3. Fractional Factorial Designsprem.uprm.edu/Forms/DoE-R/day3.pdf · Session 3 Fractional Factorial...
Transcript of 3. Fractional Factorial Designsprem.uprm.edu/Forms/DoE-R/day3.pdf · Session 3 Fractional Factorial...
3. Fractional Factorial Designs• Two-level fractional factorial designs
• Confounding
• Blocking
E. Barrios Design and Analysis of Engineering Experiments 3–1E. Barrios Design and Analysis of Engineering Experiments 3–1
Session 3 Fractional Factorial Designs 2Session 3 Fractional Factorial Designs 2
Two-Level Fractional Factorial Designs24 Full Factorial Design
I 1 2 3 4 12 13 14 23 24 34 123 124 134 234 1234 conversion+ − − − − + + + + + + − − − − + 70+ + − − − − − − + + + + + + − − 60+ − + − − − + + − − + + + − + − 89+ + + − − + − − − − + − − + + + 81+ − − + − + − + − + − + − + + − 69+ + − + − − + − − + − − + − + + 62+ − + + − − − + + − − − + + − + 88+ + + + − + + − + − − + − − − − 81+ − − − + + + − + − − − + + + − 60+ + − − + − − + + − − + − − + + 49+ − + − + − + − − + − + − + − + 88+ + + − + + − + − + − − + − − − 82+ − − + + + − − − − + + + − − + 60+ + − + + − + + − − + − − + − − 52+ − + + + − − − + + + − − − + − 86+ + + + + + + + + + + + + + + + 79
We can accommodate 16 estimates: 1 mean; 4 main effects; 6 two-factors interactioneffects; 4 three-factors interaction effects; 1 four-factors interaction effect
E. Barrios Design and Analysis of Engineering Experiments 3–2E. Barrios Design and Analysis of Engineering Experiments 3–2
Session 3 Fractional Factorial Designs 3Session 3 Fractional Factorial Designs 3
Two-Level Fractional Factorial Designsa
Modification of a Bearing Example
Two-Level Eight Run Orthogonal Array
a b c ab ac bc abc Failure Raterun A B C D y1 − − − + + + − 162 + − − − − + + 73 − + − − + − + 144 + + − + − − − 55 − − + + − − + 116 + − + − − + − 77 − + + − − + − 138 + + + + + + + 4
Last column, abc estimates two effects: D + ABC.Factors D and ABC are confounded: lD → D + ABC, and D and ABC are aliases.
aBHH2e Chapter 6.
E. Barrios Design and Analysis of Engineering Experiments 3–3E. Barrios Design and Analysis of Engineering Experiments 3–3
Session 3 Fractional Factorial Designs 4Session 3 Fractional Factorial Designs 4
Two-Level Fractional Factorial DesignsModification of a Bearing Example
E. Barrios Design and Analysis of Engineering Experiments 3–4E. Barrios Design and Analysis of Engineering Experiments 3–4
Session 3 Fractional Factorial Designs 5Session 3 Fractional Factorial Designs 5
Two-Level Fractional Factorial DesignsModification of a Bearing Example
Two-Level Eight Run Orthogonal Arraya b c ab ac bc abc
run A B C D
1 − − − + + + −2 + − − − − + +
3 − + − − + − +
4 + + − + − − −5 − − + + − − +
6 + − + − − + −7 − + + − − + −8 + + + + + + +
Confounding pattern:
lA → A + (BCD) lAB → AB + CDlB → B + (ACD) lAC → AC + BDlC → C + (ABD) lBC → BC + ADlD → D + (ABC)
Sometimes 3rd and higher order interactions are small enough to be ignored.
