Highly Fractional Factorial Designs - Revision history

Kate Racaza at 22:51, 10 August 2017

2017-08-10T22:51:12Z

Richard House: /* Example */

2017-08-10T17:23:16Z

Example

← Older revision		Revision as of 17:23, 10 August 2017
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	Based on this requirement, Taguchi OA L16(2^6*4^3) can be used since it can handle both 2 level and 4 level factors. It has 9 columns. The first 6 columns are used for 2 level factors and the last 3 columns are used for 4 level factors. We need to assign factor A and B to two of the first 6 columns, and assign factor C to one of the last 3 columns.		Based on this requirement, Taguchi OA L16(2^6*4^3) can be used since it can handle both 2 level and 4 level factors. It has 9 columns. The first 6 columns are used for 2 level factors and the last 3 columns are used for 4 level factors. We need to assign factor A and B to two of the first 6 columns, and assign factor C to one of the last 3 columns.

	In ~~DOE~~++, we can choose L16(2^6*4^3) in the Additional Settings.		In a Weibull++ DOE folio, we can choose L16(2^6*4^3) in the Additional Settings.

	[[Image:doe8_9.png\|center\|842px\|Selecting the Taguchi OA design type.\|link=]]		[[Image:doe8_9.png\|center\|842px\|Selecting the Taguchi OA design type.\|link=]]
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	[[Image:doe8_11.png\|center\|842px\|The interaction terms of interest have been specified.\|link=]]		[[Image:doe8_11.png\|center\|842px\|The interaction terms of interest have been specified.\|link=]]

	However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, DOE++ will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.		However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, the DOE folio will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.

	====Example====		====Example====

Richard House: /* Example */

2017-08-10T17:22:18Z

Example

← Older revision		Revision as of 17:22, 10 August 2017
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	And the results from DOE++ for the design are shown in the next figure.		And the results from the DOE folio for the design are shown in the next figure.

Richard House: /* Taguchi's Orthogonal Arrays */

2017-08-10T17:21:51Z

Taguchi's Orthogonal Arrays

← Older revision		Revision as of 17:21, 10 August 2017
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	Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments; they can also investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. As in the case of Placket-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in DOE++ are included in [[Taguchi_Orthogonal_Arrays\|Appendix D]].		Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments; they can also investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. As in the case of Placket-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in DOE++ are included in [[Taguchi_Orthogonal_Arrays\|Appendix D]].

	Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 <math>_{\text{III}}^{k-f}\,\!</math> designs. For example, consider the L4 array shown in figure (a) below. The L4 array is denoted as L4(2^3) in DOE++. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure (b) below shows the 2 <math>_{\text{III}}^{3-1}\,\!</math> design (defining relation <math>I=-ABC\,\!</math> ) which also requires four runs and can be used to estimate up to three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In figure (c) below, columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 <math>_{\text{III}}^{3-1}\,\!</math> design.		Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 <math>_{\text{III}}^{k-f}\,\!</math> designs. For example, consider the L4 array shown in figure (a) below. The L4 array is denoted as L4(2^3) in a DOE folio. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure (b) below shows the 2 <math>_{\text{III}}^{3-1}\,\!</math> design (defining relation <math>I=-ABC\,\!</math> ) which also requires four runs and can be used to estimate up to three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In figure (c) below, columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 <math>_{\text{III}}^{3-1}\,\!</math> design.

Richard House: /* Example */

2016-04-21T22:13:33Z

Example

← Older revision		Revision as of 22:13, 21 April 2016
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	[[Image:doe8_5.png\|center\|~~842px~~\|Design properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example\| example]].]]		[[Image:doe8_5.png\|center\|700px\|Design properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example\| example]].\|link=]]


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	[[Image:doe8_6.png\|center\|~~828px~~\|Factor properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example\| example]].]]		[[Image:doe8_6.png\|center\|700px\|Factor properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example\| example]].\|link=]]


