Home > Articles > Programming

  • Print
  • + Share This
This chapter is from the book

Alternative Approaches to Robust Design

Any design that includes replicate runs at each combination of settings of the controllable inputs can be used as a robust design, even if the noise variables creating the variation among the replicate runs are unknown. The analysis of results is similar to that just described for the data in Table 31-5. The mean response and the variation in response, evaluated using the logarithm of the standard deviation of the response values, are calculated for each combination of settings of the controllable inputs. The type of design used for the controllable inputs will determine the degree of detail provided to describe the individual and joint influences of the controllable inputs on the average response and variation in response.

Borror, Montgomery, and Myers (2002)3 proposed the use of response surface designs as robust designs in which both the controllable inputs and the noise variables are treated as factors. This approach requires the noise variables to be controlled at specified settings, which may be possible in a special environment such as a laboratory. The response surface approach is now illustrated using a modified version of the candy wrapper example employed in the preceding section of this chapter.

Robust Design with Response Surface Techniques

This illustration involves three controllable inputs over the ranges shown in Table 31-6 and two noise variables over the ranges shown in Table 31-7.

Table 31-6. Controllable Inputs for Response Surface Design

Controllable Input

Range of Interest

% Additive

5%

to

25%

Temperature

160 °F

to

200 °F

Belt Speed

50 ft/min

to

70 ft/min

Table 31-7. Noise Variables for Response Surface Design

Noise Variable

Range of Interest

Relative Humidity

50%

to

70%

Ambient Particulate Level

1%

to

5%

The performance variable is, again, film thickness. The objective is to achieve a film thickness of 1.00 with minimum variation.

As described in Chapter 30, Minitab offers two types of response surface designs, Central Composite Designs and Box-Behnken Designs. For five factors (three controllable inputs and two noise variables), the 32-run Central Composite Design shown in Table 31-8 is the more economical choice.

Table 31-8. 32-Run Central Composite Design

% Additive

Temperature

Belt Speed

Relative Humidity

Particulate Level

10

170

55

55.0

4

20

170

55

55.0

2

10

190

55

55.0

2

20

190

55

55.0

4

10

170

65

55.0

2

20

170

65

55.0

4

10

190

65

55.0

4

20

190

65

55.0

2

10

170

55

70.0

2

20

170

55

70.0

4

10

190

55

70.0

4

20

190

55

70.0

2

10

170

65

70.0

4

20

170

65

70.0

2

10

190

65

70.0

2

20

190

65

70.0

4

5

180

60

62.5

3

25

180

60

62.5

3

15

160

60

62.5

3

15

200

60

62.5

3

15

180

50

62.5

3

15

180

70

62.5

3

15

180

60

47.5

3

15

180

60

77.5

3

15

180

60

62.5

1

15

180

60

62.5

5

15

180

60

62.5

3

15

180

60

62.5

3

15

180

60

62.5

3

15

180

60

62.5

3

15

180

60

62.5

3

15

180

60

62.5

3

Additional economy can be realized by omitting the axial points (shown in italics in Table 31-8) for the two noise variables. The effect of these deletions is to give up information about the quadratic effects of the noise variables, which is not a serious loss. The resulting design is shown in Table 31-9 with the measured film thickness for each run. The order of execution of the runs is randomized.

Table 31-9. Response Surface Design with Experimental Results

%Additive

Temp

Belt Speed

Rel.Humidity

Particulate

Thickness

10

170

55

55.0

4

1.03

20

170

55

55.0

2

0.87

10

190

55

55.0

2

1.26

20

190

55

55.0

4

1.08

10

170

65

55.0

2

0.83

20

170

65

55.0

4

1.47

10

190

65

55.0

4

0.94

20

190

65

55.0

2

1.03

10

170

55

70.0

2

0.75

20

170

55

70.0

4

0.78

10

190

55

70.0

4

1.24

20

190

55

70.0

2

1.31

10

170

65

70.0

4

1.11

20

170

65

70.0

2

0.44

10

190

65

70.0

2

1.06

20

190

65

70.0

4

0.93

5

180

60

62.5

3

1.15

25

180

60

62.5

3

1.17

15

160

60

62.5

3

0.45

15

200

60

62.5

3

0.95

15

180

50

62.5

3

0.99

15

180

70

62.5

3

0.91

15

180

60

62.5

3

1.15

15

180

60

62.5

3

1.16

15

180

60

62.5

3

1.14

15

180

60

62.5

3

1.21

15

180

60

62.5

3

1.18

15

180

60

62.5

3

1.11

The settings of the five factors are coded as follows:

A

=

(% Additive - 15)/5

B

=

(Temperature - 180)/10

C

=

(Belt Speed - 60)/5

Z1

=

(Relative Humidity - 62.5)/7.5

Z2

=

(Ambient Particulate Level - 3)/1

and a full quadratic equation (without quadratic terms in the two noise variables) is fitted to the coded data. After deleting nonsignificant terms, the fitted equation for measured film thickness, expressed in the coded units of the five factors, is:

Predicted Thickness

=

1.165 - 0.011 A + 0.107 B - 0.028 C - 0.113 B2 - 0.051 C2 - 0.084 BC - 0.056 Z1 + 0.064 Z2 - 0.068 AZ1 + 0.084 B Z1 - 0.123 B Z2 - 0.036 C Z1 + 0.072 C Z2

Minitab's output for this fitted model is shown in Figure 31-6.

