Mixture Design - Revision history

Richard House: /* Example */

2017-08-10T17:00:35Z

Example

← Older revision		Revision as of 17:00, 10 August 2017
Line 828:		Line 828:
	The above table shows both process factors have significant effects on the texture of the patties. Since the model is pretty complicate, the best settings for the process variables and for components cannot be easily identified.		The above table shows both process factors have significant effects on the texture of the patties. Since the model is pretty complicate, the best settings for the process variables and for components cannot be easily identified.

	The optimization tool in DOE++ is used for the above model. The target texture value is <math>3\times {{10}^{3}}\,\!</math> grams with an acceptable range of <math>2.5-3.5\times {{10}^{3}}\,\!</math> grams.		The optimization tool in the DOE folio is used for the above model. The target texture value is <math>3\times {{10}^{3}}\,\!</math> grams with an acceptable range of <math>2.5-3.5\times {{10}^{3}}\,\!</math> grams.

	[[Image:doe_14.21.png\|600 px\|center]]		[[Image:doe_14.21.png\|600 px\|center]]

Richard House: /* Model with Process Variables */

2017-08-10T17:00:03Z

Model with Process Variables

← Older revision		Revision as of 17:00, 10 August 2017
Line 519:		Line 519:
	===Model with Process Variables===		===Model with Process Variables===

	In DOE++, regression models including both mixture components and process variables are available. For mixture components, we use L-pseudocomponent values, and for process variables coded values are used.		In a DOE folio, regression models including both mixture components and process variables are available. For mixture components, we use L-pseudocomponent values, and for process variables coded values are used.

	Assume a design has 3 mixture components and 2 process variables, as illustrated in the above figure. We can use the following models for them.		Assume a design has 3 mixture components and 2 process variables, as illustrated in the above figure. We can use the following models for them.

Richard House: /* Example */

2017-08-10T16:59:45Z

Example

← Older revision		Revision as of 16:59, 10 August 2017
Line 495:		Line 495:


	In order to find the best proportion of each ingredient, the optimization tool in ~~DOE~~++ can be utilized. Set the settings as		In order to find the best proportion of each ingredient, the optimization tool in the Weibull++ DOE folio can be utilized. Set the settings as

	[[Image:doe_14.18.png\|500 px\|center]]		[[Image:doe_14.18.png\|500 px\|center]]

Richard House: /* Example */

2017-08-10T16:58:58Z

Example

← Older revision		Revision as of 16:58, 10 August 2017
Line 321:		Line 321:


	Different blends of the three-juice recipe were evaluated by a panel. A value from 1 (extremely poor) to 9 (very good) is used for the response [John Cornell, page 74]. A {3, 2} simplex lattice design is used with one center point and three axial points. Three replicates were conducted for each ingredient combination. The settings for creating this design in DOE++ is		Different blends of the three-juice recipe were evaluated by a panel. A value from 1 (extremely poor) to 9 (very good) is used for the response [John Cornell, page 74]. A {3, 2} simplex lattice design is used with one center point and three axial points. Three replicates were conducted for each ingredient combination. The settings for creating this design in a DOE folio is

	[[Image:doe_14.14.png\|500 px\|center]]		[[Image:doe_14.14.png\|500 px\|center]]

Richard House: /* Check Constraint Consistency */

2017-08-10T16:58:18Z

Check Constraint Consistency

← Older revision		Revision as of 16:58, 10 August 2017
Line 289:		Line 289:
	Step 3: Whenever a bound is changed, restart from Step 1 to use the new bound to check if all the constraints are consistent. Repeat this until all the limits are consistent.		Step 3: Whenever a bound is changed, restart from Step 1 to use the new bound to check if all the constraints are consistent. Repeat this until all the limits are consistent.

	For extreme vertex design where linear constraints are allowed, DOE++ will give a warning and stop creating the design if inconsistent linear combination constraints are found. No adjustment will be conducted for linear constraints.		For extreme vertex design where linear constraints are allowed, the DOE folio will give a warning and stop creating the design if inconsistent linear combination constraints are found. No adjustment will be conducted for linear constraints.

