A great tasting Lemonade is a result of an experimentation.
2 level factorial designs are the first resort for anybody making their first attempt at Design of Experiments (DoE). Basically, these designs consist of all combinations of each factor at its high and low levels (Full factorial). However, these designs do not work out for some industries in which the product under investigation is made up of several components or ingredients e.g. Chemicals, Foods, Pharmaceuticals, etc. In these cases, the response depends on proportions of different ingredients in the mixture. The quantities of components, measured in weights, volumes, or some other units, add up to a common total. In contrast, in a factorial design, the amount of each factor is varied, and the sum of the amounts for each run need not be constant.
Let us understand this with a simple example of making Lemonade.
Lemonade components and their ranges:
|Mixture component||Units||Lower level (-)||Upper level (+)|
The response is the taste rating scaled from 1 to 10, 1 being worst to 10 being the best.
Let us first try to do a Factorial experiment. The design layout along with the results of the taste ratings are given in the table below.
Design Layout: Factorial design
|FACTOR 1||FACTOR 2||
Ratio of B/A
Volume of final Lemonade
|ml||(1 to 10)|
|1||(-) 200||(-) 200||1:1||400||7||This is OK. Could have been diluted more.|
|2||(+) 300||(-)200||3:2||500||3||Oh! So Sour. Can’t drink this.|
|3||(-) 200||(+) 300||2:3||500||6||Now this is too diluted!|
|4||(+)300||(+)300||1:1||600||7||Did he give me the first drink again!|
Have a look atthe experiments numbered 1 and 4. Do you think Experiment 1 and 4 are the same? Well, they only differ in the volume of the lemonade as a result of mixing the two ingredients. The proportions remain the same. Obviously, they will taste the same as well. Thus, it makes more sense to look at taste as a function of the proportion of lemonade concentrate to water, not the amount.
Mixture design accounts for the dependence of response on proportionality of ingredients. If you experiment on Compositions/formulations/mixtures where proportions matter, not the amount, factorial designs won’t help you. You will have to use mixture designs.
Now, the first thing you need to decide before using the mixture design is the total quantity, in our example, the total volume of the lemonade. Let us assume a serving of 500 ml should suffice. So, now we have fixed the total to 500 ml of Lemonade. We will now vary the proportion of concentrate and water to give a fixed total of 500 ml. This is what a mixture design does and a factorial cannot. The design layout along with the results of the taste ratings are given in the table below.
(Also note, I am calling concentrate and water as components and not factors as I did in the previous table. An important rule, when you do a mixture design).
Design Layout: Mixture (Linear model, without replicates)
|Component 1||Component 2||Ratio of B/A||Volume of final Lemonade||Taste rating||Emoji|
|ml||ml||ml||(1 to 10)|
|1||200||300||2:3||500||6||This is too diluted!|
|2||225||275||2.25:2.75||500||9||Wow! One more please|
|3||250||250||1:1||500||7||This is OK. Could have been diluted more.|
|4||275||225||2.75:2.25||500||6||This is sour!|
|5||300||200||3:2||500||4||Oh! Too Sour. Can’t drink this.|
In the diagram below you can see, how the proportion changes but the total remains constant at 500.
With 225 ml of Concentrate and 275 ml of Water (Experiment 2), we get a taste rating of 9 for the Lemonade. Bingo! Youjust found the optimum proportions of the two components for the perfect lemonade. Cheers!
Well, you can start living the proverb “When life gives you Lemons, make lemonade”, now that you know the perfect formula.
Points to ponder upon
1. In a factorial design we denote the lower and upper level of a factor as “-” and “+” respectively. What is used to denote the upper and lower levels of the components in Mixture designs?
2. If you have noticed both the components (Lemonade concentrate and water) have the same ranges. Do you think it possible to perform mixture design when ranges are not same?
Let us see if you can come up with different possible applications of mixture designs in your domain/sector! Please feel free to post queries or comments.
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