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	[[Image:doe8_7.png\|center\|650px\|Experiment design for the [[Highly_Fractional_Factorial_Designs#Example\| example]].]]		[[Image:doe8_7.png\|center\|650px\|Experiment design for the [[Highly_Fractional_Factorial_Designs#Example\| example]].\|link=]]


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	[[Image:doe8_8_1.png\|center\|~~613px~~\|Results for the experiment in the example.]]		[[Image:doe8_8_1.png\|center\|700px\|Results for the experiment in the example.\|link=]]

Richard House: /* Example */

2016-04-21T21:34:32Z

Example

← Older revision		Revision as of 21:34, 21 April 2016
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	In DOE++, we can choose L16(2^6*4^3) in the Additional Settings.		In DOE++, we can choose L16(2^6*4^3) in the Additional Settings.

	[[Image:doe8_9.png\|center\|842px\|Selecting the Taguchi OA design type.]]		[[Image:doe8_9.png\|center\|842px\|Selecting the Taguchi OA design type.\|link=]]

	Click '''Specify Interaction Terms''' to specify the interaction terms that are of interest to the experimenters.		Click '''Specify Interaction Terms''' to specify the interaction terms that are of interest to the experimenters.

	[[Image:doe8_10.png\|center\|420px\|Specifying the interaction terms of interest.]]		[[Image:doe8_10.png\|center\|420px\|Specifying the interaction terms of interest.\|link=]]

	Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.		Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.

	[[Image:doe8_11.png\|center\|842px\|The interaction terms of interest have been specified.]]		[[Image:doe8_11.png\|center\|842px\|The interaction terms of interest have been specified.\|link=]]

	However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, DOE++ will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.		However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, DOE++ will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.

Richard House: /* Example */

2016-04-21T21:34:01Z

Example

← Older revision		Revision as of 21:34, 21 April 2016
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	Assume again we want to use Taguchi OA L16(2^6*4^3). Click '''Specify Interaction Terms''' as shown above to specify the interaction effects that you want to estimate in the experiment.		Assume again we want to use Taguchi OA L16(2^6*4^3). Click '''Specify Interaction Terms''' as shown above to specify the interaction effects that you want to estimate in the experiment.

	[[Image:doe8_12.png\|center\|420px\|Specifying the interaction terms of interest.]]		[[Image:doe8_12.png\|center\|420px\|Specifying the interaction terms of interest.\|link=]]

	When you click '''OK''', you will see a message warning that some of the specified interaction effects are aliased with main effects. These means that it is not possible to clearly estimate all the main effects and the specified interaction effects AC and BD.		When you click '''OK''', you will see a message warning that some of the specified interaction effects are aliased with main effects. These means that it is not possible to clearly estimate all the main effects and the specified interaction effects AC and BD.

Richard House: /* Example */

2016-01-06T22:34:48Z

Example

← Older revision		Revision as of 22:34, 6 January 2016
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	Assume again we want to use Taguchi OA L16(2^6*4^3). Click '''Specify Interaction Terms''' as shown above to specify the interaction effects that you want to estimate in the experiment.		Assume again we want to use Taguchi OA L16(2^6*4^3). Click '''Specify Interaction Terms''' as shown above to specify the interaction effects that you want to estimate in the experiment.

	[[Image:doe8_12.png~~\|thumb~~\|center\|420px\|Specifying the interaction terms of interest.]]		[[Image:doe8_12.png\|center\|420px\|Specifying the interaction terms of interest.]]

	When you click '''OK''', you will see a message warning that some of the specified interaction effects are aliased with main effects. These means that it is not possible to clearly estimate all the main effects and the specified interaction effects AC and BD.		When you click '''OK''', you will see a message warning that some of the specified interaction effects are aliased with main effects. These means that it is not possible to clearly estimate all the main effects and the specified interaction effects AC and BD.

Richard House: /* Example */

2016-01-06T22:34:17Z

Example

← Older revision		Revision as of 22:34, 6 January 2016
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	In DOE++, we can choose L16(2^6*4^3) in the Additional Settings.		In DOE++, we can choose L16(2^6*4^3) in the Additional Settings.

