 | 43,938 | 6,056 | 99.98% |
Units missing per year are σ × G(z) × D ÷ Q, where G is the standard normal loss function: 589 at 98 per cent, 251 at the optimum. Moving the board's 98 per cent to the optimum adds 1,102 units of safety stock worth 31,531, and saves 2,169 a year net. The curve is flat on the high side and steep on the low side: 99.5 per cent costs only 788 more than the optimum, 95 per cent costs 16,946 more.
The same 2,169 appears from another direction in the forecast bias calculation for AF-7, where a forecast running 10 per cent high quietly lifts the 98 per cent to almost exactly this optimum.
| Shortage cost per unit | Stock-out probability | Optimal CSL | z |
|---|---|---|---|
| 5.00 | 3.512% | 96.49% | 1.810 |
| 10.00 | 1.756% | 98.24% | 2.107 |
| 18.90 | 0.929% | 99.07% | 2.354 |
| 30.00 | 0.585% | 99.41% | 2.521 |
| 47.50 | 0.370% | 99.63% | 2.679 |
Nearly a tenfold range in shortage cost moves the optimum by little more than three points, and z by less than one. That is useful: the shortage cost is the input everybody argues about, and the answer is not very sensitive to it. The holding rate and the order quantity matter just as much, because they sit in the same fraction.
Using the newsvendor critical ratio. The textbook ratio p ÷ (p + h) is right for a one-period product such as a newspaper or a seasonal order, where a unit is held once and either sold or not. Applied to AF-7 it gives 18.90 ÷ (18.90 + 3.8324) = 83.1 per cent, a safety stock of 3,527 units, a fill rate of 92.47 per cent and a cost 111,879 EUR a year above the optimum. The error is mixing an annual holding cost with a shortage that can happen only once per cycle. In a continuous-review system the cycle count D ÷ Q has to be in the formula.
Confusing the cycle service level with the fill rate. The optimum above is a probability per cycle. Read as a fill rate, 99.07 per cent would need much less stock. The conversion is worked in how to convert a cycle service level into a fill rate.
The service level table in the free workbook for this case prices every level for AF-7, with the demand, lead time and holding rate in blue cells you can replace with your own.
That is the newsvendor ratio, built for a single selling period where the holding cost is paid once. In a reorder-point system stock is held all year but exposed to a stock-out only once per cycle. For AF-7 the ratio gives 83.1 per cent, a fill rate of 92.47 per cent and a cost 111,879 EUR a year above the optimum.
What a missing unit actually costs: the lost margin if the sale goes elsewhere, or expediting and penalties if it is backordered. For AF-7 the lost margin is 18.90 EUR. At 10.00 the optimum falls to 98.24 per cent; at 47.50, the full selling price, it rises to 99.63 per cent. The answer moves slowly because it sits deep in the tail of the normal curve.
Little above it, more below it. For AF-7, 99.00 per cent costs 15 EUR a year more than the 99.07 optimum, and 99.50 per cent costs 788 more. Going down, 98 per cent costs 2,169 more and 95 per cent 16,946 more, as shortages climb faster than holding cost falls.
This article is one calculation from 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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