Stochastic Planning: Beyond Point Estimates

The Lie of Averages

Consider two suppliers, both with a 10-day average lead time:

• Supplier A: Delivers in 9-11 days, consistently. Normal distribution, σ = 0.5 days.
• Supplier B: Usually delivers in 7 days, but occasionally takes 25 days. Lognormal distribution, heavy right tail.

"Planning with averages is planning to be wrong. The entire field of stochastic optimization exists because point estimates discard the very information that makes supply chain decisions hard: the shape and extent of uncertainty."

A traditional planning system treats them identically, average lead time: 10 days. But the safety stock required for 95% service level is dramatically different. Supplier A needs minimal buffer. Supplier B needs a large buffer to protect against those 25-day outliers.

Using the average for both means you're either over-investing in buffer for Supplier A or under-protecting against Supplier B. Both waste money.

Distribution Fitting

"In my experience, fewer than 5% of supply chain variables are truly normally distributed. Lead times tend to be lognormal or Weibull. Demand often follows negative binomial or Poisson patterns. Using a Normal assumption when the data says otherwise is not simplification; it is systematic error."

Autonomy fits actual distributions to operational variables automatically. Instead of assuming one distribution shape fits all, the system identifies the best-fit distribution for each variable from your historical data, and stores the fitted parameters in the shared world model so every agent uses the same view of uncertainty.

This changes every downstream calculation: safety stock, reorder points, ATP availability, and capacity requirements all become more accurate when they use the right distribution instead of a Normal approximation.

What If the Distribution Is Wrong?

It is the right objection, and it has an empirical answer. At DecisionCAMP 2026, Meinolf Sellmann put a scheduling problem through both approaches and then deliberately broke the distribution to see what it cost.

The problem was surgery scheduling, where procedure times are not deterministic. Solved on minimum makespan with point estimates: roughly $953,000 a year, and 1.75 patients bumped to the next day every week. Solved stochastically against a fitted Gamma: roughly $332,000, and 0.6 bumps. Then he varied the assumed distribution's mean, its variance and its shape, and re-measured.

The roughly two-times improvement held, and the deterministic plan never won. That is the result worth carrying: the gap between a wrong distribution and a point estimate is much larger than the gap between a wrong distribution and a right one. Fitting the distribution well is worth doing, and it is the second-order decision. Carrying one at all is the first-order decision.

Which is why every agent in the substrate conditions on a band rather than a number - and why we audit for the ones that still do not, rather than assuming the wiring is complete.

"Average 2x improvement. Deterministic never beats stochastic."

The Twin Is Where the Agents Learn

Monte Carlo runs upstream of the planning engine, and what it feeds is a reinforcement-learning loop. The digital twin is not a reporting tool that produces scenarios for people to read. It is the environment an agent learns in.

The loop is ordinary reinforcement learning with a supply chain in the middle of it. A scenario sampler draws a stressed world - a supplier goes dark, capacity is cut, demand moves, a lane closes - on a curriculum that starts with quiet baselines and escalates to compound shocks as the policy matures. The agent acts through the episode. The twin returns what happened. The policy updates. Repeat across thousands of futures the business has never actually lived through.

That is the answer to an objection worth taking seriously: a policy fitted to history has learned a world that has already moved on. This one is not fitted to history. Its training distribution is authored - and because the causes of disruption are unlistable while their effects on a network are a small closed set, you can author it honestly without pretending to forecast the unforecastable.

Two things come out of the same rollouts, which is why the loop earns its cost. The first is a trained policy. The second is a calibration set for conformal prediction, which is what lets a band exist before a deployment has accumulated a year of its own production history.

And a trained policy does not ship because training finished. It ships only if it beats the incumbent - the heuristic the business runs today - scored on the tenant's balanced scorecard over held-out runs it has never seen. If it loses, nothing is saved and the previous policy stands. If the evidence is too thin to call, that is also a refusal.

This is the part that makes the rest of it checkable, and it is deliberately the unglamorous half. Training is cheap and confident; it will always produce a new set of weights, and those weights will always look like progress on the training curve. The gate is the only thing standing between "the model changed" and "the decision got better", and the two are not the same claim.

From "What's the Plan?" to "How Achievable — and How Fragile?"

Conformal prediction then wraps that output in distribution-free coverage. Instead of raw percentiles, every statement carries a bound whose accuracy is measured rather than asserted:

• "P50 cost: $2.4M, P90 cost: $2.8M", at a 90% coverage rate the substrate is built to check against realised outcomes
• "85% probability OTIF exceeds 95%", a calibrated bound, not an estimate
• "Service level risk: 12% chance of dropping below 90% in Q3", with no assumption about the shape of the distribution

This transforms the planning conversation. Instead of debating whether the plan is "right," you discuss whether the guaranteed probability distribution of outcomes is acceptable, and every agent's achievability score on the Decision Stream is trustworthy by construction. Achievability drives whether an action is automated, surfaced to inform, or escalated to inspect under AI·IO·ML.