	===Response Trace Plot===		===Response Trace Plot===

Richard House: /* L-Pseudocomponent, Proportion, and Actual Values */

2017-08-10T16:57:54Z

L-Pseudocomponent, Proportion, and Actual Values

← Older revision		Revision as of 16:57, 10 August 2017
Line 232:		Line 232:
	===L-Pseudocomponent, Proportion, and Actual Values===		===L-Pseudocomponent, Proportion, and Actual Values===

	In mixture designs, the total amount of the mixture is usually given. For example, we can make either a one-pound or a two-pound cake. Regardless of whether the cake is one or two pounds, the proportion of each ingredient is the same. When the total amount is given, the upper and lower limits for each ingredient are usually given in amounts, which is easier for the experimenter to understand. Of course, if the limits or other constraints are given in terms of proportions, these proportions need be converted to the real amount values when conducting the experiment. To keep everything consistent, all the constraints in DOE++ are treated as amounts.		In mixture designs, the total amount of the mixture is usually given. For example, we can make either a one-pound or a two-pound cake. Regardless of whether the cake is one or two pounds, the proportion of each ingredient is the same. When the total amount is given, the upper and lower limits for each ingredient are usually given in amounts, which is easier for the experimenter to understand. Of course, if the limits or other constraints are given in terms of proportions, these proportions need be converted to the real amount values when conducting the experiment. To keep everything consistent, all the constraints in a DOE folio are treated as amounts.

	In regular factorial design and response surface methods, the regression model is calculated using coded values. Coded values scale all the factors to the same magnitude, which makes the analysis much easier and reduces convergence error. Similarly, the analysis in mixture design is conducted using the so-called L-pseudocomponent value. L-pseudocomponent values scale all the components' values within 0 and 1. In DOE++ all the designs and calculations for mixture factors are based on L-pseudocomponent values. The relationship between L-pseudocomponent values, proportions and actual amounts are explained next.		In regular factorial design and response surface methods, the regression model is calculated using coded values. Coded values scale all the factors to the same magnitude, which makes the analysis much easier and reduces convergence error. Similarly, the analysis in mixture design is conducted using the so-called L-pseudocomponent value. L-pseudocomponent values scale all the components' values within 0 and 1. In a DOE folio all the designs and calculations for mixture factors are based on L-pseudocomponent values. The relationship between L-pseudocomponent values, proportions and actual amounts are explained next.

	====Example for L-Pseudocomponent Value====		====Example for L-Pseudocomponent Value====

Richard House: /* Extreme Vertex Design */

2017-08-10T16:57:16Z

Extreme Vertex Design

← Older revision		Revision as of 16:57, 10 August 2017
Line 136:		Line 136:


	The CONSIM method described in [Snee 1979] is used in ~~DOE~~++ to check the consistency of all the constraints and to get the vertices defined by them.		The CONSIM method described in [Snee 1979] is used in a Weibull++ DOE folio to check the consistency of all the constraints and to get the vertices defined by them.

	Extreme vertex designs by default use the vertices at the boundary. Additional points such as the centroid of spaces of different dimensions, axial points and the overall center point can be added. In extreme vertex designs, axial points are between the overall center point and the vertices. For the above example, if the axial points and the overall center point are added, then all the runs in the experiment will be:		Extreme vertex designs by default use the vertices at the boundary. Additional points such as the centroid of spaces of different dimensions, axial points and the overall center point can be added. In extreme vertex designs, axial points are between the overall center point and the vertices. For the above example, if the axial points and the overall center point are added, then all the runs in the experiment will be:

Richard House: /* Simplex Axial Design */

2017-08-10T16:56:16Z

Simplex Axial Design

← Older revision		Revision as of 16:56, 10 August 2017
Line 113:		Line 113:
	Points 4, 5 and 6 are the axial blends.		Points 4, 5 and 6 are the axial blends.

