Response Surface Methods for Optimization - Revision history

Kate Racaza: /* Box-Behnken Designs */

2017-08-10T22:51:58Z

Box-Behnken Designs

← Older revision		Revision as of 22:51, 10 August 2017
Line 199:		Line 199:


	The advantages of Box-Behnken designs include the fact that they are all spherical designs and require factors to be run at only three levels. The designs are also rotatable or nearly rotatable. Some of these designs also provide orthogonal blocking. Thus, if there is a need to separate runs into blocks for the Box-Behnken design, then designs are available that allow blocks to be used in such a way that the estimation of the regression parameters for the factor effects are not affected by the blocks. In other words, in these designs the block effects are orthogonal to the other factor effects. Yet another advantage of these designs is that there are no runs where all factors are at either the <math>+1\,\!</math> or <math>-1\,\!</math> levels. For example, in the figure below the representation of the Box-Behnken design for three factors clearly shows that there are no runs at the corner points. This could be advantageous when the corner points represent runs that are expensive or inconvenient because they lie at the end of the range of the factor levels. A few of the Box-Behnken designs available in DOE++ are presented in [[Box-Behnken_Designs\|Appendix F]].		The advantages of Box-Behnken designs include the fact that they are all spherical designs and require factors to be run at only three levels. The designs are also rotatable or nearly rotatable. Some of these designs also provide orthogonal blocking. Thus, if there is a need to separate runs into blocks for the Box-Behnken design, then designs are available that allow blocks to be used in such a way that the estimation of the regression parameters for the factor effects are not affected by the blocks. In other words, in these designs the block effects are orthogonal to the other factor effects. Yet another advantage of these designs is that there are no runs where all factors are at either the <math>+1\,\!</math> or <math>-1\,\!</math> levels. For example, in the figure below the representation of the Box-Behnken design for three factors clearly shows that there are no runs at the corner points. This could be advantageous when the corner points represent runs that are expensive or inconvenient because they lie at the end of the range of the factor levels. A few of the Box-Behnken designs available in a DOE folio are presented in [[Box-Behnken_Designs\|Appendix F]].

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

Richard House: /* Desirability Functions */

2017-08-10T19:46:23Z

Desirability Functions

← Older revision		Revision as of 19:46, 10 August 2017
Line 538:		Line 538:


	[[Image:doe9_27.png\|center\|650px\|Optimum solution from ~~DOE~~++ for the three responses of yield, cost, and pH.]]		[[Image:doe9_27.png\|center\|650px\|Optimum solution from Weibull++ for the three responses of yield, cost, and pH.]]

Richard House: /* Desirability Functions */

2017-08-10T17:19:17Z

Desirability Functions

← Older revision		Revision as of 17:19, 10 August 2017
Line 529:		Line 529:


	The objective of the experimenter is to find the settings of <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}}\,\!</math> such that the overall desirability, <math>D\,\!</math>, is maximum. In DOE++, the settings for the desirability functions for each of the three responses can be entered as shown in the next figure.		The objective of the experimenter is to find the settings of <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}}\,\!</math> such that the overall desirability, <math>D\,\!</math>, is maximum. In a DOE folio, the settings for the desirability functions for each of the three responses can be entered as shown in the next figure.


Line 535:		Line 535:


	Based on these settings, ~~DOE~~++ solves this optimization problem to obtain the following solution:		Based on these settings, the Weibull++ DOE folio solves this optimization problem to obtain the following solution:


Line 589:		Line 589:


	This is the same as the Global Desirability displayed by ~~DOE~~++ in the figure above. At times, a number of solutions may be obtained from DOE++, and it is up to the experimenter to choose the most feasible one.		This is the same as the Global Desirability displayed by the Weibull++ DOE folio in the figure above. At times, a number of solutions may be obtained from the DOE folio, and it is up to the experimenter to choose the most feasible one.

Richard House: /* Multiple Responses */

2017-08-10T17:18:03Z

Multiple Responses

← Older revision		Revision as of 17:18, 10 August 2017
Line 421:		Line 421:


	The problem in dealing with multiple responses is that now there might be conflicting objectives because of the different requirements of each of the responses. The experimenter needs to come up with a solution that satisfies each of the requirements as much as possible without compromising too much on any of the requirements. The approach used in ~~DOE~~++ to deal with optimization of multiple responses involves the use of desirability functions that are discussed next (for details see [[DOE_References\|[Derringer and Suich, 1980]]]).		The problem in dealing with multiple responses is that now there might be conflicting objectives because of the different requirements of each of the responses. The experimenter needs to come up with a solution that satisfies each of the requirements as much as possible without compromising too much on any of the requirements. The approach used in a Weibull++ DOE folio to deal with optimization of multiple responses involves the use of desirability functions that are discussed next (for details see [[DOE_References\|[Derringer and Suich, 1980]]]).

