Which Quantile Do You Order For a 95 Percent Service Level?
A planner asks for "a 95 percent service level" on every SKU. The order covers a 9-day protection interval (7 days between reviews plus a 2-day supplier lead time). You have a model for the 9-day demand sum, with these realised results on 19,600 held-out series-cycles (mean demand 18.1 units per cycle):
| Requested quantile | Cycle service level | Fill rate | Units left over per cycle |
|---|---|---|---|
| 0.80 | 78.7% | 95.0% | 4.66 |
| 0.90 | 87.7% | 97.2% | 6.57 |
| 0.95 | 92.5% | 98.2% | 8.38 |
| 0.98 | 96.0% | 98.9% | 10.59 |
A colleague proposes summing the nine daily 95th percentiles instead. That policy gives a 99.8 percent cycle service level and 24.51 units left over per cycle.
- Which request meets "95 percent service" if it means cycle service, and which if it means fill rate? What do the two cost in stock?
- Why does the colleague's sum over-deliver?
- For a thin-margin perishable item the cost of one unit short is 25 percent of price and the cost of one unit wasted is 75 percent. What quantile minimises expected cost? Is 95 percent sensible there?
- What is the cost ratio hidden behind a 95 percent target, and how would you fix the shortfall at the 0.95 request?
1. Two definitions. Cycle service level is the share of cycles with no stockout; fill rate is the share of units served. For 95 percent fill the 0.80 request is enough (fill 95.0 percent, 4.66 units left over). For 95 percent cycle service even the 0.95 request misses (92.5 percent); by linear interpolation between 0.95 and 0.98 you need about 0.97, and the 0.98 request delivers 96.0 percent with 10.59 units left over, more than twice the stock of the fill-rate answer. Ask which one the business means before quoting a number, and report both.
2. Quantiles do not add. The daily demands are not perfectly correlated, so errors partly cancel across nine days and the 95th percentile of the sum is much smaller than the sum of nine 95th percentiles. The sum policy delivers 99.8 percent cycle service but holds 24.51 units left over against 8.38 for the direct 9-day quantile, 2.9 times the stock for a target that was 95 percent. Model the lead-time sum directly (or simulate the joint distribution) instead.
3. The newsvendor critical ratio. The optimal order-up-to level is the quantile at c_u/(c_u+c_o), here 0.25/(0.25+0.75)=0.25. A 95 percent quantile is wildly wrong for this class. In the lesson's experiment (EUR per series-cycle, same 19,600 cycles) the 0.30 level cost EUR 31.74 against EUR 107.29 at 0.95, 3.4 times as much, and ordering the mean forecast cost EUR 37.00.
4. The hidden ratio and the fix. A 95 percent target is a 19 to 1 ratio of shortage cost to overstock cost, 0.95/0.05, which is only plausible for a few strategic items. As for calibration, the 0.95 request delivers 92.5 percent, so fix it with split-conformal recalibration using earlier origins (normalised score (A-S)/(S+1); offsets near 0.064 to 0.080 in the lesson), which raised pooled realised service from 92.8 to 95.5 percent at the price of 8.50 to 10.39 left over per cycle. Always check coverage on slices (promo days covered 92.6 percent against 94.3 percent on other days).
Share this question