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DOE with lots of constraints cannot be set up

martindemel

My customer has the following situation to set up a DOE:

3 components of a mixture (S, X, M, could be treated as mixture or continous) plus a filler component (W), as well two suppliers for each of the ingredients S and X, called OS and OX (each categorical with two levels).

The difficulties raise with the constraints, first the ranges:

Range S 0 -0.04

Range X 0-0.006

Range M 0-0.07

Range W 0.884-1

OS L1, L2

OX L1, L2

Or as continuous, where the range expresses the percentage and the filler is not necessary anymore as they are not interested at all in the effect of the filler itself

Range S 0 -4 [%]

Range X 0-0.6 [%]

Range M 0-7 [%]

OS L1, L2

OX L1, L2

In addition there are the following requirements:

S+X >0, so that S and X cannot be 0 at the same time
If X==0, S must be >0.5% (but if X!=0, S can be 0)
If S ==0, X must be >0.05% (but if S!=0, X can be 0)
If S==0, OS is not allowed/makes no sense
If X==0, OX is not allowed/makes no sense

The issue is that there cannot be constraints with the mixture factors other than the ranges or a simple linear constraint to ensure S+X>0. And in the continuous case we could not manage to set the constraints in a way to fulfill all constraints. Not with linear ones, nor with disallowed combinations, and a mixture of both is not possible either.

Any ideas how I could create a design with the above settings?

Attached some things I tried so far. In one I have created a design close to the requirements but thought it’s not balanced enough as there is no run where S or X alone occurs. Please see the three scripts to illustrate the behaviours, and especial the one with the 3 level categorical, as there are multiple designs included. Factors are saved in tables to your convenience

Thanks for your support,

Martin

Script_DOE_Mixture_simpleLinConstr.jsl

Factors_continuous.jmp

Skript_DOE_Cont_DisallCombi.jsl

Skript_DOE_Cont_DisallCombi_3LevelCat.jsl

Factors_mixture.jmp

Factors_continuous_Cat3Level_SplitPlot.jmp

Factors_continuous_Cat3Level.jmp

Accepted answers

Ryan_Lekivetz

Oops, I hadn't looked closely enough before... I think there's a couple of things going on here - first, the discrete numeric terms for higher orders are being added as If Possible (these have to be removed manually). The bigger issue is the S*OS interaction term, as it can't be estimated with the disallowed combinations. The designer should have caught this, but didn't, I'll have to look into it further. Running the below and removing those if possible terms, things should look better. I'll see if we can fix up these issues in a future release.

You could try removing that interaction and returning to the continuous route... but instead of using S==0, you might want something like S < 0.01. (n.b. this can also be an issue with discrete numeric, due to the storage of doubles, so even then I've needed to use abs(S)<.0001 instead of S==0).

Cheers,

Ryan

DOE(Custom Design,

{Add Response( Maximize, "Y", ., ., . ),

Add Factor( Discrete Numeric, {0, 1, 2, 3, 4}, "S", 0 ),

Add Factor( Discrete Numeric, {0, 0.15, 0.3, 0.45, 0.6}, "X", 0 ),

Add Factor( Discrete Numeric, {0, 1.75, 3.5, 5.25, 7}, "M", 0 ),

Add Factor( Categorical, {"L1", "L2", "None"}, "OS", 0 ),

Add Factor( Categorical, {"L1", "L2"}, "OX", 1 ), Set Random Seed( 891554152 ),

Number of Starts( 1000 ), Add Term( {1, 0} ), Add Term( {1, 1} ),

Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ), Add Term( {5, 1} ),

Add Term( {1, 1}, {2, 1} ), Add Term( {1, 1}, {3, 1} ),

Add Term( {1, 1}, {5, 1} ), Add Term( {2, 1}, {3, 1} ),

Add Term( {2, 1}, {4, 1} ), Add Term( {2, 1}, {5, 1} ),

Add Term( {3, 1}, {4, 1} ), Add Term( {3, 1}, {5, 1} ),

Add Term( {4, 1}, {5, 1} ), Add Alias Term( {1, 1}, {4, 1} ),

Set N Whole Plots( 4 ), Set Sample Size( 28 ),

0 < S <= 0.5 | S > 0 & OS == 3 | S == 0 & (OS == 1 | OS == 2))});

All comments

Phil_Kay

Hi Martin,

Have you tried using Disallowed Combinations Script? This gives you more flexibility for defining constraints.

http://www.jmp.com/support/help/Define_Factor_Constraints.shtml#205355

Regards,

Phil

Ryan_Lekivetz

I think you're running into problems using the disallowed combinations with equalities on continuous factors. In particular, when using X==0 or S==0, something that's really close to 0 (even displayed as 0) can end up being used. Have you tried with discrete numeric for the continuous? Using one of your last scripts, something like this:

DOE(

{Add Response( Maximize, "Y", ., ., . ),

//Add Factor( Continuous, 0, 4, "S", 0 ),

//Add Factor( Continuous, 0, 0.6, "X", 0 ), Add Factor( Continuous, 0, 7, "M", 0 ),

Add Factor( Discrete Numeric, {0, 1, 2, 3, 4}, "S", 0),

Add Factor( Discrete Numeric, {0, .15, .3, .45, .6}, "X", 0),

Add Factor( Discrete Numeric, {0, 1.75, 3.5, 5.25, 7}, "M", 0),

Add Factor( Categorical, {"L1", "L2", "None"}, "OS", 0 ),

Add Factor( Categorical, {"L1", "L2"}, "OX", 1 ), Set Random Seed( 513965188 ),

Number of Starts( 1000 ), Add Term( {1, 0} ), Add Term( {1, 1} ),

Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ), Add Term( {5, 1} ),

Add Term( {1, 1}, {2, 1} ), Add Term( {1, 1}, {3, 1} ),

Add Term( {1, 1}, {4, 1} ), Add Term( {1, 1}, {5, 1} ),

Add Term( {2, 1}, {3, 1} ), Add Term( {2, 1}, {4, 1} ),

Add Term( {2, 1}, {5, 1} ), Add Term( {3, 1}, {4, 1} ),

Add Term( {3, 1}, {5, 1} ), Add Term( {4, 1}, {5, 1} ), Set N Whole Plots( 4 ),

Set Sample Size( 28 ), Disallowed Combinations(

X < 0.05 | 0 < S <= 0.5 | S > 0 & OS == 3 | S == 0 & (OS == 1 | OS == 2)

martindemel

Hi Ryan,

no, I haven't. It seems to work, I will have to check with the customer if that's good enough and ok to use the discrete values.

One thing I got looking at the design evaluation and power analysis, I cannot estimate the power, but I guess this is due to the discrete numeric, right? The colormap looks quite ok to me :

The Fraction of Design Space Plot looks strange after 0,6:

Is that because of the constraints set?

Thanks,

Martin

Ryan_Lekivetz

Cheers,

Ryan

DOE(Custom Design,

{Add Response( Maximize, "Y", ., ., . ),

Add Factor( Discrete Numeric, {0, 1, 2, 3, 4}, "S", 0 ),

Add Factor( Discrete Numeric, {0, 0.15, 0.3, 0.45, 0.6}, "X", 0 ),

Add Factor( Discrete Numeric, {0, 1.75, 3.5, 5.25, 7}, "M", 0 ),

Add Factor( Categorical, {"L1", "L2", "None"}, "OS", 0 ),

Add Factor( Categorical, {"L1", "L2"}, "OX", 1 ), Set Random Seed( 891554152 ),

Number of Starts( 1000 ), Add Term( {1, 0} ), Add Term( {1, 1} ),

Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ), Add Term( {5, 1} ),

Add Term( {1, 1}, {2, 1} ), Add Term( {1, 1}, {3, 1} ),

Add Term( {1, 1}, {5, 1} ), Add Term( {2, 1}, {3, 1} ),

Add Term( {2, 1}, {4, 1} ), Add Term( {2, 1}, {5, 1} ),

Add Term( {3, 1}, {4, 1} ), Add Term( {3, 1}, {5, 1} ),

Add Term( {4, 1}, {5, 1} ), Add Alias Term( {1, 1}, {4, 1} ),

Set N Whole Plots( 4 ), Set Sample Size( 28 ),

0 < S <= 0.5 | S > 0 & OS == 3 | S == 0 & (OS == 1 | OS == 2))});

martindemel

Hi Ryan,

please read first the second reply as that one probably makes this one obsolete.

So here is my final script which has decent power and prediction variance if one adds more runs to it (e.g. 36):

// 24 runs

DOE(

Custom Design,

{Add Response( Maximize, "Y", ., ., . ),

Add Factor( Discrete Numeric, {0, 1, 2, 3, 4}, "S", 0 ),

Add Factor( Discrete Numeric, {0, 0.15, 0.3, 0.45, 0.6}, "X", 0 ),

Add Factor( Discrete Numeric, {0, 1.75, 3.5, 5.25, 7}, "M", 0 ),

Add Factor( Categorical, {"L1", "L2", "None"}, "OS", 0 ),

Add Factor( Categorical, {"L1", "L2"}, "OX", 1 ), Set Random Seed( 984492505 ),

Number of Starts( 1000 ), Add Term( {1, 0} ), Add Term( {1, 1} ),

Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ), Add Term( {5, 1} ),

Add Term( {1, 1}, {2, 1} ), Add Term( {1, 1}, {3, 1} ),

Add Term( {1, 1}, {5, 1} ), Add Term( {2, 1}, {3, 1} ),

Add Term( {2, 1}, {4, 1} ), Add Term( {2, 1}, {5, 1} ),

Add Term( {3, 1}, {4, 1} ), Add Term( {3, 1}, {5, 1} ),

Add Term( {4, 1}, {5, 1} ), Optimize Number of Whole Plots,

Set Sample Size( 24 ), Disallowed Combinations(

0 < S <= 0.5 | S > 0 & OS == 3 | S == 0 & (OS == 1 | OS == 2) | X < 0.05

)}

);