	[[Image:doe8_9.png~~\|thumb~~\|center\|842px\|Selecting the Taguchi OA design type.]]		[[Image:doe8_9.png\|center\|842px\|Selecting the Taguchi OA design type.]]

	Click '''Specify Interaction Terms''' to specify the interaction terms that are of interest to the experimenters.		Click '''Specify Interaction Terms''' to specify the interaction terms that are of interest to the experimenters.

	[[Image:doe8_10.png~~\|thumb~~\|center\|420px\|Specifying the interaction terms of interest.]]		[[Image:doe8_10.png\|center\|420px\|Specifying the interaction terms of interest.]]

	Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.		Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.

	[[Image:doe8_11.png~~\|thumb~~\|center\|842px\|The interaction terms of interest have been specified.]]		[[Image:doe8_11.png\|center\|842px\|The interaction terms of interest have been specified.]]

	However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, DOE++ will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.		However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, DOE++ will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.

Richard House: /* Taguchi's Orthogonal Arrays */

2016-01-06T22:33:19Z

Taguchi's Orthogonal Arrays

← Older revision		Revision as of 22:33, 6 January 2016
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	[[Image:doe8.3.png~~\|thumb~~\|center\|400px\|Taguchi's L4 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{3-1} \,\!</math> design with the defining relation <math>I=-ABC \,\!</math> and (c) marks the columns of the L4 array with the corresponding columns of the design in (b).]]		[[Image:doe8.3.png\|center\|400px\|Taguchi's L4 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{3-1} \,\!</math> design with the defining relation <math>I=-ABC \,\!</math> and (c) marks the columns of the L4 array with the corresponding columns of the design in (b).]]


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	[[Image:doe8.4.png~~\|thumb~~\|center\|400px\|Taguchi's L8 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{7-4} \,\!</math> design with the defining relation <math>I=ABD=ACE=BCF=ABCG \,\!</math> and (c) marks the columns of the L8 array with the corresponding columns of the design in (b).]]		[[Image:doe8.4.png\|center\|400px\|Taguchi's L8 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{7-4} \,\!</math> design with the defining relation <math>I=ABD=ACE=BCF=ABCG \,\!</math> and (c) marks the columns of the L8 array with the corresponding columns of the design in (b).]]


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	[[Image:doet8.2.png~~\|thumb~~\|center\|400px\|Alias relations for the L8 array.]]		[[Image:doet8.2.png\|center\|400px\|Alias relations for the L8 array.]]


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	[[Image:doet8.3.png~~\|thumb~~\|center\|639px\|Using Taguchi's L8 array to investigate factors affecting the surface finish of automobile brake drums.]]		[[Image:doet8.3.png\|center\|639px\|Using Taguchi's L8 array to investigate factors affecting the surface finish of automobile brake drums.]]


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	[[Image:doe8_5.png~~\|thumb~~\|center\|842px\|Design properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example\| example]].]]		[[Image:doe8_5.png\|center\|842px\|Design properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example\| example]].]]


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	[[Image:doe8_6.png~~\|thumb~~\|center\|828px\|Factor properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example\| example]].]]		[[Image:doe8_6.png\|center\|828px\|Factor properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example\| example]].]]


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	[[Image:doe8_7.png~~\|thumb~~\|center\|650px\|Experiment design for the [[Highly_Fractional_Factorial_Designs#Example\| example]].]]		[[Image:doe8_7.png\|center\|650px\|Experiment design for the [[Highly_Fractional_Factorial_Designs#Example\| example]].]]


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	[[Image:doe8_8_1.png~~\|thumb~~\|center\|613px\|Results for the experiment in the example.]]		[[Image:doe8_8_1.png\|center\|613px\|Results for the experiment in the example.]]

← Older revision		Revision as of 22:51, 10 August 2017
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	Due to the complex aliasing, Plackett-Burman designs involving a large number of factors should be used with care. Some of the Plackett-Burman designs available in DOE++ are included in [[Plackett-Burman_Designs\|Appendix C]].		Due to the complex aliasing, Plackett-Burman designs involving a large number of factors should be used with care. Some of the Plackett-Burman designs available in the DOE folio are included in [[Plackett-Burman_Designs\|Appendix C]].

