Industrial Engineering Chapter 14 Homework Which effects appear important?

Reserve Problems Chapter 14 Section 6 Problem 1

An article in the Annals of Nuclear Energy (“Statistical Analysis of the Fort Calhoun Reactor

Coolant Pump System,” Vol. 24(3), 1997) considered the effect of five factors on a recirculating

pump motor power input (kW).

The factors were temperature settings at the sites: A = Upper guide bearing, B = Upper thrust

bearing, C = Lower thrust bearing, D = Lower guide bearing, E = Seal winding.

The levels were coded as A: -1 = 140, +1 = 160; B: -1 = 140, +1 = 160; C: -1 = 140, +1 = 160; D:

-1 = 140, +1 = 160; E: -1 = 140, +1 = 160.

A

B

C

D

E

Power

-1

2000

1

-1

2039

-1

1

-1

1998

1

-1

1990

-1

1

-1

2008

1

-1

1

-1

2364

-1

1

-1

2163

1

-1

2360

-1

1

-1

2013

1

-1

1

-1

2013

-1

1

-1

1

-1

1959

1

-1

1

-1

1964

-1

1

-1

2367

1

-1

1

-1

2310

-1

1

-1

2090

1

-1

2072

-1

1

1993

1

-1

1

1998

-1

1

-1

1

1959

1

-1

1

2057

-1

1

-1

1

2098

1

-1

1

-1

1

2364

-1

1

-1

1

2034

1

-1

1

2160

-1

1

1998

1

-1

1

1997

-1

1

-1

1

2092

1

-1

1

2094

-1

1

2044

1

-1

1

2212

-1

1

2010

1

2036

(a) Estimate the factor effects.

(b) Which effects appear important?

(c) Analyze the residuals from this experiment.

SOLUTION

(a)

Factor

Effect

A

75.25

B

-48.75

E

-35.25

A*B

-21.75

A*C

57.75

A*B*E

-1.5

A*C*E

2.5

A*D*E

22.12

B*C*D

-45.13

B*C*E

-30.25

B*D*E

58.88

C*D*E

-34.38

A*B*C*D

14.63

A*B*C*E

-18.75

A*B*D*E

-21.38

A*C*D*E

31.63

B*C*D*E

31.62

A*B*C*D*E

-3.62

(b)

A*D

-59.63

A*E

11

B*C

-56.5

B*D

-30.87

B*E

16

C*D

-31.63

C*E

-61.75

D*E

-2.87

A*B*C

-28.5

A*B*D

9.87

A*C*D

-43.63

(c)

Reserve Problems Chapter 14 Section 6 Problem 2

An article in the Fuel (“Optimization of Reactive Extraction of Castor Seed to Produce Biodiesel

Using Response Surface Methodology,” Vol. 2012) considered the effects of the following four

factors on FAME (fatty acid methyl esters) yield (%) of biodiesel.

A

B

C

D

FAME Yield

(%)

Catalyst Concentration

(%)

Methanol/Oil

Ratio

Mixing Intensity

(rpm)

Temperature

(C)

0.75

162.5

225

50

51.2

1.25

162.5

225

50

61

0.75

287.5

225

50

78.2

1.25

287.5

225

50

75.5

0.75

162.5

475

50

54

1.25

162.5

475

50

65.9

0.75

287.5

475

50

81.1

1.25

287.5

475

50

82.2

0.75

162.5

225

60

49.5

1.25

162.5

225

60

0.75

287.5

225

60

90.2

1.25

287.5

225

60

86

0.75

162.5

475

60

53.2

1.25

162.5

475

60

64.4

0.75

287.5

475

60

97.2

1.25

287.5

475

60

93

(a) Estimate the factor effects.

(b) Which effects appear important? Use a normal probability plot.

(c) How should the process variables B, C, D should be changed to increase yield?

