I think Taguhi is the wrong approach here. I tend to find Taguchi is used incorrectly and people think they are using it right.

Someone correct me if I am wrong, but if you do not have the outer array of noise factors then you are missing a crucial piece of the taguchi design.

For your example lets say you have environment noise factors of ambient temperature and ambient pressure. You would want to run all 36 runs where the ambient temperature and ambient pressure are expected to reduce the response. Then run another set of 36 where the ambient temperature and pressure are expected to increase the response.

By comparing the results of the runs at the different noise levels you can discover combinations of inputs that minimise the variation introduced by noise on the environment.

That is the situation where I would use Taguchi, if it isn't being used there that way then you can get a more optimum design with the factorial designs.

You have several choices, if the factors with more than 2 levels are numeric, continous factors then I would use a 2 level factorial design. You do assume a linear model on the response. This is where entering centre points can be used to see if there is curvature.

If the factors with more than 2 levels are categorical you could use a general full factorial. This will let you set the numbers of levels on each factor.

Is this design a screening design or do you want to see all interactions?

Using a 2-level factorial design, a full design that see's all interactions needs 16 runs. If you want to check for curvature add 2 centre points. These will be replicated for each categorical factor, so if you have 1 categorical factor you get 4 centre points. Replicating the whole design increases the power of the design.

If this is a screening design consider the half fraction 2 level factorial, it needs 8 runs, but will confound the 2 way interactions with each other. It is resolution IV.

If you need 3 levels on a categorical factor, then a general full factorial will need 36 runs in total. 2*2*3*3, This design will allow you to see all interactions, but will be saturated without replicates, and you can probably just remove all the 3 and 4 way interactions to start with to get an error estimate. Unless you expect 3 way interactions to be present.

Now if you are only interested in a few interactions, and you know certain ones cannot exist you can use the D-optimal design tool to specify which interactions you want to see. You do need to calculate how many degrees of freedom you are using, but this is the number of levels on each factor -1 for the main effects.

eg: A has 2 levels, D has 3 levels, A needs 1 DF and D needs 2 DF

The interaction df are just multiplying the main effects df together so A*D would need 2 DF.

Count up the number of df you need, so A (2L) =1DF, B(2L) = 1DF, C(3L) = 2DF, D(3L) = 2DF. These need 6df, then look at interactions and lets say you want to see AB and CD. AB = 1df, CD = 4df. In total then you need 11df.

That would be 12 runs, if you left no runs to find an error with. Then you can ask how many runs you can add to find an error, as many as possible is better but you may get away with 3. So total is 15 runs to find a model of A B C D AB and CD.

This is just given as an example of how it can be used, and the designs you can pick. Compare the above to the L36 you are proposing, which doesn't see any interactions.

A 36 run design that really should be run at 2 different noise levels, giving 72 runs total.

This is why I always question the use of taguchi designs.

Each DOE has a particular are of use, so it would be interesting to know if there is a good reason why your company is using taguchi or wants you to use taguchi over the other options above.

Where I am talking about position of factors in the design I really need a link to some tables or diagrams. The taguchi designs are all generated in a specific structure, you can put factors into different positions of the table. I find them a bit confusing but the idea is that it is supposed to represent which positions are confounded, and you can choose where to put items to make interactions visible, or confound them with another factor.

For instance the diagram may show a line where it starts

Position 1 - - - - Position 3 - - - - position 2

If you put A in 1 and B in 2, position 3 would be your interaction AB. but if you put factor C in position 3 it is confounded with AB.

Does anyone have a good link for these diagrams? and a better explanation of them?