	==Taguchi's Orthogonal Arrays==		==Taguchi's Orthogonal Arrays==

	Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments; they can also investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. As in the case of Placket-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in DOE++ are included in [[Taguchi_Orthogonal_Arrays\|Appendix D]].		Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments; they can also investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. As in the case of Placket-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in a DOE folio are included in [[Taguchi_Orthogonal_Arrays\|Appendix D]].

	Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 <math>_{\text{III}}^{k-f}\,\!</math> designs. For example, consider the L4 array shown in figure (a) below. The L4 array is denoted as L4(2^3) in a DOE folio. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure (b) below shows the 2 <math>_{\text{III}}^{3-1}\,\!</math> design (defining relation <math>I=-ABC\,\!</math> ) which also requires four runs and can be used to estimate up to three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In figure (c) below, columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 <math>_{\text{III}}^{3-1}\,\!</math> design.		Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 <math>_{\text{III}}^{k-f}\,\!</math> designs. For example, consider the L4 array shown in figure (a) below. The L4 array is denoted as L4(2^3) in a DOE folio. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure (b) below shows the 2 <math>_{\text{III}}^{3-1}\,\!</math> design (defining relation <math>I=-ABC\,\!</math> ) which also requires four runs and can be used to estimate up to three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In figure (c) below, columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 <math>_{\text{III}}^{3-1}\,\!</math> design.
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	=== Preferred Columns in Taguchi OA===		=== Preferred Columns in Taguchi OA===
	One of the difficulties of using Taguchi OA is to assign factors to the appropriate columns of the array. For example, take a simple Taguchi OA L8(2^7), which can be used for experiments with up to 7 factors. If you have only 3 factors, which 3 columns in this array should be used? DOE++ provides a simple utility to help users utilize Taguchi OA more effectively by assigning factors to the appropriate columns.		One of the difficulties of using Taguchi OA is to assign factors to the appropriate columns of the array. For example, take a simple Taguchi OA L8(2^7), which can be used for experiments with up to 7 factors. If you have only 3 factors, which 3 columns in this array should be used? The DOE folio provides a simple utility to help users utilize Taguchi OA more effectively by assigning factors to the appropriate columns.
	Let’s use Taguchi OA L8(2^7) as an example. The design table for this array is:		Let’s use Taguchi OA L8(2^7) as an example. The design table for this array is:

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	For experiments using the above design table, all the effects will be alias free. Therefore, this design is much better than the previous one which used column 1, 2, and 3 of L8(2^7). Although both designs have the same number of runs, more information can be obtained from this design since it is alias free.		For experiments using the above design table, all the effects will be alias free. Therefore, this design is much better than the previous one which used column 1, 2, and 3 of L8(2^7). Although both designs have the same number of runs, more information can be obtained from this design since it is alias free.

	Clearly, it is very important to assign factors to the right columns when applying Taguchi OA. DOE++ can help users automatically choose the right columns when the number of factors is less than the number of columns in a Taguchi OA. The selection is based on the specified model terms by users. Let’s use an example to explain this.		Clearly, it is very important to assign factors to the right columns when applying Taguchi OA. The DOE folio can help users automatically choose the right columns when the number of factors is less than the number of columns in a Taguchi OA. The selection is based on the specified model terms by users. Let’s use an example to explain this.

	====Example====		====Example====
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	[[Image:doe8_10.png\|center\|420px\|Specifying the interaction terms of interest.\|link=]]		[[Image:doe8_10.png\|center\|420px\|Specifying the interaction terms of interest.\|link=]]

	Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.		Based on the specified interaction effects, the DOE folio will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.

	[[Image:doe8_11.png\|center\|842px\|The interaction terms of interest have been specified.\|link=]]		[[Image:doe8_11.png\|center\|842px\|The interaction terms of interest have been specified.\|link=]]