SOLUTION

(a)

Term

Effect

Coef

Constant

71.412

A

4.175

2.087

B

28.025

14.012

C

4.925

2.463

D

5.550

2.775

A*B

-6.675

-3.337

A*C

0.825

0.413

A*D

-0.850

-0.425

B*C

0.975

0.487

A*B*D

-0.850

-0.425

(b)

(c)

Effects B, C, D and interaction BD are important and positive. Effect A is important and

Reserve Problems Chapter 14 Section 6 Problem 3

A

4

2

factorial design was run in a chemical process. The design factors are A = time, B =

concentration, C = pressure, and D = temperature. The response variable is yield. The data

follow:

Yield

(pounds)

Factor levels

Run

A

B

C

D

–

+

1

–

12

A (hours)

2

3

2

+

–

18

B (%)

14

18

A*C*D

-0.650

-0.325

B*C*D

0.500

0.250

A*B*C*D

-0.300

-0.150

3

–

+

–

13

C (psi)

60

80

4

+

–

16

D (°C)

200

250

5

–

+

–

17

6

+

–

+

–

15

7

–

+

–

20

8

+

–

15

9

–

+

10

+

–

+

25

11

–

+

–

+

13

12

+

–

+

24

13

–

+

19

14

+

–

+

21

15

–

+

17

16

+

23

(a) Estimate the factor effects. Based on a normal probability plot of the effect estimates, identify

which factors and interactions are significant?

(b) Conduct an ANOVA, use the hierarchical model, which includes A, C, D factors and AC and

AD interactions. Which factors and interactions are significant at

0.05



=

?

(c) Find a regression model to predict yield in terms of the actual factor levels.

(d) Can this design be projected into a

3

2

design with two replicates?

SOLUTION

(a)

Term

Effect

Coef

Constant

17.375

A

4.500

2.250

B

0.500

0.250

C

2.000

1.000

D

3.250

1.625

-0.750

-0.375

AD

4.000

2.000

BC

0.250

0.125

BD

0.000

(b)

Analysis of Variance for yield, using Adjusted SS for Tests

Source

DF

Seq SS

Adj SS

Adj MS

F

P

ABC

1.000

0.500

ABD

0.750

0.375

-0.250

-0.125

-0.750

-0.375

1.000

0.500

C

1

16.000

9.85

0.011

D

1

42.250

26.00

0.000

(c)

The regression equation is

(d)

Reserve Problems Chapter 14 Section 6 Problem 4

A two-level factorial experiment in four factors was conducted by Chrysler and described in the

article “Sheet Molded Compound Process Improvement” by P. I. Hsieh and D. E. Goodwin

(Fourth Symposium on Taguchi Methods, American Supplier Institute, Dearborn, MI, 1986, pp.

13–21). The purpose was to reduce the number of defects in the finish of sheet-molded grill

opening panels. A portion of the experimental design, and the resulting number of defects,

i

y

observed on each run are shown in the following table. This is a single replicate of the

4

2

design.

Grill Defects Experiment

Run

A

B

C

D

y

1

–

56

7.48

2

+

–

17

4.12

3

–

+

–

2

1.41

4

+

–

4

2

1

72.250

44.46

0.000

1

64.000

39.38

0.000

16.250

291.750

5

–

+

–

3

1.73

6

+

–

+

–

4

2

7

–

+

–

50

7.07

8

+

–

2

1.41

9

–

+

1

10

+

–

+

0

11

–

+

–

+

3

1.73

12

+

–

+

12

3.46

13

–

+

3

1.73

14

+

–

+

4

2

15

–

+

0

16

+

0

(a) Estimate the factor effects and use a normal probability plot to tentatively identify the

important factors.

(b) Fit an appropriate model using the factors identified in part (a)

(c) Use the residuals versus the predicted number of defects plot and a normal probability plot of

the residuals to comment the adequacy of this model.

(d) The table also shows the square root of the number of defects. Use a normal probability plot

to identify the important factors using the square root of the number of defects as the response.

Use the residuals versus square root of the number of defects plot and a normal probability plot

of the residuals to comment the adequacy of this model.

