DOE viscosité : mélange ou facteurs continus ? Gestion des contraintes et additifs catégoriels

[#1]

Nous nous demandons si notre manière de poser le problème de DOE dans un custom design est appropriée.

Notre objectif est d’étudier l’influence sur la viscosité de trois familles d’additifs épaississants ainsi que leurs interactions.

Les additifs sont regroupés selon leur effet déclaré lorsqu’ils sont utilisés seuls : bas, moyen et haut gradient de cisaillement. Les interactions entre additifs agissant au même gradient ne nous intéressent pas, car nous n’en utiliserions qu’un seul à la fois.

Concernant les taux, nous avons initialement envisagé de traiter le plan comme un plan de mélange, mais nous avons finalement opté pour des facteurs continus. En effet, nous avons estimé que le reste de la formule influence également la viscosité et que la valeur de l’intercept n’est donc pas nulle.

Dans notre première version, comprenant les facteurs suivants, nous avons obtenu un plan de 90 expériences. Toutefois, nous avons constaté que combiner plusieurs additifs à leur taux maximal (0,4) conduirait à des formules trop épaisses pour être mesurées. Au moins 11 points avec les trois niveaux à 0,4 seraient ainsi non mesurables. De plus, il y aurait 11 points correspondant à des catégories différentes mais avec un taux à 0, soit 11 répétitions du produit sans additif.

Nous avons donc envisagé d’ajouter des contraintes.

 En termes de faisabilité, nous sommes désormais plus confiants quant à la possibilité de mesurer chaque point. Le produit sans additif est également mieux représenté.

Nous nous posons néanmoins plusieurs questions :

• Avons-nous eu raison d’imposer un minimum de 0,1 pour le taux ?
• Si oui, y a-t-il un intérêt à ajouter manuellement les 11 points avec un taux à 0, en réalisant 11 mesures supplémentaires du produit sans additif en plus du plan actuel ?
• Existe-t-il une meilleure manière de représenter ce problème dans sa globalité ?
• Si nous souhaitons introduire un nouvel épaississant dans l’une des catégories, comment procéder ? Nous n’avons pas trouvé de moyen de transformer un facteur catégoriel à trois niveaux en un facteur à quatre niveaux.

if relevant i'm on JMP 19.0.0 and this is the script for the final table:

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New Table( "Custom Design",
	Add Rows( 90 ),
	New Table Variable( "Design", "Custom Design" ),
	New Table Variable( "Criterion", "D Optimal" ),
	New Script(
		"Model",
		Fit Model(
			Effects(
				:low sheer additive, :mid sheer additive, :high sheer additive,
				:low sheer amount, :mid sheer amount, :high sheer amount,
				:low sheer additive * :mid sheer additive,
				:low sheer additive * :high sheer additive,
				:low sheer additive * :low sheer amount,
				:low sheer additive * :mid sheer amount,
				:low sheer additive * :high sheer amount,
				:mid sheer additive * :high sheer additive,
				:mid sheer additive * :low sheer amount,
				:mid sheer additive * :mid sheer amount,
				:mid sheer additive * :high sheer amount,
				:high sheer additive * :low sheer amount,
				:high sheer additive * :mid sheer amount,
				:high sheer additive * :high sheer amount,
				:low sheer amount * :mid sheer amount,
				:low sheer amount * :high sheer amount,
				:mid sheer amount * :high sheer amount,
				:low sheer additive * :mid sheer additive * :high sheer additive,
				:low sheer additive * :mid sheer additive * :low sheer amount,
				:low sheer additive * :mid sheer additive * :mid sheer amount,
				:low sheer additive * :mid sheer additive * :high sheer amount,
				:low sheer additive * :high sheer additive * :low sheer amount,
				:low sheer additive * :high sheer additive * :mid sheer amount,
				:low sheer additive * :high sheer additive * :high sheer amount,
				:low sheer additive * :low sheer amount * :mid sheer amount,
				:low sheer additive * :low sheer amount * :high sheer amount,
				:low sheer additive * :mid sheer amount * :high sheer amount,
				:mid sheer additive * :high sheer additive * :low sheer amount,
				:mid sheer additive * :high sheer additive * :mid sheer amount,
				:mid sheer additive * :high sheer additive * :high sheer amount,
				:mid sheer additive * :low sheer amount * :mid sheer amount,
				:mid sheer additive * :low sheer amount * :high sheer amount,
				:mid sheer additive * :mid sheer amount * :high sheer amount,
				:high sheer additive * :low sheer amount * :mid sheer amount,
				:high sheer additive * :low sheer amount * :high sheer amount,
				:high sheer additive * :mid sheer amount * :high sheer amount,
				:low sheer amount * :mid sheer amount * :high sheer amount
			),
			Y( :Y )
		)
	),
	New Script(
		"Evaluate Design",
		DOE(
			Evaluate Design,
			X(
				:low sheer additive, :mid sheer additive, :high sheer additive,
				:low sheer amount, :mid sheer amount, :high sheer amount
			)
		)
	),
	New Script(
		"Constraint",
		{1 * :low sheer amount + 1 * :mid sheer amount + 1 * :high sheer amount <=
		0.40000000000000002, 1 * :low sheer amount + 1 * :mid sheer amount + 1 *
		:high sheer amount >= 0.10000000000000001}
	),
	New Script(
		"DOE Dialog",
		DOE(
			Custom Design,
			{Add Response( Maximize, "Y", ., ., . ),
			Add Factor( Categorical, {"L1", "L2", "L3"}, "low sheer additive", 0 ),
			Add Factor( Categorical, {"L1", "L2", "L3"}, "mid sheer additive", 0 ),
			Add Factor( Categorical, {"L1", "L2"}, "high sheer additive", 0 ),
			Add Factor( Continuous, 0, 0.40000000000000002, "low sheer amount", 0 ),
			Add Factor( Continuous, 0, 0.40000000000000002, "mid sheer amount", 0 ),
			Add Factor( Continuous, 0, 0.40000000000000002, "high sheer amount", 0 ),
			Set Random Seed( 538775543 ), Number of Starts( 195 ),
			Add Constraint(
				[1 1 1 0.40000000000000002, -1 -1 -1 -0.10000000000000001]
			), 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( {6, 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( {1, 1}, {6, 1} ),
			Add Term( {2, 1}, {3, 1} ), Add Term( {2, 1}, {4, 1} ),
			Add Term( {2, 1}, {5, 1} ), Add Term( {2, 1}, {6, 1} ),
			Add Term( {3, 1}, {4, 1} ), Add Term( {3, 1}, {5, 1} ),
			Add Term( {3, 1}, {6, 1} ), Add Term( {4, 1}, {5, 1} ),
			Add Term( {4, 1}, {6, 1} ), Add Term( {5, 1}, {6, 1} ),
			Add Term( {1, 1}, {2, 1}, {3, 1} ), Add Term( {1, 1}, {2, 1}, {4, 1} ),
			Add Term( {1, 1}, {2, 1}, {5, 1} ), Add Term( {1, 1}, {2, 1}, {6, 1} ),
			Add Term( {1, 1}, {3, 1}, {4, 1} ), Add Term( {1, 1}, {3, 1}, {5, 1} ),
			Add Term( {1, 1}, {3, 1}, {6, 1} ), Add Term( {1, 1}, {4, 1}, {5, 1} ),
			Add Term( {1, 1}, {4, 1}, {6, 1} ), Add Term( {1, 1}, {5, 1}, {6, 1} ),
			Add Term( {2, 1}, {3, 1}, {4, 1} ), Add Term( {2, 1}, {3, 1}, {5, 1} ),
			Add Term( {2, 1}, {3, 1}, {6, 1} ), Add Term( {2, 1}, {4, 1}, {5, 1} ),
			Add Term( {2, 1}, {4, 1}, {6, 1} ), Add Term( {2, 1}, {5, 1}, {6, 1} ),
			Add Term( {3, 1}, {4, 1}, {5, 1} ), Add Term( {3, 1}, {4, 1}, {6, 1} ),
			Add Term( {3, 1}, {5, 1}, {6, 1} ), Add Term( {4, 1}, {5, 1}, {6, 1} ),
			Set Sample Size( 90 ), Simulate Responses( 0 ), Save X Matrix( 0 ),
			Make Design}
		)
	),
	New Column( "low sheer additive",
		Character,
		"Nominal",
		Set Property( "Value Order", {Custom Order( {"L1", "L2", "L3"} )} ),
		Set Property( "Design Role", DesignRole( Categorical ) ),
		Set Property( "Factor Changes", Easy ),
		Set Values(
			{"L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1",
			"L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1",
			"L1", "L1", "L1", "L1", "L1", "L2", "L2", "L2", "L2", "L2", "L2", "L2",
			"L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2",
			"L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2",
			"L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3",
			"L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3",
			"L3", "L3", "L3", "L3", "L3", "L3"}
		)
	),
	New Column( "mid sheer additive",
		Character,
		"Nominal",
		Set Property( "Value Order", {Custom Order( {"L1", "L2", "L3"} )} ),
		Set Property( "Design Role", DesignRole( Categorical ) ),
		Set Property( "Factor Changes", Easy ),
		Set Values(
			{"L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L2", "L2",
			"L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L3", "L3", "L3", "L3",
			"L3", "L3", "L3", "L3", "L3", "L1", "L1", "L1", "L1", "L1", "L1", "L1",