E. Barrios Design and Analysis of Engineering Experiments 3–5E. Barrios Design and Analysis of Engineering Experiments 3–5
Session 3 Fractional Factorial Designs 6Session 3 Fractional Factorial Designs 6
Two-Level Fractional Factorial DesignsThe Anatomy of the Half Replicate
Generation: A = a; B = b; C = c and D = abc. Thus
D = ABC
is the generating relation of the design. Note that
I = D ×D = D2 = ABC ·DThus, for this design:
I = ABCD, B = ACD, C = ABD, D = ABCAB = CD, AC = BD, AD = BC
This design is resolution 4, since the length of the defining relation has four letters(factors) I = ABCD. It is denoted as
24−1IV
where, the 2 means that factors of the design have 2 levels each; 4-1 because there are4 factors and we are running only one have of the full factorial: 8 = 24−1 = 16/2; andIV because the design is resolution 4.E. Barrios Design and Analysis of Engineering Experiments 3–6E. Barrios Design and Analysis of Engineering Experiments 3–6
Session 3 Fractional Factorial Designs 7Session 3 Fractional Factorial Designs 7
Two-Level Fractional Factorial DesignsThe Anatomy of the Half Replicate
If instead, we use the column ab to accommodate factor D, then AB = D and there-fore I = ABD. Then, for this design,
A = BD, B = AD, C = ABCD, D = ABAC = BCD, BC = ACD, CD = ABC
Note:lA → A + BD; lB → B + AD; lD → D + AB
The defining relation contains three letters I = ABD, thus the design is of resolutionIII, 24−1
III .
E. Barrios Design and Analysis of Engineering Experiments 3–7E. Barrios Design and Analysis of Engineering Experiments 3–7
Session 3 Fractional Factorial Designs 8Session 3 Fractional Factorial Designs 8
Two-Level Fractional Factorial DesignsJustification for the use of fractional factorials:
• Redundancy: When high order interactions are considered negligible lower ordereffects are arranged to be confounded with them and thus are estimable.
• Parsimony: Effect sparsity; vital few, trivial many; Pareto effect.
• Projectivity: A 3D design project onto a 22 factorial design in all 3 subspacesof dimension 2. Then 3D designs are of projectivity 2. Similarly, 24−1 designsare of projectivity 3 since after dropping any factor a full 23 design is left for theremaining three factors.
In general, for fractional factorials designs of resolution R, the projectivity P =R − 1. Every subset of P = R − 1 factors is a complete factorial (possiblyreplicated) in P factors.
E. Barrios Design and Analysis of Engineering Experiments 3–8E. Barrios Design and Analysis of Engineering Experiments 3–8
Session 3 Fractional Factorial Designs 9Session 3 Fractional Factorial Designs 9
Two-Level Fractional Factorial Designs3D Projectivity:
A 23−1III design showing projections into three 22 factorials.
E. Barrios Design and Analysis of Engineering Experiments 3–9E. Barrios Design and Analysis of Engineering Experiments 3–9
Session 3 Fractional Factorial Designs 10Session 3 Fractional Factorial Designs 10
Two-Level Fractional Factorial DesignsNote
It is recommended to dedicate just a modest amount of the budget to the first stages ofthe experimentation.
• Find or determine which factors to consider and appropriate responses
• Determine proper experimental region and factor ranges.
Then you can dedicate to study deeper your experiment
• Estimate better factor effects
• Confirmatory experimentation
• Optimize product or process.
E. Barrios Design and Analysis of Engineering Experiments 3–10E. Barrios Design and Analysis of Engineering Experiments 3–10
Session 3 Fractional Factorial Designs 11Session 3 Fractional Factorial Designs 11
Two-Level Fractional Factorial DesignsSequential Experimentation
In sequential experimentation, unless the total number of runs is necessary to achievea desired level of precision, it is usually best to start with a fractional factorial. Thedesign could be later augmented if necessary
• To cover “more interesting” regions.
• To resolve ambiguities.
E. Barrios Design and Analysis of Engineering Experiments 3–11E. Barrios Design and Analysis of Engineering Experiments 3–11
Session 3 Fractional Factorial Designs 12Session 3 Fractional Factorial Designs 12
Two-Level Fractional Factorial DesignsEight-run Designsa
aBHH2e Chapter 6, BHH Chapter 10.
E. Barrios Design and Analysis of Engineering Experiments 3–12E. Barrios Design and Analysis of Engineering Experiments 3–12
Session 3 Fractional Factorial Designs 13Session 3 Fractional Factorial Designs 13
Two-Level Fractional Factorial DesignsEight-run nodal designs
E. Barrios Design and Analysis of Engineering Experiments 3–13E. Barrios Design and Analysis of Engineering Experiments 3–13
Session 3 Fractional Factorial Designs 14Session 3 Fractional Factorial Designs 14
Two-Level Fractional Factorial DesignsA Bicycle Example
E. Barrios Design and Analysis of Engineering Experiments 3–14E. Barrios Design and Analysis of Engineering Experiments 3–14
Session 3 Fractional Factorial Designs 15Session 3 Fractional Factorial Designs 15
Two-Level Fractional Factorial DesignsSign switching, Foldover and Sequential Assembly
After running a fractional factorial further runs may be necessary to resolve ambigui-ties.