SOLUTION

(a)

Estimated Effects and Coefficients for y (coded units)

Term

Effect

Coef

Constant

10.063

C

-3.625

-1.813

D

-14.375

-7.188

0.125

0.063

11.625

5.812

11.375

5.688

3.625

1.813

1.375

0.688

-12.625

-6.313

2.125

1.062

ACD

0.375

0.188

BCD

-16.625

-8.313

ABCD

9.875

4.938

(b)

(c)

(d)

Estimated Effects and Coefficients for

y

(coded units)

Term

Effect

Coef

Constant

2.323

A

-0.895

-0.448

AB

0.061

0.030

AC

-0.385

-0.192

AD

1.145

0.573

BC

0.627

0.314

BD

0.488

0.244

0.042

0.021

ABC

-1.609

-0.804

ABD

0.555

0.278

ACD

0.269

0.134

BCD

-2.609

-1.305

ABCD

0.859

0.429

B

-0.372

-0.186

C

D

-2.164

-1.082

From the normal probability plot, only BCD is a significant effect.

Reserve Problems Chapter 14 Section 7 Problem 1

An article in the Fuel (“Optimization of Reactive Extraction of Castor Seed to Produce Biodiesel

Using Response Surface Methodology,” Vol. 2012) considered the effects of the following four

factors on FAME (fatty acid methyl esters) yield (%) of biodiesel.

A

B

C

D

Catalyst Concentration

(%)

Methanol/Oil

Ratio

Mixing Intensity

(rpm)

Temperature

(C)

FAME Yield

(%)

0.75

162.5

225

50

51.2

1.25

162.5

225

50

61

0.75

287.5

225

50

78.2

1.25

287.5

225

50

75.5

0.75

162.5

475

50

54

1.25

162.5

475

50

65.9

0.75

287.5

475

50

81.1

1.25

287.5

475

50

82.2

0.75

162.5

225

60

49.5

1.25

162.5

225

60

0.75

287.5

225

60

90.2

1.25

287.5

225

60

86

0.75

162.5

475

60

53.2

1.25

162.5

475

60

64.4

0.75

287.5

475

60

97.2

1.25

287.5

475

60

93

The article provided additional center point runs with the following values: 88.3, 88.2, 88.4,

88.2, 88.4, 88.2.

Compute an ANOVA with the sum of squares for curvature and conduct an F-test for curvature.

Use α = 0.05.

SOLUTION

Analysis of Variance for Yield (coded units)

Source

DF

Seq SS

Adj SS

Adj MS

F

P

Main Effects

4

3431.56

857.89

88747.18

0.000

Reserve Problems Chapter 14 Section 7 Problem 2

2-Way Interactions

6

374.04

62.34

6448.92

0.000

3-Way Interactions

4

5.64

1.41

145.93

0.000

4-Way Interactions

1

0.36

37.24

0.002

Curvature

1

1242.00

128482.77

0.000

Residual Error

5

0.05

0.01

Pure Error

5

0.05

0.01

Total

21

An experiment was run in a semiconductor fabrication plant in an effort to increase yield. Five

factors, each at two levels, were studied. The factors (and levels) were A = aperture setting

(small, large), B = exposure time (20% below nominal, 20% above nominal), C = development

time (30 and 45 seconds), D = mask dimension (small, large), and E = etch time (14.5 and 15.5

minutes). The following unreplicated

5

2

design was run:

( )

1

=

7

e

=

8

a

=

9

ae

=

12

b

=

34

be

=

35

ab

=

55

abe

=

52

c

=

16

ce

=

15

ac

=

20

ace

=

22

bc

=

40

bce

=

45

abc

=

60

abce

=

65

d

=

8

de

=

6

ad

=

10

ade

=

10

bd

=

32

bde

=

30

abd

=

50

abde

=

53

cd

=

18

cde

=

15

acd

=

21

acde

=

20

bcd

=

44

bcde

=

41

abcd

=

61

abcde

=

63

Suppose that a center point with five replicates is added to the factorial runs and the responses

are 45, 40, 41, 47, and 43.