			"L1", "L1", "L1", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2",
			"L2", "L2", "L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3", "L3",
			"L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L1", "L2",
			"L2", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L3", "L3", "L3", "L3",
			"L3", "L3", "L3", "L3", "L3", "L3"}
		)
	),
	New Column( "high sheer additive",
		Character,
		"Nominal",
		Set Property( "Value Order", {Custom Order( {"L1", "L2"} )} ),
		Set Property( "Design Role", DesignRole( Categorical ) ),
		Set Property( "Factor Changes", Easy ),
		Set Values(
			{"L1", "L1", "L1", "L1", "L1", "L2", "L2", "L2", "L2", "L2", "L1", "L1",
			"L1", "L1", "L1", "L2", "L2", "L2", "L2", "L2", "L1", "L1", "L1", "L1",
			"L1", "L2", "L2", "L2", "L2", "L1", "L1", "L1", "L1", "L1", "L1", "L2",
			"L2", "L2", "L2", "L1", "L1", "L1", "L1", "L1", "L2", "L2", "L2", "L2",
			"L2", "L2", "L1", "L1", "L1", "L1", "L2", "L2", "L2", "L2", "L2", "L2",
			"L1", "L1", "L1", "L1", "L2", "L2", "L2", "L2", "L2", "L2", "L2", "L1",
			"L1", "L1", "L1", "L1", "L2", "L2", "L2", "L2", "L1", "L1", "L1", "L1",
			"L1", "L2", "L2", "L2", "L2", "L2"}
		)
	),
	New Column( "low sheer amount",
		Numeric,
		"Continuous",
		Format( "Best", 12 ),
		Set Property( "Coding", {0, 0.40000000000000002} ),
		Set Property( "Design Role", DesignRole( Continuous ) ),
		Set Property( "Factor Changes", Easy ),
		Set Values(
			[0, 0, 0.10000000000000005, 0.1501747410225249, 0.40000000000000002, 0,
			0, 0, 0.20000000000000001, 0.40000000000000002, 0, 0, 0,
			0.19999999999999998, 0.40000000000000002, 0, 0, 0.079999999999999988,
			0.20000000000000001, 0.40000000000000002, 0, 0, 0, 0.20000000000000001,
			0.40000000000000002, 0, 0, 0.06802583802886511, 0.40000000000000002, 0,
			0, 0, 0.15999999999999998, 0.20000000000000001, 0.39999999999999997, 0,
			0, 0.080000000000000029, 0.40000000000000002, 0, 0, 0.10000000000000003,
			0.13883780667167989, 0.40000000000000002, 0, 0, 0, 0.10000000000000001,
			0.20000000000000001, 0.39999999999999997, 0, 0, 0.070000000000000034,
			0.40000000000000002, 0, 0, 0, 0.19999999999999998, 0.20000000000000001,
			0.40000000000000002, 0, 0, 0.063073352199479821, 0.40000000000000002, 0,
			0, 0, 0.10000000000000001, 0.19999999999999996, 0.19999999999999998,
			0.40000000000000002, 0, 0, 0, 0.20000000000000004, 0.40000000000000002,
			0, 0, 0.050000000000000017, 0.40000000000000002, 0, 0, 0,
			0.19999999999999998, 0.40000000000000002, 0, 0, 0.090000000000000024,
			0.13715600638057818, 0.40000000000000002]
		)
	),
	New Column( "mid sheer amount",
		Numeric,
		"Continuous",
		Format( "Best", 12 ),
		Set Property( "Coding", {0, 0.40000000000000002} ),
		Set Property( "Design Role", DesignRole( Continuous ) ),
		Set Property( "Factor Changes", Easy ),
		Set Values(
			[0, 0.40000000000000002, 0, 0.10495025895846891, 0, 0,
			0.10000000000000003, 0.23999999999999996, 0.20000000000000001, 0, 0,
			0.10000000000000003, 0.23999999999999999, 0.19999999999999998, 0, 0,
			0.40000000000000002, 0.019999999999999962, 0, 0, 0, 0.16000000000000003,
			0.40000000000000002, 0, 0, 0, 0.39999999999999997, 0.015747420986803484,
			0, 0, 0.20000000000000004, 0.40000000000000002, 0, 0.20000000000000007,
			0, 0, 0.40000000000000002, 0, 0, 0, 0.40000000000000002, 0,
			0.11324531058065736, 0, 0, 0.19999999999999996, 0.39999999999999997, 0,
			0.20000000000000001, 0, 0, 0.40000000000000002, 0.029999999999999999, 0,
			0, 0.15999999999999998, 0.39999999999999997, 0, 0.19999999999999998, 0,
			0, 0.40000000000000002, 0.029216023765151894, 0, 0, 0.19999999999999998,
			0.40000000000000002, 0, 0.20000000000000004, 0, 0, 0,
			0.15999999999999998, 0.40000000000000002, 0, 0, 0, 0.40000000000000002,
			0.049999999999999989, 0, 0, 0.10000000000000001, 0.23999999999999999,
			0.19999999999999998, 0, 0, 0.40000000000000002, 0.0099999999999999811,
			0.10660542813245405, 0]
		)
	),
	New Column( "high sheer amount",
		Numeric,
		"Continuous",
		Format( "Best", 12 ),
		Set Property( "Coding", {0, 0.40000000000000002} ),
		Set Property( "Design Role", DesignRole( Continuous ) ),
		Set Property( "Factor Changes", Easy ),
		Set Values(
			[0.39999999999999997, 0, 0, 0.14487500001900624, 0, 0.40000000000000002,
			0, 0.16, 0, 0, 0.40000000000000002, 0, 0.16, 0, 0, 0.40000000000000002,
			0, 0, 0.20000000000000004, 0, 0.10000000000000005, 0.23999999999999999,
			0, 0.19999999999999998, 0, 0.39999999999999991, 0, 0.016226740984330218,
			0, 0.10000000000000003, 0.20000000000000004, 0, 0.24000000000000005, 0,
			0, 0.39999999999999997, 0, 0.020000000000000018, 0, 0.40000000000000002,
			0, 0, 0.14791688274766282, 0, 0.39999999999999991, 0.20000000000000001,
			0, 0, 0, 0, 0.40000000000000002, 0, 0, 0, 0.10000000000000003,
			0.24000000000000005, 0, 0.19999999999999996, 0, 0, 0.40000000000000002,
			0, 0.0077106240353676525, 0, 0.40000000000000002, 0.20000000000000004, 0,
			0, 0, 0.20000000000000004, 0, 0.10000000000000002, 0.24000000000000002,
			0, 0.20000000000000001, 0, 0.40000000000000002, 0, 0, 0,
			0.40000000000000002, 0, 0.16, 0, 0, 0.39999999999999997, 0, 0,
			0.15623856548696785, 0]
		)
	),
	New Column( "Y",
		Numeric,
		"Continuous",
		Format( "Best", 12 ),
		Set Property(
			"Response Limits",
			{Goal( Maximize ), Lower( . ), Upper( . ), Importance( . )}
		),
		Set Values(
			[., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., .,
			., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., .,
			., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., .,
			., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., .]
		)
	),
	New Column( "Run Order",
		Numeric,
		"Continuous",
		Format( "Best", 12 ),
		Set Values(
			[50, 66, 33, 43, 45, 62, 73, 76, 5, 20, 79, 17, 18, 60, 84, 23, 72, 13,
			38, 82, 4, 27, 6, 55, 58, 24, 3, 2, 80, 16, 32, 64, 28, 26, 53, 11, 47,
			29, 59, 7, 83, 37, 49, 21, 68, 25, 57, 40, 63, 89, 54, 10, 81, 67, 1, 48,
			77, 39, 44, 42, 22, 9, 36, 8, 52, 51, 87, 19, 30, 74, 86, 65, 46, 78, 31,
			75, 70, 71, 34, 61, 85, 69, 15, 88, 14, 90, 56, 12, 41, 35]
		)
	)
)