Folding over (changing signs) one column (main effect, say D) provide unaliased es-timates of the main effect and all two-factor interactions involving factor D.
E. Barrios Design and Analysis of Engineering Experiments 3–15E. Barrios Design and Analysis of Engineering Experiments 3–15
Session 3 Fractional Factorial Designs 16Session 3 Fractional Factorial Designs 16
Two-Level Fractional Factorial DesignsA Bicycle Example. Second fraction
E. Barrios Design and Analysis of Engineering Experiments 3–16E. Barrios Design and Analysis of Engineering Experiments 3–16
Session 3 Fractional Factorial Designs 17Session 3 Fractional Factorial Designs 17
Two-Level Fractional Factorial DesignsA Bicycle Example. Resulting 16-run design
Main effect D and two-factor interactions involving D are free of aliasing.
For any given fraction one-column foldover will “dealias” a particular main effect andall its interactionsE. Barrios Design and Analysis of Engineering Experiments 3–17E. Barrios Design and Analysis of Engineering Experiments 3–17
Session 3 Fractional Factorial Designs 18Session 3 Fractional Factorial Designs 18
Two-Level Fractional Factorial DesignsAn Investigation Using Multiple-Column FoldoverFiltration Example
E. Barrios Design and Analysis of Engineering Experiments 3–18E. Barrios Design and Analysis of Engineering Experiments 3–18
Session 3 Fractional Factorial Designs 19Session 3 Fractional Factorial Designs 19
Two-Level Fractional Factorial DesignsFiltration Example
E. Barrios Design and Analysis of Engineering Experiments 3–19E. Barrios Design and Analysis of Engineering Experiments 3–19
Session 3 Fractional Factorial Designs 20Session 3 Fractional Factorial Designs 20
Two-Level Fractional Factorial DesignsFiltration Example
E. Barrios Design and Analysis of Engineering Experiments 3–20E. Barrios Design and Analysis of Engineering Experiments 3–20
Session 3 Fractional Factorial Designs 21Session 3 Fractional Factorial Designs 21
Two-Level Fractional Factorial DesignsSecond Fraction: a 27−3
III . (foldover all columns [mirror image])
E. Barrios Design and Analysis of Engineering Experiments 3–21E. Barrios Design and Analysis of Engineering Experiments 3–21
Session 3 Fractional Factorial Designs 22Session 3 Fractional Factorial Designs 22
Two-Level Fractional Factorial DesignsAnalysis of the resulting sixteen-run design: a 27−3
IV fractional factorial
E. Barrios Design and Analysis of Engineering Experiments 3–22E. Barrios Design and Analysis of Engineering Experiments 3–22
Session 3 Fractional Factorial Designs 23Session 3 Fractional Factorial Designs 23
Two-Level Fractional Factorial Designs2D Projection over the [AE] subspace. A 22 design replicated 4 times
E. Barrios Design and Analysis of Engineering Experiments 3–23E. Barrios Design and Analysis of Engineering Experiments 3–23
Session 3 Fractional Factorial Designs 24Session 3 Fractional Factorial Designs 24
Two-Level Fractional Factorial DesignsIncreasing Design Resolution from III to IV by Foldover
In general, any design of resolution III plus its mirror image becomes a design ofresolution IV.