	By default, a simple axial design in DOE++ only has vertices, axial blends, centroid of the constraint planes and the overall centroid. For a design with q components, constraint plane centroids are the center points of dimension of q-1 space. One component is 0, and the remaining components have the same values for the center points of constraint planes. The number of the constraint plane centroids is the number of components q. The total number of runs in a simple axial design will be 3q+1. They are q vertex runs, q centroids of constraint planes, q axial blends and 1 overall centroid.		By default, a simple axial design in a DOE folio only has vertices, axial blends, centroid of the constraint planes and the overall centroid. For a design with q components, constraint plane centroids are the center points of dimension of q-1 space. One component is 0, and the remaining components have the same values for the center points of constraint planes. The number of the constraint plane centroids is the number of components q. The total number of runs in a simple axial design will be 3q+1. They are q vertex runs, q centroids of constraint planes, q axial blends and 1 overall centroid.

	A simplex axial design for 3 components has 10 points as given below.		A simplex axial design for 3 components has 10 points as given below.

Richard House: /* Mixture Design with Process Variables */

2016-02-12T20:04:23Z

Mixture Design with Process Variables

← Older revision		Revision as of 20:04, 12 February 2016
Line 513:		Line 513:
	For example, a {3, 2} simplex lattice design is used for a mixture with 3 components. Together with the center point, it has total of 7 runs or 7 different ingredient combinations. Assume 2 process variables are potentially important and a two level factorial design is used for them. It has a total of 4 combinations for these 2 process variables. If the 7 different mixtures are made under each of the 4 process variable combinations, then the experiment has a total of 28 runs. This is illustrated in the figure below.		For example, a {3, 2} simplex lattice design is used for a mixture with 3 components. Together with the center point, it has total of 7 runs or 7 different ingredient combinations. Assume 2 process variables are potentially important and a two level factorial design is used for them. It has a total of 4 combinations for these 2 process variables. If the 7 different mixtures are made under each of the 4 process variable combinations, then the experiment has a total of 28 runs. This is illustrated in the figure below.

	[[Image:doe_14.20.png\|500 px\|center]]		[[Image:doe_14.20.png\|500 px\|center\|link=]]

	Of course, if it is possible, all the 28 experiments should be conducted in a random order.		Of course, if it is possible, all the 28 experiments should be conducted in a random order.

Richard House: /* Extreme Vertex Design */

2016-02-12T19:59:17Z

Extreme Vertex Design

← Older revision		Revision as of 19:59, 12 February 2016
Line 133:		Line 133:
	Then the feasible region is defined by the six points in the following simplex plot. To meet the above constraints, all the runs conducted in the experiment should be in the feasible region or on its boundary.		Then the feasible region is defined by the six points in the following simplex plot. To meet the above constraints, all the runs conducted in the experiment should be in the feasible region or on its boundary.

	[[Image:doe_14.7.png\|500 px\|center]]		[[Image:doe_14.7.png\|500 px\|center\|link=]]


Line 140:		Line 140:
	Extreme vertex designs by default use the vertices at the boundary. Additional points such as the centroid of spaces of different dimensions, axial points and the overall center point can be added. In extreme vertex designs, axial points are between the overall center point and the vertices. For the above example, if the axial points and the overall center point are added, then all the runs in the experiment will be:		Extreme vertex designs by default use the vertices at the boundary. Additional points such as the centroid of spaces of different dimensions, axial points and the overall center point can be added. In extreme vertex designs, axial points are between the overall center point and the vertices. For the above example, if the axial points and the overall center point are added, then all the runs in the experiment will be:

	[[Image:doe_14.8.png\|500 px\|center]]		[[Image:doe_14.8.png\|500 px\|center\|link=]]

	Point 0 in the center of the feasible region is the overall centroid. The other red points are the axial points. They are at the middle of the lines connecting the center point with the vertices.		Point 0 in the center of the feasible region is the overall centroid. The other red points are the axial points. They are at the middle of the lines connecting the center point with the vertices.