[#2 · Accepted solution]

Hi @Pierrick,

Welcome in the Community !

I'm not sure about the reason you gave to not use a Mixture design ? Mixture designs are used when the factors are components of a mixture and the total quantity should sum up to a specific value. The mixture design tells you about the effects of different ratios of the components. There is no assumption for mixture or factorial designs that the rest of the formulation should have an influence or not. If the rest of the formulation may influence the response, why not including these components as well ? Unless the rest of the formulation is some kind of solvent(s), so it's normal that viscosity may increase or decrease based on the concentration/dilution, and you would be more interested by the quantities of components rather than their ratios ?

In mixture designs, there is no intercept, as it is part of the main effects due to the linear constraint (all factors should sum up to a specific value). It is a different modeling scenario than having an intercept equal to zero.

I can't discuss the constraints, ranges, settings rationales of the design, as there is too little info and I'm not a domain expert of your topic to discuss if it sounds reasonable or not. I think there might be a mistake in your screenshot for the second constraint, as the inequality sign should be reversed for the lower sum limit (or opposite factors signs), but the JSL looks ok.

Why did you choose a minimum rate of 0,1 ? I would have let 0 to have all situations in the same design (absence and presence of components), without the need to add extra runs "manually" (and it ensured experiments are randomized and make the analysis easier). If you expect non-linear effects of the factors, just add quadratic effects in your model terms for the continuous quantities factors, so you'll end up with three levels for the factors, which avoids the trivial absence/presence design levels settings.

Adding levels in a categorical factor (or adding factors in an augmented design) requires a simple trick: add a new row in your table, with missing values everywhere except for the categorical factor you're interested to expand. Write the new level for this factor, and it should be considered when augmenting the design. See Add new levels to a design.

Hope this answer will help you,