	===Desirability Functions===		===Desirability Functions===

Richard House: /* Example */

2017-08-10T17:17:16Z

Example

← Older revision		Revision as of 17:17, 10 August 2017
Line 406:		Line 406:


	In DOE++, the maximum response can be obtained by entering the required values as shown in the figure below. In the figure, the goal is to maximize the response and the limits of the search range for maximizing the response are entered as 90 and 100. The value of the maximum response and the corresponding values of the factors obtained are shown in the second figure following. These values match the values calculated in this example.		In a DOE folio, the maximum response can be obtained by entering the required values as shown in the figure below. In the figure, the goal is to maximize the response and the limits of the search range for maximizing the response are entered as 90 and 100. The value of the maximum response and the corresponding values of the factors obtained are shown in the second figure following. These values match the values calculated in this example.

Richard House: /* Example */

2017-08-10T17:16:45Z

Example

← Older revision		Revision as of 17:16, 10 August 2017
Line 208:		Line 208:


	Such a design can be set up in ~~DOE~~++ using the properties shown in the figure below.		Such a design can be set up in a Weibull++ DOE folio using the properties shown in the figure below.

Richard House: /* Example */

2017-08-10T17:14:51Z

Example

← Older revision		Revision as of 17:14, 10 August 2017
Line 30:		Line 30:


	In DOE++, this design can be set up using the properties shown next.		In a DOE folio, this design can be set up using the properties shown next.

Richard House: /* Desirability Functions */

2016-03-23T22:03:59Z

Desirability Functions

← Older revision		Revision as of 22:03, 23 March 2016
Line 532:		Line 532:


	[[Image:doe9_26.png\|center\|~~625px~~\|Optimization settings for the three responses of yield, cost, and pH.]]		[[Image:doe9_26.png\|center\|750px\|Optimization settings for the three responses of yield, cost, and pH.]]


Line 538:		Line 538:


	[[Image:doe9_27.png\|center\|~~800px~~\|Optimum solution from DOE++ for the three responses of yield, cost, and pH.]]		[[Image:doe9_27.png\|center\|650px\|Optimum solution from DOE++ for the three responses of yield, cost, and pH.]]

Richard House: /* Example */

2016-03-23T22:01:39Z

Example

← Older revision		Revision as of 22:01, 23 March 2016
Line 409:		Line 409:


	[[Image:doe9_21.png\|center\|~~625px~~\|Settings to obtain the maximum value of the response in the [[Response_Surface_Methods_for_Optimization#Example_3\| example]].]]		[[Image:doe9_21.png\|center\|543px\|Settings to obtain the maximum value of the response in the [[Response_Surface_Methods_for_Optimization#Example_3\| example]].\|link=]]


	[[Image:doe9_22.png\|center\|~~762px~~\|Plot of the maximum response in the [[Response_Surface_Methods_for_Optimization#Example_3\| example]] against the factors, temperature and time.]]		[[Image:doe9_22.png\|center\|650px\|Plot of the maximum response in the [[Response_Surface_Methods_for_Optimization#Example_3\| example]] against the factors, temperature and time.\|link=]]

	==Multiple Responses==		==Multiple Responses==

Richard House: /* Example */

2016-03-23T21:59:40Z

Example

← Older revision		Revision as of 21:59, 23 March 2016
Line 234:		Line 234:
	The response surface and the contour plot for this model, in terms of the actual variables, are shown in the below figures (a) and (b), respectively.		The response surface and the contour plot for this model, in terms of the actual variables, are shown in the below figures (a) and (b), respectively.

	[[Image:doe9_18_a.png\|center\|~~588px~~]]		[[Image:doe9_18_a.png\|center\|650px\|link=]]


	[[Image:doe9_18_b.png\|center\|~~588px~~\|Response surface and countour plot for the experiment in the [[Response_Surface_Methods_for_Optimization#Example_2\| example]].]]		[[Image:doe9_18_b.png\|center\|650px\|Response surface and countour plot for the experiment in the [[Response_Surface_Methods_for_Optimization#Example_2\| example]].\|link=]]

	==Analysis of the Second Order Model==		==Analysis of the Second Order Model==