Sailing Strategies

An Application Involving Stochastics, Optimization, and Statistics (SOS)

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Introduction

Consider a sailor who wishes to sail from one point (A) to another (B) in the shortest possible time (for example, in a race). To create a simple model of such a situation, suppose that the lake is represented by a 2-dimensional square n-by-n grid of waypoints. Every path from A to B consists of a sequence of legs, the endpoints of which must be waypoints. Suppose further that the only legs that the sailor considers are legs connecting one waypoint to one of its 8 neighbors (found by sailing N, S, E, W, NE, SE, SW, or NW from the current point).

Time as a Function of Wind Angle

The boat's speed depends on the relative angle between the boat's heading and the wind's direction. For example, the boat cannot sail directly into the wind. If, however, the wind is at 45 degrees to the boat's heading, we say that the boat is on an upwind tack. On such a tack, it takes 4 minutes to sail from one waypoint to a nearest neighbor. But, if the wind is at 90 degrees to the boat's heading, the boat moves faster through the water and can reach the next waypoint in only 3 minutes. Such a tack is called a crosswind tack. If the wind is a quartering tailwind, we say that the boat is on a downwind tack---such a tack takes 2 minutes. Finally, if the boat is sailing directly downwind, we say that it is on an away tack.

Starboard Vs. Port Tacks

If the wind is hitting the left side of the sails (as in the picture) we say that the boat is on a port tack. If, on the other hand, the wind is on the right-hand side, we say that it is a starboard tack. Changing from a port to a starboard tack (or vice versa) takes time since it involves stearing the boat straight into the wind and moving the sails from one side of the boat to the other. Expert sailors can do this with great finesse, but, for beginners, the boat might come dead in the water and it can take awhile to get it going again. We assume that our sailor wastes 3 minutes for every such change of tack. This time is refered to as the tacking delay.

Deterministic Vs. Stochastic Models

If the wind blows constantly from one direction, then it is fairly straightforward to use the information given above to find the best path from A to B. Such a path would involve just one change of direction. But, winds have a tendency to change (both in direction and in intensity). To keep our model simple, we assume that the intensity is constant but its direction can change from one leg to the next (we assume, rather artificially, that the wind only changes when the boat is at a waypoint). The new wind comes from one of three directions: either from the same direction as the old wind, or from 45 degrees to the left or to the right of the old wind. We allow the user (i.e., you) to set the probabilities for each of these scenarios.

Stochastic Optimization

The data described above can be put into a precise mathematical model, called a dynamic programming model, and solved on a computer. By pushing the button at the bottom of this page you can solve this problem on your computer.

Simulation

Once the problem has been solved, it remains an issue as to how to present it since the optimal strategy depends on four independent pieces of information: (i) the current latitude, (ii) the current longitude, (iii) the wind direction for the next leg, and (iv) whether the previous leg was a port tack or a starboard tack. A table showing all possibilities would be huge even for a small lake. Therefore, we prefer to show some specific realizations generated using your computer's random number generator. The process of generating these realizations is called a simulation. The simulations are produced in such a manner that, if a large number of them are generated, the empirically observed wind change probabilities will eventually approach those given as input to the model. When you press the button below, your computer will calculate the optimal solution to the dynamic programming problem and then produce one simulation which will be shown graphically. To generate subsequent simulations, press either the "Simulate" button to generate them 1 at a time or the "Simulate 100" button to do 100 all at once.

Statistics

At any time, you can push the "View Stats" button to get a graphical summary of all the simulations done since the last "Calculate". In addition, the following statistics are shown: the minimum time, the maximum time, the mean time (i.e., average time), and the median time.

Stochastics, Optimization, and Statistics at Princeton

If this example seems interesting to you and you are a student looking for a place to do your undergraduate or your graduate studies, then you might want to consider our undergraduate program in Engineering Managment Systems or our graduate program in Statistics and Operations Research. In addition to learning about these neat things, you can also take up sailing on nearby Lake Carnegie.