Industry: Transportation Product: ModelRisk Application: Freight Scheduling
At a six-berth mid-size U.S. east-coast container terminal, vessel demurrage — the per-hour penalty charged when a ship sits at anchor waiting for a berth — is the single largest controllable operating cost line. The terminal's deterministic appointment-spacing model said the current 22-hour spacing between scheduled vessel arrivals produced an expected $155,000 a week in demurrage. The model couldn't say what the week-to-week distribution of that spend looked like, which weeks would breach the operations budget, or whether tightening or relaxing the spacing actually helped.
A terminal-operating company rebuilt the appointment-spacing decision in ModelRisk against 18 months of actual arrival, dwell, and crane-productivity data. The simulated weekly demurrage spend is the whole story — a violently right-skewed distribution where the mean is set by a handful of brutal weeks, not the typical one.
The chart above shows why an "expected cost" line is the wrong planning instrument here. The mean weekly spend is $155k, but that average sits well out in the right tail of a distribution whose body is far lower: the 90th-percentile week is roughly $429k, and the worst 1% of weeks run above $1.1M. The reframing — from "what is the expected demurrage cost?" to "what is the distribution of weekly costs and which spacing flattens the right tail?" — changed the recommended policy and put a defensible number on the marginal value of a contract concession from one of the terminal's two big shipping-line tenants.
Vessel arrivals/week were fitted as Negative Binomial with mean 18 and dispersion parameter r=8 from 18 months of actual berth-window allocations. The over-dispersion is real: variance is roughly 56 against a mean of 18 — three times what a plain Poisson would give. The over-dispersion source is structural — feeder service schedule shifts, ENSO weather windows, North-Atlantic storm reroutes — and using Poisson here understates the 90th-percentile week by 5 vessels (a NegBinomial P90 of 28 against a Poisson P90 of 23).
Per-vessel TEU count is LogNormal with mean 2,400 TEU and CV 0.55 — the terminal sees a mix of 1,200-TEU feeders and 4,000-TEU mainliners on the same berths. LogNormal rather than Normal because the right tail of large-vessel calls is the real source of berth-time variability.
Crane productivity is 28 TEU/crane-hour mean with a shared weekly Gamma shock (mean 1, CV 0.35) representing weather, labor (longshore call-out variability), and equipment-availability shocks that affect all cranes on the same week. Treating crane productivity as a per-vessel independent draw — the deterministic model's implicit assumption — understated the bad-week tail by roughly 40%.
The model assigns 3 cranes per vessel working in parallel on the same berth. A 2,400-TEU vessel under the mean productivity needs 2,400 / (3 × 28) = 28.6 berth-hours. Sum across the week's vessels and you have demand in berth-hours. Capacity is 6 berths × 168 hours = 1,008 berth-hours.
Reading the demurrage distribution above in detail: mean weekly spend at the current 22-hour spacing is about $155k, but the distribution is heavy-right. The 90th-percentile (VaR90) week is roughly $429k, and the 99th-percentile week — about five weeks a year — runs above $1.1M. The probability of a single week exceeding the $150k operations alert threshold is roughly 29%. A single "expected cost" line is doubly misleading here: it overstates the typical week (whose spend sits below the mean) yet gives no warning of the tail weeks that actually consume the budget.
This is the "Why Monte Carlo" beat for terminal operations: the cost line is the right tail. The fat-right shape of weekly demurrage comes from the interaction of over-dispersed arrivals with correlated productivity shocks — a textbook compound-distribution problem that a deterministic spreadsheet cannot solve.
Three spacing policies were evaluated on identical input draws:
The CDFs cross only at the extreme right. Current (22-hour) is cheapest from the left tail through roughly the 90th percentile; Relaxed (28-hour) edges ahead only beyond the P90, where its larger buffer absorbs the worst arrival surges for slightly less than the demurrage they would otherwise trigger ($418k vs $429k at P90). The decision is therefore a genuine tail-vs-mean trade: an operator chasing the lowest expected cost stays at 22 hours, while one managing a hard weekly-cost covenant pays about $60k/week of extra mean cost to shave roughly $11k off the P90 — a trade that only pencils out if the covenant breach is expensive enough.
The baseline weekly demurrage the tornado decomposes is $155k. The largest mover is the vessel arrival rate itself — the dispersion shock in the NegBinomial drives much of the tail. Crane productivity shock ranks second; appointment spacing ranks fourth. Critically, the spacing decision is roughly half as powerful as the productivity-shock decision — which is what made the terminal's investment case for a port-weather-and-labor pre-warning system: a tool that shifted the productivity-shock distribution from CV 0.35 to CV 0.25 would compress the tail more than any spacing tweak the schedulers could make.
Container-terminal economics are a compound-distribution problem: over-dispersed arrivals, log-normal vessel sizes, correlated productivity shocks, and a steep right-tail cost line called demurrage. Monte Carlo simulation in ModelRisk surfaces the shape of that compound distribution — and turns the appointment-spacing decision from "what is the expected cost?" into "which spacing has the tail we can live with?" That reframing is what separates the cost-minimising policy (22 hours) from the tail-managing override (28 hours on surge weeks), justifies the productivity-improvement spend, and gives the terminal a defensible number for every contract conversation about scheduling concessions.