When I think back to Christmas times of my youth, I recall the cookies my grandmother used to make. They were great! Now she wants to mass-produce the cookies and sell them. Unfortunately, the recipe has been lost. We remember all of the recipe except the “secret sauce” she added to the dough. The only thing we know is that the recipe called for 1 cup of secret sauce, which is a mixture of three ingredients, call them A, B, and C. But, as luck would have it, we forgot the proportion of the ingredients in the secret sauce. To have the optimal taste, the 1 cup worth of secret sauce has to be composed of the right proportions of the ingredients -- or else tragedy could result.
What can I do? I can run an experiment to study the “taste” of cookies made by sauce with varying proportions of the ingredients. This is a mixture experiment. Since the secret sauce is 1 cup, I know the ingredients have to add up to 1 cup. The factors in the experiment are the proportions of the ingredients, and the response is the goodness of the cookies.
Luckily, I have JMP 8. Below I use a ternary plot to show the design points of the mixture experiment.
The point in the middle corresponds to equal proportions of the three ingredients. In other words, the secret sauce is composed of 1/3 cup of each ingredient. The point to the lower left of the middle corresponds to the following proportions: 2/3 cup of A, 1/6 cup of B, and 1/6 cup of C. Points on the edges of the ternary plot correspond to secret sauce for which only two ingredients are used, and the proportion of the third ingredient is set to 0.
We proceed to make secret sauce and cookies corresponding to the design points of the experiment, and have people taste test the cookies. The cookies are ranked on a scale from 1 to 100, with 100 being the best cookie ever. Using the data from the experiment, we fit a model that predicts goodness (taste of cookie) as a function of the varying proportion of the ingredients. We can use this model to find the proportions of ingredients in the secret sauce that lead to the best tasting cookie. In other word, we can find the spot with a Taste value as close as possible to 100.
The data and model are found in Ingredient Data.jmp, and can be downloaded from JMP’s file exchange. New to JMP 8 is the Mixture Profiler, which allows the user to interactively visualize a response surface for a model with mixture factors.
The platform generates a contour (the red line) plot. The value of Taste at the contour shown is 54.44. Any recipe with ingredient proportions that fall on that contour will produce cookies with average Taste value of 54.44. For example, I placed the three-way cross hairs on the contour. The ingredient proportions at that spot are read from the top of the window under Current X.
In this case, the values are 15% for ingredient A, 27.5% for ingredient B, and 57.5 % for ingredient C. Notice the proportions add to 1 as they should: 15% + 27.5% + 57.5% = 100%. The little dots running along the contour represent the direction of increasing Taste. I can’t show you in a screenshot, but you can use the Response slider to move the contour to different parts of the response surface. Doing this allows you to locate the spot with a value of Taste as close to 100 as possible, which is what we want. To visualize many contours at once, we can add a contour grid to the plot, as shown below.
The three-way cross hairs can be used to investigate the value of Taste at various contours. I use the Response slider to study how the contours change as I move them, and as the dots show, the highest value of Taste occurs in the region of the center of the smallest contour.
As shown below, place the cross hairs in the middle of the contour and read the ingredient proportions from the top of the window. The ideal secret sauce (the one that yields the best tasting cookie) has the following proportions of the ingredients: 41% for A, 34% for B, and 25% for C.
Now that we know the best recipe for the cookies, we are ready mass-produce them and sell them. What price should we charge for a cookie? Stay tuned -- in my next blog post I will use JMP’s Simulator to investigate what price I should charge for the cookies.
The functionality of the Mixture Profiler goes far beyond what I have shown here. Multiple responses can be contoured at once, and more than three ingredients can be investigated at the same time. Constraints can be added, so that unfeasible regions are identified on the plot.
Data referenced in this entry can be found in the JMP File Exchange.
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