Wax selection: an alternate method using statistical techniques.
By: Steffan Lloyd (2014/07/07)
Wax selection: an alternate method using statistical techniques.
Waxing is tricky business. The difficulty is that time is short and there are a multitude of possibilities to test. Each wax combination can take quite a few minutes to prepare and test. Testing too far before the competition isn’t always effective – especially if the conditions are changing. So being able to intuitively choose the best wax from a limited number of tests is important. In this article, I propose an alternative method for testing waxes which minimizes the number of tests needed, and gives us additional information on the importance of each element of waxing. Let me also preface this by saying that I am neither an experienced wax technician, nor a statistician – I'm just an engineering student with an interest in optimization problems.
Let us characterize this problem mathematically. Since grip testing and glide testing are generally independent of one another, let’s just look at glide testing for the purpose of this article, and we'll see later that the concepts apply equally well to grip testing. We generally have 4 parameters we are trying to optimize for:
- Glide wax
- Glide powder
- Block/liquid cover (if any)
- Structure (if any)
Each one of these parameters can have a pre-determined number of options. The goal is to find the optimal option for each parameter with the least amount of testing possible.
Enter math. A method for optimizing this problem was outlined by a genius named Genichi Taguchi. His method is used extensively in manufacturing, engineering and marketing, and involves carefully choosing options on each trial, in such a way to reduce the number of trials significantly and analyze the effect of each option, for each parameter, mathematically.
His method is extremely effective when optimizing a large number of parameters, each with only a few options. Let us choose a sample problem for which this method will be optimal. Suppose on a given day the wax technicians are able to narrow down potential candidates for waxes to 3 glide waxes, 3 powders, 3 cover options (2 covers + no cover), and 2 structure options (yes or no).
One way of thoroughly testing this would be to test every possible permutation of options for each parameter. However, this is obviously unfeasible as the number of trials need is far too large (3 glide options × 3 powder options × 3 cover options × 2 structure options = 54 trials needed).
What is commonly done, however, is to test each parameter individually, while keeping the other parameters constant. This allows the wax tech to quickly see the effect of each option. This requires only 11 tests in this case. However, it will give poor results when the performance effect of one parameter is closely linked to how it works with another parameter. For example, a certain cover could work well with one powder or wax, but not with others, and thus skew the results if all the tests for covers were done in conjunction with that powder or wax.
Using Taguchi’s methods, however, we can choose the options for each test carefully in order to find the effect, and importance, of each option of each parameter, with a similar number of tests, but without the bias of closely linked options – essentially getting the simplicity of the first method with the accuracy of the second method.
To implement this method, we must first find the appropriate “Orthogonal Array” for our problem. For our case of 4 parameters and 3 options, or “levels” for each parameter, the appropriate array has already been created, and can be adapted to the current problem as follows. For the yes no options, I’ve placed a number beside it so we can reference that choice numerically later on.
By conducting these 9 tests as described above, we can determine the optimal choice for each parameter, as well as the relative importance for each choice of parameter. The method for doing so is as follows. First, we calculate what is known as the “Signal Noise” for each experiment. Skipping the derivation of this formula, the signal noise for each experiment can be calculated as:
Where SNi is the signal noise for experiment i, x is the experiment performance (on 10), and log is the logarithm function base 10.
With these numbers calculated, a decision table can then be made. The decision table will look like this:
Each element of this table is calculated as the average of the “Signal Noise” for every experiment that that option for that parameter was used. For example, with the table above, SNglide, option 1, could be calculated as:
The rest of the table can be calculated in the same way.
This table then allows us to easily to read off the results. The option for each parameter with the highest average signal noise is the optimal choice for that parameter. Furthermore, the greater the range between the highest, and lowest value for each parameter, the greater importance that parameter has on the overall performance of the ski.
These calculations may seem daunting, but they are very easy to automate. I have created a simple excel document which can be used to calculate these values quickly, after simply inputting the performance numbers for each test.
Take for example the following sample performance data, for which the above spreadsheet has automatically done the number-crunching and has highlighted the relevant boxes:
With this sample data inputed, we can deduce the following information for these conditions:
- Option 1 glide wax was optimal, since it's average signal noise was highest. Similarly, Option 3 powder was best, and Option 2 cover was best. Lastly, no structure had better performance than structure.
- Furthermore, since the range of the signal noises for glide wax was the highest, we can deduce that the choice of glide wax was by far the most critical component in the skis' performance. Further testing in that area could be beneficial.
- Structure had little or no effect on the results, as the signal range was extremely small.
So while this method clearly requires more thought and planning before testing, it is interesting to note that, using this method, and only 9 tests, we were able to determine the optimal setting for each parameter, without any bias from linked parameters, and as well determine the relative importance of each parameter!
This method can also be easily adapted to other problems, such as grip testing, ski manufacturing, or the same glide testing with different numbers of wax options (perhaps 4 instead of 3). All that is required is to select the appropriate orthogonal array for the problem, knowing the number of parameters and the number of levels for each parameter. A full list exists here: https://www.mne.psu.edu/me345/Lectures/Taguchi_orthogonal_arrays.pdf
Design of Experiments via Taguchi Methods
Good overview of entire process.