Figure 31-6. Analysis of Data in Table 31-9

Response Surface Regression: thickness versus A, B, C, Z1, Z2

The analysis was done using coded units.

Estimated Regression Coefficients for thickness

Term

Coef

SE Coef

T

P

Constant

1.16525

0.011794

98.796

0.000

A

-0.01125

0.007613

-1.478

0.162

B

0.10708

0.007613

14.065

0.000

C

-0.02792

0.007613

-3.667

0.003

Z1

-0.05562

0.009324

-5.966

0.000

Z2

0.06437

0.009324

6.904

0.000

B*B

-0.11306

0.007223

-15.654

0.000

C*C

-0.05056

0.007223

-7.001

0.000

A*Z1

-0.06813

0.009324

-7.306

0.000

B*C

-0.08438

0.009324

9.049

0.000

B*Z1

0.08437

0.009324

9.049

0.000

B*Z2

-0.12313

0.009324

-13.205

0.000

C*Z1

-0.03562

0.009324

-3.821

0.002

C*Z2

0.07187

0.009324

7.708

0.000

S = 0.03730

R-Sq = 98.7%

R-Sq(adj) = 97.4%

The corresponding fitted equation expressed in the original units of the five factors is:

Predicted Thickness

=

1.165 - 0.005 (% Add) 1 0.054 (Temp) - 0.014(Speed)-

0.028 (Temp)2 - 0.013 (Speed)2- 0.021 (Temp)(Speed)

- 0.056 (Humidity) + 0.064 (Particulate)

- 0.034 (% Add)(Humidity) + 0.042 (Temp)(Humidity)

- 0.062 (Temp)(Particulate) - 0.018 (Speed)(Humidity)

+ 0.036 (Speed)(Particulate)

To use the fitted equation in the coded factors, A, B, C, Z1 and Z2 as a predictor, the controllable inputs are regarded as fixed factors, whereas the noise variables are regarded as random variables, each with mean 0 and variances σ2Z1 and σ2Z2, respectively. Estimates of the variances σ2Z1 and σ2Z2 must be available from previous experience. In fact, in this example the coded values of -1 and 1 for each coded noise variable are assumed to be about two standard deviations away from the average value of that noise variable.

Under these assumptions, the predictive equation for the average thickness consists of terms from the fitted equation in the coded factors that involve only the coded controllable inputs A, B, and C. In coded units this is:

Predicted Thickness

=

1.165 - 0.011 A + 0.107 B - 0.028 C - 0.113 B2 - 0.051 C2 - 0.084 BC

An equation describing how the variance of the film thickness depends on the settings of the controllable inputs can also be obtained from the fitted equation in the coded factors by evaluating the variance of each term and adding an estimate of the random error variance from the fitted equation, in this case s2 = (0.0373)2 = 0.00139.

Estimated Variance (predicted thickness)

=

(- 0.056 - 0.068 A 1 0.084 B 2 0.036 C)2σ2Z1

+ (0.064 - 0.123 B + 0.072 C)2 σ2Z2 + 0.00139

If the coded values of -1 and 1 for each noise variable are assumed to span a range of approximately four standard deviations for that coded noise variable, then in this example σZ1 ≈ 0.5 and σZ2 ≈ 0.5. Using these estimates,

Estimated Variance of Predicted Thickness

=

(- 0.056 - 0.068 A + 0.084 B - 0.036 C)2 (0.5)2

+ (0.064 - 0.123 B + 0.072 C)2 (0.5)2 + 0.00139

The optimal solution is found by minimizing the estimated variance of the predicted thickness subject to the constraint that the predicted mean thickness = 1.00, the target value. A grid search over values of A, B, and C indicates an approximately optimal solution at the following settings:

A

=

-1.0 (% Additive = 10 %)

B

=

-0.6 (Temperature = 174 °F)

C

=

-1.0 (Belt Speed = 55 ft/min)

  • + Share This
  • 🔖 Save To Your Account

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

Overview


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information


To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

Newsletters

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information


Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.

Security


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

Children


This site is not directed to children under the age of 13.

Marketing


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information


If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

Choice/Opt-out


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information


Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents


California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure


Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

Links


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact


Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice


We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020