Reflection Final Paper

Johnnetwork
DOEbatteryexample.pptx

IE 590/490

Lean Manufacturing in Systems Engineering

Dr. Delia Valles-Rosales

Industrial Engineering Department

1

Lean Six Sigma-Tools

Project Selection, NPV, SIPOC Diagram, VOC, CTQ, Pareto Chart, Process Mapping

C&E Analysis, Process Capability Analysis, Sampling, Basic Statistics

Hypothesis Test, Correlation, Regression, Multi-Vari, CI’s, ANOVA

DOE, Poka-Yoke, FMEA, RSM

Control Plans, SPC, SOP’s

In this type of design we want to study the effect of two or more factors. In each complete trial or replication of the experiment, all possible combinations of the levels of the factors are investigated.

Basic idea: Treatments are a combination of multiple factors with different levels (i.e. settings)

Factorial Designs

3

The effect of a factor is defined to be as the change in the response produced by a change in the level of the factor (main effect).

Interaction between factors is present when the difference in response between the levels of one factor is not the same at all levels of the other factors (i.e. the effect of factor A depends on the level selected for factor B).

Factorial Designs: Main Concepts

4

One-Factor ANOVA

Separates total variation observed in a set of measurements into:

Variation within one system

Due to random measurement errors

Variation between systems

Due to real differences + random error

One-factor experimental design

ANOVA Summary

Generalized Design of Experiments

Goals

Isolate effects of each input variable.

Determine effects of interactions.

Determine magnitude of experimental error

Obtain maximum information for a given effort

Basic idea

Expand 1-factor ANOVA to m factors

Terminology

Response variable

Measured output value

E.g. total execution time

Factors

Input variables that can be changed

E.g. cache size, clock rate, bytes transmitted

Levels

Specific values of factors (inputs)

Continuous (~bytes) or discrete (type of system)

Terminology

Replication

Completely re-run experiment with same input levels

Used to determine impact of measurement error

Interaction

Effect of one input factor depends on level of another input factor

Two-factor Experiments

Two factors (inputs)

A, B

Separate total variation in output values into:

Effect due to A

Effect due to B

Effect due to interaction of A and B (AB)

Experimental error

Two-factor ANOVA

Factor A – a input levels

Factor B – b input levels

n measurements for each input combination

abn total measurements

Two Factors, n Replications

Factor A
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Factor B 1
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One-factor ANOVA

Each individual measurement is a composition of

Overall mean

Effect of alternatives

Measurement errors

Two-factor ANOVA

Each individual measurement is composition of

Overall mean

Effects

Interactions

Measurement errors

Sum-of-Squares

As before, use sum-of-squares identity

SST = SSA + SSB + SSAB + SSE

Degrees of freedom

df(SSA) = a – 1

df(SSB) = b – 1

df(SSAB) = (a – 1)(b – 1)

df(SSE) = ab(n – 1)

df(SST) = abn - 1

Two-Factor ANOVA

Need for Replications

If n=1

Only one measurement of each configuration

Can then be shown that

SSAB = SST – SSA – SSB

Since

SSE = SST – SSA – SSB – SSAB

We have

SSE = 0

Need for Replications

Thus, when n=1

SSE = 0

→ No information about measurement errors

Must replicate each experiment at least twice

Generalized m-factor Experiments

Effects for 3 factors:

A

B

C

AB

AC

BC

ABC

An engineer is designing a battery that will be used in a device that will be subject to extreme variations in temperature.

She is interested in examining three different materials for this battery at three different temperatures (15, 70 and 125 °F) in order to determine how battery life is affected by these conditions.

* Example from Montgomery (2009)

Factorial Designs Example: Battery Design

20

Response: Battery life

Treatment: All combinations the factors:

Material: 3 levels (1, 2 and 3)

Temperature: 3 levels (15, 70 and 125 °F)

Factorial Design Example: Study

21

An appropriate statistical model to describe this design is:

The life time data are (Data source: Montgomery, 1991):

Row Matype Temp Life

1 1 15 74

2 1 15 130

3 1 15 155

4 1 15 180

5 2 15 126

6 2 15 150

7 2 15 159

8 2 15 188

9 3 15 110

10 3 15 138

11 3 15 160

12 3 15 168

13 1 70 34

14 1 70 40

15 1 70 75

16 1 70 80

17 2 70 106

18 2 70 115

19 2 70 122

20 2 70 126

21 3 70 120

22 3 70 139

23 3 70 150

24 3 70 174

25 1 125 20

26 1 125 58

27 1 125 70

28 1 125 82

29 2 125 25

30 2 125 45

31 2 125 58

32 2 125 70

33 3 125 60

34 3 125 82

35 3 125 96

36 3 125 104

A typical procedure to conduct analysis for two-factor experiment

Conduct descriptive summary using both graphical and numerical techniques for detecting unusual observations and for demonstrating some interesting patterns that will be useful during the analysis.

Conduct the preliminary ANOVA analysis using raw data and residual analysis to check for the adequacy of assumptions, especially the constant variance and normality. Graphical methods are particularly useful here.