Consider for example, the 28−5III (= 18 : 27−4
III ) design and its mirror image. Their gener-ation relations are respectively:
I8 = 8 = 124 = 135 = 236 = 1237 (1)
andI8 = −8 = −124 = −135 = −236 = 1237 (2)
then, combining (1) and (2)I16 = 1237
Also, from (1), I8 = (8)(124) = 1248, and from (2), I8 = (−8)(−124) = 1248. Thus,I16 = 1248. The four generators for this 28−4
III design are:
I16 = 1237 = 1248 = 1358 = 2368
E. Barrios Design and Analysis of Engineering Experiments 3–24E. Barrios Design and Analysis of Engineering Experiments 3–24
Session 3 Fractional Factorial Designs 25Session 3 Fractional Factorial Designs 25
Two-Level Fractional Factorial DesignsSixteen-Run DesignsNodal Designs:
E. Barrios Design and Analysis of Engineering Experiments 3–25E. Barrios Design and Analysis of Engineering Experiments 3–25
Session 3 Fractional Factorial Designs 26Session 3 Fractional Factorial Designs 26
Two-Level Fractional Factorial DesignsDesign Matrix and Alias Patterns
E. Barrios Design and Analysis of Engineering Experiments 3–26E. Barrios Design and Analysis of Engineering Experiments 3–26
Session 3 Fractional Factorial Designs 27Session 3 Fractional Factorial Designs 27
Two-Level Fractional Factorial DesignsThe 25−1
V Nodal Design. Reactor Example
E. Barrios Design and Analysis of Engineering Experiments 3–27E. Barrios Design and Analysis of Engineering Experiments 3–27
Session 3 Fractional Factorial Designs 28Session 3 Fractional Factorial Designs 28
Two-Level Fractional Factorial DesignsAlias Pattern
E. Barrios Design and Analysis of Engineering Experiments 3–28E. Barrios Design and Analysis of Engineering Experiments 3–28
Session 3 Fractional Factorial Designs 29Session 3 Fractional Factorial Designs 29
Two-Level Fractional Factorial DesignsNormal plots for full and half fraction factorial designs.
−10 0 10 20
−2
−1
01
2
Full Factorial
effects
norm
al s
core
A
B
C
D
E
A:B
A:C B:C
A:D
B:D
C:D
A:E
B:E
C:E
D:E
A:B:C A:B:D
A:C:D
B:C:D
A:B:E A:C:E
B:C:E A:D:E
B:D:E
C:D:E A:B:C:D
A:B:C:E
A:B:D:E
A:C:D:E
B:C:D:E A:B:C:D:E
−10 −5 0 5 10 15 20 25
−1
01
Fractional Factorial
effects
norm
al s
core
A
B
C
D
E
A:B
A:C
B:C
A:D
B:D
C:D
A:E B:E
C:E
D:E
E. Barrios Design and Analysis of Engineering Experiments 3–29E. Barrios Design and Analysis of Engineering Experiments 3–29
Session 3 Fractional Factorial Designs 30Session 3 Fractional Factorial Designs 30
Two-Level Fractional Factorial Designs3D Projectivity of 25−1
V design.
E. Barrios Design and Analysis of Engineering Experiments 3–30E. Barrios Design and Analysis of Engineering Experiments 3–30
Session 3 Fractional Factorial Designs 31Session 3 Fractional Factorial Designs 31
Two-Level Fractional Factorial DesignsThe 28−4
IV Nodal Design. Paint Trial Example
E. Barrios Design and Analysis of Engineering Experiments 3–31E. Barrios Design and Analysis of Engineering Experiments 3–31
Session 3 Fractional Factorial Designs 32Session 3 Fractional Factorial Designs 32
Two-Level Fractional Factorial DesignsNormal plot of effects for glossiness and abrasion.
E. Barrios Design and Analysis of Engineering Experiments 3–32E. Barrios Design and Analysis of Engineering Experiments 3–32
Session 3 Fractional Factorial Designs 33Session 3 Fractional Factorial Designs 33
Two-Level Fractional Factorial DesignsContour plots for glossiness and abrasion.
E. Barrios Design and Analysis of Engineering Experiments 3–33E. Barrios Design and Analysis of Engineering Experiments 3–33
Session 3 Fractional Factorial Designs 34Session 3 Fractional Factorial Designs 34
Two-Level Fractional Factorial DesignsThe 215−11
III Nodal Design. Shrinkage of Speedometer Examplea.
aQuinlan (1985)
E. Barrios Design and Analysis of Engineering Experiments 3–34E. Barrios Design and Analysis of Engineering Experiments 3–34
Session 3 Fractional Factorial Designs 35Session 3 Fractional Factorial Designs 35
Two-Level Fractional Factorial DesignsNormal plot of effects for location and dispersion responses.