(a) Estimate the experimental error using the center points. Compare this to the estimate obtained

by pooling out apparently nonsigninficant effects.

(b) Test for curvature with

0.05



=

.

SOLUTION

(a)

Estimated Effects and Coefficients for y (coded units)

Term

Effect

Coef

SE Coef

T

P

Constant

30.5313

0.5062

60.31

0.000

A

11.8125

5.9063

0.5062

11.67

0.000

B

33.9375

16.9687

0.5062

33.52

0.000

C

9.6875

4.8438

0.5062

9.57

0.001

D

0.5062

-0.80

0.467

E

0.4375

0.2188

0.5062

0.43

0.688

A*B

7.9375

3.9687

0.5062

7.84

0.001

A*C

0.4375

0.2187

0.5062

0.43

0.688

A*D

0.5062

-0.06

0.954

A*E

0.9375

0.4688

0.5062

0.93

0.407

C*D

0.8125

0.4062

0.5062

0.80

0.467

C*E

0.3125

0.1563

0.5062

0.31

0.773

D*E

-1.1875

-0.5938

0.5062

-1.17

0.306

Analysis of Variance for y (coded units)

Source

DF

Seq SS

Adj SS

Adj MS

F

P

Main Effects

5

11087.9

2217.58

270.44

0

A

1

1116.3

1116.28

136.13

0

B

1

9214.0

9214.03

1123.66

0

C

1

750.8

750.78

91.56

0.001

D

1

5.3

5.28

0.64

0.467

E

1

1.5

1.53

0.19

0.688

2-Way Interactions

10

536.3

53.63

6.54

0.043

A*B

1

504.0

504.03

61.47

0.001

A*C

1

1.5

1.53

0.19

0.688

A*D

1

0.0

0.03

0.00

0.954

A*E

1

7.0

7.03

0.86

0.407

B*C

1

0.0

0.03

0.00

0.954

B*D

1

3.8

3.78

0.46

0.534

B*E

1

2.5

2.53

0.31

0.608

C*D

1

5.3

5.28

0.64

0.467

C*E

1

0.8

0.78

0.10

0.773

D*E

1

11.3

11.28

1.38

0.306

3-Way Interactions

10

24.3

2.43

0.30

0.946

A*B*C

1

1.5

1.53

0.19

0.688

A*B*D

1

0.8

0.78

0.10

0.773

A*B*E

1

0.3

0.28

0.03

0.862

A*C*D

1

1.5

1.53

0.19

0.688

A*C*E

1

0.8

0.78

0.10

0.773

A*D*E

1

5.3

5.28

0.64

0.467

B*C*D

1

1.5

1.53

0.19

0.688

B*C*E

1

7.0

7.03

0.86

0.407

A*B*C

-0.4375

-0.2187

0.5062

-0.43

0.688

A*B*D

0.3125

0.1563

0.5062

0.31

0.773

A*B*E

-0.1875

-0.0937

0.5062

-0.19

0.862

A*C*D

-0.4375

-0.2187

0.5062

-0.43

0.688

A*C*E

0.3125

0.1562

0.5062

0.31

0.773

A*D*E

0.8125

0.4062

0.5062

0.80

0.467

B*C*D

0.4375

0.2187

0.5062

0.43

0.688

B*C*E

0.9375

0.4688

0.5062

0.93

0.407

B*D*E

0.1875

0.0938

0.5062

0.19

0.862

C*D*E

-0.8125

-0.4063

0.5062

-0.80

0.467

A*B*C*D

-0.0625

-0.0312

0.5062

-0.06

0.954

A*B*C*E

0.1875

0.0938

0.5062

0.19

0.862

A*B*D*E

0.9375

0.4688

0.5062

0.93

0.407

A*C*D*E

-0.3125

-0.1562

0.5062

-0.31

0.773

B*C*D*E

-0.9375

-0.4688

0.5062

-0.93

0.407

A*B*C*D*E

-0.1875

-0.0937

0.5062

-0.19

0.862

Ct Pt

12.6688

1.3770

9.20

0.001