A typical procedure to conduct analysis for two-factor experiment

3. Determine if further analysis is needed:

If interaction is significant (also closely examine the interaction plot to learn about the interaction pattern), an analysis of simple effects of factor A (or B) at each level of factor B (or A) is recommended.

If main effect of a factor is significant, one should decide what further comparisons should be useful: Pairwise comparison, contrasts, trend analysis, comparison with control , and so on. (consult the one-way analysis for more details).

The corresponding ANOVA Table for Two-Factor Model

Source Df SS MS F P-value EMS
Treatment A a-1 SSA MSA=SSA/(a-1) MSA/MSE
Treatment B b-1 SSB MSB=SSB/(b-1) MSB/MSE
AB Interaction (a-1)(b-1) SSAB MSAB=SSAB/ [(a-1)(b-1)] MSAB/MSE
Error ab(r-1) SSE MSE=SSE/[ab(r-1)] s2
Total abr-1 SSTO

The F-statistics test the following hypothesis:

In analyzing the ANOVA results for two-factor models, we need to examine the Interaction effect before analyzing main effects. If there exists interaction effect, sometimes the results of main effects may be misleading.

Factor Type Levels Values

Matype fixed 3 1 2 3

Temp fixed 3 15 70 125

Analysis of Variance for Life, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P

Matype 2 10633.2 10633.2 5316.6 7.98 0.002

Temp 2 39083.2 39083.2 19541.6 29.34 0.000

Matype*Temp 4 9437.7 9437.7 2359.4 3.54 0.019

Error 27 17980.7 17980.7 666.0

Total 35 77134.8

Unusual Observations for Life

Obs Life Fit SE Fit Residual St Resid

1 74.000 134.750 12.903 -60.750 -2.72R

4 180.000 134.750 12.903 45.250 2.02R

All three types of material show much longer life time at low temperature, at much shorter life time at high temperature environment. However, the performance varies greatly at middle temperature. Material #3 seems to perform well at middle temperature as well.

The trend of the life time is clearly going shorter when temperature increases. However, the trends are different for different types of material. Material #3 maintains its high life time at very high levels of temperatures. Material 1 and 2 sharply decrease at low temperature and middle temperature.

When temperature increases, Material 2 is affected by temperature #2

There is a slight evidence that there is larger variation at higher life time. When temperature is low, life time is higher

There is a slight evidence, material one seems to have a larger life time variation.

Normality seems to be fine.

Two-Factor Design – Random effect model

Through cause-effect diagram and team discussion, it was suspected that the surface finish of a metal part is influenced by the feed rate and the depth of cut. The variability is of a particular concern, since uneven metal finish will result leaking of the finish products using this metal. Three feed rates and four depths of cuts are randomly chosen. A factorial experiment with three replications is performed. The surface roughness is measured and recorded. The lower the roughness, the better the surface.

Row Depth Feed Rough

1 0.12 0.20 74

2 0.12 0.20 68

3 0.12 0.20 60

4 0.12 0.25 90

5 0.12 0.25 85

6 0.12 0.25 89

7 0.12 0.30 99

8 0.12 0.30 106

9 0.12 0.30 103

10 0.14 0.20 79

11 0.14 0.20 67

12 0.14 0.20 75

Row Depth Feed Rough

13 0.14 0.25 99

14 0.14 0.25 102

15 0.14 0.25 94

16 0.14 0.30 107

17 0.14 0.30 106

18 0.14 0.30 98

19 0.16 0.20 80

20 0.16 0.20 84

21 0.16 0.20 86

22 0.16 0.25 91

23 0.16 0.25 95

24 0.16 0.25 98

Row Depth Feed Rough

25 0.16 0.30 99

26 0.16 0.30 105

27 0.16 0.30 100

28 0.18 0.20 99

29 0.18 0.20 102

30 0.18 0.20 103

31 0.18 0.25 108

32 0.18 0.25 111

33 0.18 0.25 99

34 0.18 0.30 111

35 0.18 0.30 110

36 0.18 0.30 106

The analysis of the Surface Roughness data

General Linear Model: Rough versus Depth, Feed

Factor Type Levels Values

Depth random 4 0.12 0.14 0.16 0.18

Feed random 3 0.20 0.25 0.30

Analysis of Variance for Rough, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P

Depth 3 1801.56 1801.56 600.52 4.60 0.053

Feed 2 3230.72 3230.72 1615.36 12.37 0.007

Depth*Feed 6 783.28 783.28 130.55 6.84 0.000

Error 24 458.00 458.00 19.08

Total 35 6273.56

Unusual Observations for Rough

Obs Rough Fit SE Fit Residual St Resid

3 60.000 67.333 2.522 -7.333 -2.06R

The ANOVA results indicate all three uncertainty components are statistically significant when compared to the random error. Different Feeding Rates, different Depth all introduces huge variation. In particular, the significance of Depth*Feed interaction component indicates there is a huge inconsistence of roughness due to different feeding rates for different level of depths.

Thank you!!!

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