−10 0 10 20 30
−1
0
1
Location Effects
effects
norm
al s
core A
B
C
D
E
F
G
H
J
K
L
M
N
O
P
−1.0 −0.5 0.0 0.5 1.0 1.5
−1
0
1
Dispersion Effects
effects
norm
al s
core
A
B
C D
E
F
G
H
J
K
L
M
N
O
P
Location Effects
factors
effe
cts
A C E G J L N P
−20
−10
0
10
20
ME
ME
Dispersion Effects
factors
effe
cts
A C E G J L N P
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5ME
ME
E. Barrios Design and Analysis of Engineering Experiments 3–35E. Barrios Design and Analysis of Engineering Experiments 3–35
Session 3 Fractional Factorial Designs 36Session 3 Fractional Factorial Designs 36
Elimination of Block EffectsBoys Shoes Example
2 4 6 8 10
108
110
112
114
boys
wea
r
material Amaterial B
Two−sample Comparison
2 4 6 8 10
−1.0
−0.5
0.0
0.5
1.0
boys
wea
r di
ffere
nce
Paired Comparison
10 + 10 observations2 sample means
18 degrees of freedom
10 differences1 sample mean
9 degrees of freedom
E. Barrios Design and Analysis of Engineering Experiments 3–36E. Barrios Design and Analysis of Engineering Experiments 3–36
Session 3 Fractional Factorial Designs 37Session 3 Fractional Factorial Designs 37
Elimination of Block Effects
“Block what you can, randomize what you cannot”
• Identify important extraneous factors within blocks and eliminate them.
• Representative variation between blocks should be encourage
E. Barrios Design and Analysis of Engineering Experiments 3–37E. Barrios Design and Analysis of Engineering Experiments 3–37
Session 3 Fractional Factorial Designs 38Session 3 Fractional Factorial Designs 38
Two-Level Factorial DesignsBlocking Arrangements for 2k Factorial Designsa
Number of Number of BlockVariables Runs Size Block Interactions Confounded with Blocks
3 8 4 B1 = 123 1232 B1 = 12,B2 = 13 12, 13, 23
4 16 8 B1 = 1234 12344 B1 = 124,B2 = 134 124, 134, 232 B1 = 12,B2 = 23, 12, 23, 34, 13, 1234, 24, 14
B3 = 34
5 32 16 B1 = 12345 123458 B, = 123,B2 = 345 123, 345, 12454 B1 = 125,B2 = 235, 125, 235, 345, 13, 1234, 24, 145
B3 = 3452 B1 = 12,B2 = 13, 12, 13, 34, 45, 23, 1234, 1245, 14,
B3 = 34,B4 = 45 1345, 35, 24, 2345, 1235, 15, 25,i.e., all 2fi and 4fi
6 64 32 B1 = 123456 12345616 B1 = 1236,B2 = 3456 1236, 3456, 12458 B1 = 135,B2 = 1256, 135, 1256, 1234, 236, 245, 3456, 146B3 = 12344 B1 = 126,B2 = 136, 126, 136, 346, 456, 23, 1234, 1245,
B3 = 346,B4 = 456 14, 1345, 35, 246, 23456, 12356, 156, 252 B1 = 12,B2 = 23, All 2fi, 4fi, and 6fi
B3 = 34,B4 = 45,B5 = 56
aBHH2e Table 5A.1
E. Barrios Design and Analysis of Engineering Experiments 3–38E. Barrios Design and Analysis of Engineering Experiments 3–38
Session 3 Fractional Factorial Designs 39Session 3 Fractional Factorial Designs 39
Elimination of Block Effects25−1V Design Matrix
E. Barrios Design and Analysis of Engineering Experiments 3–39E. Barrios Design and Analysis of Engineering Experiments 3–39
Session 3 Fractional Factorial Designs 40Session 3 Fractional Factorial Designs 40
Two-Level Fractional Factorial DesignsMinimal Aberration Two-Level Fractional Factorial Design for k Variables andN Runsa
(Number in Parentheses Represent Replication).
aBHH2e Table 6.22
E. Barrios Design and Analysis of Engineering Experiments 3–40E. Barrios Design and Analysis of Engineering Experiments 3–40