Explore the problem. Inspect the result.

Recover a disrupted schedule, manage energy under uncertainty or locate a hidden leak. Each demonstration connects a practical question to a working mathematical model.

Five problems. Five working models.

Every figure below is computed by the model it describes. Open a demonstration to change the scenario and inspect the result.

Expected delay cost per day over 16 unseen days, vessel six hours late: original plan €172k, first come first served €294k, optimised plan €113kOriginal€172kFirst come€294kOptimised€113k

Fictional terminal, vessel six hours late. Mean delay cost per day on 16 simulated days the optimiser never saw.

Port operations

Problem
A late vessel, a crane fault or a jammed gate spreads through berths, cranes, tractors and the yard within hours.
Result
With one vessel six hours late, the optimised berth plan cuts expected delay cost by 34%, from €172k to €113k a day, on 16 simulated days it never saw.
Method
Discrete-event simulation, simulation-based plan search, parallel evaluation
Why it matters
Test recovery plans against many possible days before committing berths and cranes.
Explore port operations
Stored battery energy over example day 69: the risk-aware controller keeps a reserve into the cloudy afternoon, the forecast controller does not600 kWh00:0024:00

Stored energy over one day with a cloudy afternoon: risk-aware controller solid, forecast controller dashed.

Energy dispatch

Problem
Run a battery behind a tight grid connection when tomorrow's prices are known and its clouds are not.
Result
Over 300 simulated days, the risk-aware controller cuts the cost of the worst days (CVaR 95%) from €1,016 with a rule-based controller to €788, close to the €774 only hindsight could reach.
Method
Stochastic dynamic programming, entropic risk, Monte Carlo
Why it matters
Choose how much protection against bad afternoons is worth paying for.
Explore energy dispatch
Route map of the fictional Sofia-based airline: 20 destinations across Europe served by ten aircraftATHLCAISTCDGFCOAMSLHRBCNWAWTLVSOF

Ten aircraft, sixty flights from Sofia. Real airports, fictional airline.

Airline recovery

Problem
Recover a regional airline's day after an aircraft fault, hub storms or an air traffic control strike.
Result
After an aircraft fault at dawn, the optimised plan cuts the day's disruption cost by 67%, from €387k to €128k, and passengers arriving three or more hours late from 926 to 274.
Method
Day propagation, passenger re-accommodation, simulated annealing
Why it matters
Keep one disruption from spreading through aircraft rotations and passenger connections.
Explore airline recovery
Leak probability across the district from four pressure loggers, with the two loggers the model suggests next

Leak probability from four loggers (larger, darker = more likely). Squares: loggers. Rings: the two loggers the model suggests next.

Water leak inference

Problem
Locate a hidden leak in a district network from night flow and a few noisy pressure readings.
Result
Four pressure loggers narrow a burst to 2.2 km of main to survey for 90% confidence; two more, placed where the model suggests, cut that to 140 m.
Method
Hydraulic network solver, Bayesian inference, information-based sensor placement
Why it matters
Prioritise excavation and decide where the next sensor pays for itself.
Explore water leak inference
Strut cross-sections of equal area: a circle with drag coefficient 2.85 and the optimised shape with 1.67flow →Drag: circle 2.85 → optimised 1.67

Same cross-sectional area, flow left to right: circle dashed, optimised shape filled.

Flow and shape optimization

Problem
Reshape a strut in a cooling channel to cut drag without giving up strength.
Result
At fixed cross-section, the optimised strut has 41% less drag than a circle; on a 1.5× finer grid the reduction is 33%.
Method
Lattice Boltzmann fluid solver, constrained shape search
Why it matters
Screen geometry changes before committing to detailed engineering analysis.
Explore flow and shape optimization

New to optimization? A step-by-step introduction to allocating limited resources.

Read the allocation tutorial

Mathematics in industry.

Our commentary on published projects by Molslinjen, Halfspace, Lufthansa Group, Google and DeepMind. These are their results, not ours.

One disruption. A system of decisions.

Lufthansa Group × Google, airline operations

Decision context

  1. Aircraft
  2. Passengers
  3. Crew

Coordinating aircraft, passengers and crew when a disrupted schedule must be repaired quickly.

Turn a forecast into a commitment.

Google × DeepMind, wind energy

Decision context

  1. Forecast
  2. Commit
  3. Evaluate

Using wind-generation forecasts to support advance commitments to the electricity grid.

Apply these methods to your operation.

Explore a model or prototype built around your scheduling, planning or engineering challenge.

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