// 36 runs, has much more power and less prediction variance

DOE(

Custom Design,

{Add Response( Maximize, "Y", ., ., . ),

Add Factor( Discrete Numeric, {0, 1, 2, 3, 4}, "S", 0 ),

Add Factor( Discrete Numeric, {0, 0.15, 0.3, 0.45, 0.6}, "X", 0 ),

Add Factor( Discrete Numeric, {0, 1.75, 3.5, 5.25, 7}, "M", 0 ),

Add Factor( Categorical, {"L1", "L2", "None"}, "OS", 0 ),

Add Factor( Categorical, {"L1", "L2"}, "OX", 1 ), Set Random Seed( 1173606031 ),

Number of Starts( 1000 ), Add Term( {1, 0} ), Add Term( {1, 1} ),

Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ), Add Term( {5, 1} ),

Add Term( {1, 1}, {2, 1} ), Add Term( {1, 1}, {3, 1} ),

Add Term( {1, 1}, {5, 1} ), Add Term( {2, 1}, {3, 1} ),

Add Term( {2, 1}, {4, 1} ), Add Term( {2, 1}, {5, 1} ),

Add Term( {3, 1}, {4, 1} ), Add Term( {3, 1}, {5, 1} ),

Add Term( {4, 1}, {5, 1} ), Add Alias Term( {1, 1}, {4, 1} ),

Set N Whole Plots( 4 ), Set Sample Size( 36 ),

Disallowed Combinations(

0 < S <= 0.5 | S > 0 & OS == 3 | S == 0 & (OS == 1 | OS == 2) | X < 0.05

)}

);

As you suggested I removed manually the higher order terms as well the S*OS interaction. Thanks for that.

I have two add-on questions:

1) I just got a reply. with the Split Plot set up as it is we are constrained to have X always in. But they would like to have also the estimation for X == 0 but S >0.5. I tried to set up a new design with OX having 3 levels. I'm having troubles to force OX "None" when X == 0 , as well if X > 0 forcing OX != "None" (3) and still getting power calculations, So I removed the interaction effect X*OX and OS*OX. Here is my design I finally came up with:

DOE(

Custom Design,

{Add Response( Maximize, "Y", ., ., . ),

Add Factor( Discrete Numeric, {0, 1, 2, 3, 4}, "S", 0 ),

Add Factor( Discrete Numeric, {0, 0.15, 0.3, 0.45, 0.6}, "X", 0 ),

Add Factor( Discrete Numeric, {0, 1.75, 3.5, 5.25, 7}, "M", 0 ),

Add Factor( Categorical, {"L1", "L2", "None"}, "OS", 0 ),

Add Factor( Categorical, {"L1", "L2", "None"}, "OX", 1 ),

Set Random Seed( 1317349002 ), Number of Starts( 1000 ), Add Term( {1, 0} ),

Add Term( {1, 1} ), Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ),

Add Term( {5, 1} ), Add Term( {1, 1}, {2, 1} ), Add Term( {1, 1}, {3, 1} ),

Add Term( {1, 1}, {5, 1} ), Add Term( {2, 1}, {3, 1} ),

Add Term( {2, 1}, {4, 1} ), Add Term( {3, 1}, {4, 1} ),

Add Term( {3, 1}, {5, 1} ), Add Alias Term( {1, 1}, {4, 1} ),

Set N Whole Plots( 6 ), Set Sample Size( 36 ),

Disallowed Combinations(

0 < S <= 0.5 | S > 0 & OS == 3 | S == 0 & (OS == 1 | OS == 2) | X > 0 & OX

== 3 | X == 0 & (OX == 1 | OX == 2)

)}

);

Is there anything that should me keep away from using this design? Except the fraction of design space plot all seems to be ok. And even the FDS plot is not toooo bad, I think. Would you change anything.

2) Would it make sense to keep the initial split plot design and do another for S/OS instead of X/OX to get an estimate also for the X==0 option? I donât think itâs a good idea to combine both results to answer the question if there are significant differences in the suppliers for the different combinations on a whole, do you agree?