Industry: Retail Product: ModelRisk Application: Inventory Management Under Uncertainty
A distribution centre ran a textbook continuous-review policy on a fast-moving SKU: when on-hand-plus-on-order inventory fell below a reorder point, it triggered a fixed 1,500-unit replenishment. The reorder point had been set the way reorder points usually are — mean lead-time demand plus a safety-stock cushion sized off an average lead time — and the planning sheet certified it for a 98% service level.
The Monte Carlo rebuild in ModelRisk ran that exact policy across 6,000 simulated years and found the certificate was fiction. The annual fill rate the policy actually delivers has a mean of 93.4%, not 98% — with a P10 of 90.3% and a P90 of 96.4% — and the probability that any given year clears the 98% target is 1.2%, roughly one year in eighty. Worse, the policy was not even cost-optimal: the cheapest reorder point sits well above the one in use, and hitting 98% reliably costs more than the planners had ever priced.
The textbook reorder point is built from two averages — average demand and average lead time — and that is exactly why it disappoints. Lead time is not a constant; it is Triangular(2, 5, 12) days, and the long right tail means the truly dangerous replenishment cycles are the slow ones, during which demand keeps drawing the shelf down. Demand is not constant either: a shared weekly factor keeps demand correlated within a week, so a hot week stays hot and lead-time demand is far more variable than an i.i.d. day-count would suggest. A safety stock sized off mean lead time covers the average cycle and stocks out in the bad ones — and because service is a property of the worst cycles, not the average, the realised fill rate lands well below the headline figure. The honest object is not a service-level label; it is a distribution of annual fill rates and a cost curve that has its own opinion about where the reorder point belongs.
Running the installed policy over 6,000 simulated years gives the distribution the planning sheet replaced with one figure:
The realised annual fill rate has a mean of 93.4%, a P10 of 90.3%, a median of 93.5% and a P90 of 96.4%. The 98% service target sits out on the right edge, beyond almost the entire distribution: the probability of meeting it in any year is 1.2%. The gap between the certified 98% and the realised 93.4% is the cost of sizing safety stock off an average lead time — the slow-replenishment cycles that the average ignores are precisely the ones that drive the stockouts. A planner who sees only "98%" budgets for a service level the policy reaches roughly once a lifetime; a planner who sees this distribution knows the SKU runs a 90% floor in a bad year and can decide whether that is acceptable.
Sweeping the safety stock from zero to nine days of supply traces holding cost up, stockout cost down, and a total cost that bottoms in between:
Holding cost climbs steadily with safety stock — from about $16,029 at zero safety stock to $37,314 at nine days. Stockout cost falls the other way, from $34,970 down to $600. Their sum, the total cost, is the U-shaped curve that matters: it falls from $50,999 at zero, bottoms at $31,822 at 5.0 days of safety stock (a reorder point of 1,360 units, a 98.5% fill rate), then rises again to $37,914 at nine days as holding cost overtakes the dwindling stockout savings. This is a real interior optimum, not a corner solution — more safety stock is genuinely wasteful past five days, and less is genuinely under-served. The installed policy, with its sub-two-day cushion, sits on the steep left wall of this curve, paying far more in stockouts than it saves in holding.
The cost-optimal point and the reliability-optimal point are not the same, and the gap between them is the chart every inventory decision should start from:
The total-cost curve bottoms at 5.0 days of safety stock ($31,822), where the probability of meeting the 98% fill target is 72%. To raise that reliability to 95% takes 8.0 days of safety stock at a total cost of $35,771 — about $3,950 a year more than the cost optimum. Pushing to a 99% chance of meeting target needs 9 days and $37,914. The two curves diverging is the whole point: the cheapest policy is not the most reliable one, and "98% service" can mean either "mean fill of 98.5%" (5 days, the cost optimum) or "98% of years actually clear 98%" (8 days, $4k dearer). The simulation forces that distinction into the open and lets the planner buy exactly as much reliability as the SKU's margin justifies — no more.
A tornado over the cost drivers ranks where the annual-cost uncertainty lives:
Replenishment lead-time variability dominates, with a half-spread of roughly ±$4,819 on annual cost — the single most valuable supply-chain improvement is a tighter, more predictable lead time, worth more than any safety-stock tuning. The shared weekly demand swing is second at ±$3,614, then the stockout penalty assumption at ±$2,650 and the reorder point/safety stock itself at ±$2,289; the holding-cost rate (±$1,446) and order quantity (±$1,084) matter least. The ranking points the buyer at the supplier first — shrinking the 2-to-12-day lead-time spread does more for cost and service than any amount of safety stock can.
Inventory management is not "pick a reorder point that the formula labels 98% service" — it is "what fill rate does this policy actually deliver across a year of variable demand and lead time, and what does each extra point of reliability cost." The deterministic sheet certified a service level the policy met one year in eighty; ModelRisk showed the real fill-rate distribution, found the cost-optimal reorder point, and put a price tag on the reliability the business actually wanted.