Modelling a yacht race

English schoolchildren these days are assessed in examinations at the ages of 15-16 and 17-18, now known as GCSEs (General Certificate of Secondary Education) and "A"-levels.  In my youth I was examined by "O"-levels and "A"-levels (Ordinary and Advanced).  I remember very few aspects of these examinations  One exception is a question in mathematics, as it was thought-provoking.  "Assuming that a yacht's sail deflects the incoming wind as in a mirror, explain why the angle of the sail affects the speed of the yacht"  Quite obviously it was a matter of vectors of force from the wind against the sail and the resistance of the water on the hull.  But this question was different from the usual questions in books, where the forces were acting on idealised objects ("a pendulum on a rod of negligible mass ... ", "a perfectly smooth bead is on a wire ... ")

I hope that the examiner realised that this question would make the examinees pause and think, and allowed extra marks for the time that this might take; and I wonder if other students can remember the question so many years later.  (I have two other memories of that year's examinations - one that I realised ten minutes after a mathematics paper that I had subtracted 530 from 600 and got the answer 50;  the other that in a physics practical involving the cooling of water, I had poured my water away before measuring the volume - so I quickly measured the warm portion of the container, refilled it to the same depth with cold water and measured that, thus getting an answer which was "near enough".)

In the May issue of the Journal of the Operational Research Society, Robert Dalang, Frederic Dumas, Sylvain Sardy, Stephan Morgenthaler and Juan Vila describe how they modelled the forces acting on a racing yacht to determine the optimum course for that boat in the America's Cup races in 2007.  It was rather more sophisticated than my few lines on the "O"-level script in the 1960s. 

Stochastic optimization of sailing trajectories in an upwind regatta appears in the Journal of the Operational Research Society(2015) volume 66, pp 807–820,doi:10.1057/jors.2014.40 (abstract: In a sailboat race, the navigator’s attempts to plot the fastest possible course are hindered by shifty winds. We present mathematical models appropriate for this situation, which use statistical analysis of wind fluctuations and are amenable to stochastic optimization methods. We describe the decision tool that was developed and used in the 2007 America’s Cup race and its impact on the races.)

As the abstract points out, the winds are not constant in direction or strength.  Therefore, the model needs analysis of the past variation as well.  One simplification (realistic) was to model the first upwind leg of the race, as the yacht that leads at the first turn is likely to stay in the lead.  In the paper there are diagrams "Boat polars" which would have been useful in the "O"-level; they show the velocity of a selected boat in a wind of given speed and with the course set at a given bearing.  These lead to the optimal bearing for the wind speed.  The paper discusses tacking (changing the direction of the boat when going upwind) and the choice of whether to start on a left-hand tack or right-hand tack.  And then this feeds into an optimisation model which uses a discrete space for the progress of the yacht.  

Boat polars for a given speed of wind showing (on the right) the optimal course to maximise speed against the wind (from the cited paper)

All well-worth reading.  Towards the end of the paper there is a wistful comment:

Given that the Swiss team implemented our strategy software into their onboard computer system, trained to use it, and finally actually won the 2007 America’s Cup race, we can consider that stochastic optimization techniques were useful to the team. However, since a typical team’s budget is on the order of 100 million dollars, it is clear that a team of a few mathematicians only makes a small contribution to the overall effort.

With hindsight, how much should the mathematicians (O.R. scientists) have charged for their services?