- A point ETA answers the wrong question. A turnaround planner needs to know the probability a shipment arrives before the day it is needed.
- Meridian composes per-leg delay distributions, learned from lane history and updated live from AIS, into an arrival distribution per shipment. Most of the uncertainty sits in transhipment and port dwell, not at sea.
- The same distribution prices the inventory decision: safety stock for critical spares from a real lead-time distribution, and an expedite rule that compares expected lateness cost with the cost of flying it.
01The date on the bill of lading
A turnaround is a schedule with a critical path, and a surprising amount of that path runs through a container. A subsea BOP stack out of Ras Tanura, 36-inch cryogenic valves from Hamburg, an exchanger bundle from Houston: each is a long-lead item whose arrival gates a scope of work with a crew and a crane booked against it. The cost of a turnaround day is well understood on the plant side. It is not usually connected to the logistics side, where the shipment is represented by a single date, the carrier's ETA, that quietly moves.
The carrier ETA is a sailing schedule. It is honest about the ocean leg and silent about everything else: dwell at the origin port, the transhipment connection at Jebel Ali or Singapore, the queue at the destination anchorage, customs, and the truck. When the date slips, it tends to slip late in the journey and late in the day, and the planner learns about it from a phone call.
The two conventional responses are both expensive. One is to hold more spares than the plant needs, which ties up capital in slow-moving inventory. The other is to expedite when the slip becomes visible, which means air freight at panic rates, sometimes for equipment that would have arrived in time. What is missing is a number in between: how likely is this shipment to be late, by how much, and what is that worth?
02ETA as a distribution
Meridian treats a journey as a sequence of legs: origin dwell, ocean transit, transhipment, destination port dwell, customs clearance, inland delivery. Each leg has a delay relative to schedule, and the shipment's total delay is their sum.
Each leg's delay is drawn from a distribution conditioned on the lane, the carrier, the season and a live port-congestion index κ. Legs are modelled as skewed, heavy-tailed families (a log-normal body with a discrete "missed connection" component for transhipment) and fitted from 12 months of lane history. Total delay is their convolution.
The distribution is not static. As the shipment moves, completed legs collapse to their observed values and the remaining legs are re-conditioned on what is now known. The ocean leg in particular is updated continuously from the vessel's AIS track: distance to go over speed over ground gives a sailing estimate whose variance shrinks with every position report, and a vessel that has slowed or diverted shows up as a widening tail hours before any schedule changes.
FD is the cumulative distribution of total delay, tsched the scheduled arrival and b the receiving buffer (inspection, staging). A shipment is at risk when r exceeds a threshold set by the criticality of what it gates.
03Where the uncertainty actually lives
Fitting the legs separately makes one thing visible that a carrier ETA hides: the ocean leg is the longest part of the journey and the least uncertain part of it. On the Europe-to-Gulf lanes in the reference deployment, the variance of total delay decomposes as follows.
| Leg | Median (days) | P90 (days) | Share of delay variance |
|---|---|---|---|
| Origin port dwell | 1.2 | 3.5 | 18% |
| Ocean transit | 14.1 | 16.0 | 9% |
| Transhipment connection | 2.4 | 6.1 | 31% |
| Destination port dwell | 2.8 | 5.9 | 24% |
| Customs clearance | 1.5 | 4.2 | 14% |
| Inland delivery | 0.8 | 1.6 | 4% |
More than half of the uncertainty comes from two legs, transhipment and destination dwell, that together account for a quarter of the median journey. Both are observable early: a missed connection is known the moment the feeder sails without the box, and destination congestion is visible in the anchorage count days before a vessel joins the queue. Meridian's watchlist is therefore not sorted by "days late", which is a lagging quantity, but by expected cost, which is a leading one.
04From arrival risk to inventory
An arrival distribution is only useful if it changes a decision. Two decisions depend on it directly.
Expedite or wait
For a shipment gating a turnaround scope, the question is whether to pay for a faster mode now. The expected cost of waiting is the probability of lateness times what lateness costs, where the cost is the turnaround day-rate of the scope the shipment gates, over the expected number of days late given that it is late:
CTA is the cost per day of the gated scope slipping; Cx the expedite premium. Both sides are recomputed on every AIS update, so the rule fires the day the distribution moves, not the day the ETA changes.
Worked through for the valves in Figure 1: on day 14, r = 0.31, the expected lateness given late is 4.2 days, and the scope's day-rate is $1.1M. The expected cost of waiting is roughly $1.4M against an air-freight premium of $180k. The instrument recommends expediting, with the arithmetic shown. On day 9, with r = 0.22 and an expected lateness of 1.6 days, the same arithmetic said wait. Neither answer is obvious from a date.
How much to hold
For spares that are replenished rather than project-shipped, the standard safety-stock formula assumes a lead time that is either fixed or normally distributed. Neither is true of a lane with a transhipment. Meridian replaces the assumed lead time with the fitted lead-time distribution for the lane the part actually travels, and sets the reorder point at the quantile that meets the service target:
α is the service level for the part's criticality class; FL the lead-time distribution including the delay D from Eq. 1. Reliable lanes earn lower stock; lanes with heavy transhipment tails earn more, and the difference is now explicit rather than hidden inside one blanket buffer.
The effect is redistribution, not simply reduction. In the reference deployment, safety-stock value across the tracked critical-spares list fell by about 12% at the same service level, because the blanket buffer had been sized to the worst lane and applied to all of them. Two part families on the least reliable lane had their stock increased.
05What this changes for a materials team
- A watchlist ranked by expected cost. Two shipments "at risk" are not equal; the one gating a $1.1M/day scope sits above the one gating a warehouse restock.
- Expedite decisions with the arithmetic attached. The recommendation carries r, the expected lateness and the two costs, so procurement can defend it and finance can audit it.
- Demurrage prevented rather than disputed. Predicted dwell drives collection scheduling; the free-time clock is visible before it starts.
- Carrier and lane scorecards on distributions. Two carriers with the same average transit can have very different P90s; the P90 is what a turnaround planner is exposed to.
06Limits
AIS coverage has gaps: transponders are switched off, coastal receivers miss vessels, and inland legs are invisible unless the forwarder shares telematics. Transhipment visibility depends on the carrier's event feed, which is good for the major lines and poor for feeders. Customs regimes change by policy, not by trend, and a regime change is a break in the data that the lane history cannot anticipate. Air freight is modelled with a much simpler leg structure. Most importantly, the model begins at the ex-works date: it says nothing about the supplier's own production delay, which is often the largest source of lateness on long-lead equipment. Connecting expediting reports and supplier milestones to the same distribution is the next piece of work, and the subject of a later note.
The claim here is narrow. A date is a poor summary of a journey with six legs and two queues in it. A distribution over the same journey is not harder to compute, and it is the only object that can answer the question the planner is actually asking.