Chalmers University of Technology
Two important inventions contribute to the successfulness of the hearing aid device
The titanium implant
The vibrator
Chalmers University of Technology
A simple test performed at Cochlear BAS to evaluate a new supplier of components
• The objective of the test was to evaluate whether washers from a new supplier could be used• Altogether 120 vibrators where produced, 60 with washersordinary used and 60 with washers from a new potential supplier
• The order in which the 120 vibrators was produced was randomised
Chalmers University of Technology
Details of the improvement work
The objective of the project is to improve the production yield of the vibrators
• The vibrator is made up by a rather large number of components• The assembly process of vibrators include a ratherlarge number of operations• The assembly process requires that measurement equipment work satisfactory
Chalmers University of Technology
Individual value plot
Washer from new supplierOrdinary used
1200
1100
1000
900
800
700
Type of washer
Individual Value Plot
Chalmers University of Technology
Individual value plot – two outliers removed
Washer from new supplierOrdinary used
800
790
780
770
760
750
740
Type of washer
Individual Value Plot
Chalmers University of Technology
Time series plot to investigate whether the process was stable during the test
10896847260483624121
800
790
780
770
760
750
740
Index
Time Series Plot
Chalmers University of Technology
Histogram of the”populations”
790780770760750740
0,04
0,03
0,02
0,01
0,00
Densi
ty
760,1 9,643 59769,9 11,82 59
Mean StDev N
Ordinary usedWasher from new supplier
Type of washer
Normal Histogram
Chalmers University of Technology
Complete randomisation
• Randomisation of run order
• Resetting of all factor levels between each experiment
Chalmers University of Technology
Randomizing
• Problem: Systematic dependence between the experiments.
• Solution: Make the experiments in random order.
order Exp.nr
A B C Y
8
5
1
2
4
7
6
3
1
2
3
4
5
6
7
8
-
+
-
+
-
+
-
+
-
-
+
+
-
-
+
+
-
-
-
-
+
+
+
+
53.8
51.8
47.4
47.8
50.6
51.8
48.2
48.6
Chalmers University of Technology
Resetting of factor levels
4
8
1
i
i
contrasts
24)(
2
8
1
i
i
VarcontrastsVar
Exp. A B C y ε
1 - - - 53.8 εA1+εB1+εC1+ε1 2 + - - 51.8 εA2+εB2+εC2+ε2 3 - + - 47.4 εA3+εB3+εC3+ε3 4 + + - 47.8 εA4+εB4+εC4+ε4 5 - - + 50.6 εA5+εB5+εC5+ε5 6 + - + 51.8 εA6+εB6+εC6+ε6 7 - + + 48.2 εA7+εB7+εC7+ε7 8 + + + 48.6 εA8+εB8+εC8+ε8
Chalmers University of Technology
If factors are not reset between each experiment, contrasts will have unequal variance!
Exp. A B C y ε
1 - - - 53.8 εA1+εB1+εC1+ε1 2 + - - 51.8 εA2+εB2+εC1+ε2 3 - + - 47.4 εA3+εB3+εC1+ε3 4 + + - 47.8 εA4+εB4+εC1+ε4 5 - - + 50.6 εA5+εB5+εC2+ε5 6 + - + 51.8 εA6+εB6+εC2+ε6 7 - + + 48.2 εA7+εB7+εC2+ε7 8 + + + 48.6 εA8+εB8+εC2+ε8
4
8
1
i
iBiAi
Acontrast
4
42
1
8
1
j
Cji
iBiAi
Ccontrast
2
222 BA
AcontrastVar
2222
22 CBA
CcontrastVar
Responses arenot independent!
Chalmers University of Technology
• Four different process conditions• Eight batches of raw material
Split-plot designs: A Composite Material Example
Manufacturing process of composite material
y – bending strength response variable
A – curing temperatureB – pressureC – holding time
control factors (process variables)
D – proportion of hardenerE – thermo-plastic contentF – proportion of epoxyG – material ageingH – process type
noise factors
y = f (A,B,C,D,E,F,G,H)?
Chalmers University of Technology
Experimental designD E F G H-1 -1 -1 1 -1 20751 -1 -1 1 1 2117-1 1 -1 -1 1 22211 1 -1 -1 -1 2227
-1 -1 1 -1 1 22011 -1 1 -1 -1 2179-1 1 1 1 -1 19881 1 1 1 1 1858-1 -1 -1 1 -1 18291 -1 -1 1 1 1978-1 1 -1 -1 1 21111 1 -1 -1 -1 2205
-1 -1 1 -1 1 2127
A B C 1 -1 1 -1 -1 2106-1 -1 1 -1 1 1 1 -1 1870
1 -1 -1 1 1 1 1 1 1879-1 1 -1 -1 -1 -1 1 -1 22451 1 1 1 -1 -1 1 1 2242
-1 1 -1 -1 1 22451 1 -1 -1 -1 2258
-1 -1 1 -1 1 22061 -1 1 -1 -1 2207-1 1 1 1 -1 20531 1 1 1 1 2188-1 -1 -1 1 -1 22191 -1 -1 1 1 2145-1 1 -1 -1 1 21741 1 -1 -1 -1 2265
-1 -1 1 -1 1 22411 -1 1 -1 -1 2187-1 1 1 1 -1 22081 1 1 1 1 2181
Process variables (control factors)A Curing temperatureB PressureC Holding time
Incoming material (noise factors)D Proportion of hardenerE Thermo-plastic contentF Proportion of epoxyG Material agingH Type of process
Process
Product
Chalmers University of Technology
1 I EFG DEFH ABC ABCEFG ABCDEFH ABCDGH DGH 2 A AEFG ADEFH BC BCEFG BCDEFH BCDGH ADGH 3 B BEFG BDEFH AC ACEFG ACDEFH ACDGH BDGH 4 D DEFG EFH ABCD ABCDEFG ABCEFH ABCGH GH 5 E FG DFH ABCE ABCFG ABCDFH ABCDEGH DEGH 6 F EG DEH ABCF ABCEG ABCDEH ABCDFGH DFGH 7 AB ABEFG ABDEFH C CEFG CDEFH CDGH ABDGH 8 AD ADEFG AEFH BCD BCDEFG BCEFH BCGH AGH 9 AE AFG ADFH BCE BCFG BCDFH BCDEGH ADEGH 10 AF AEG ADEH BCF BCEG BCDEH BCDFGH ADFGH 11 BD BDEFG BEFH ACD ACDEFG ACEFH ACGH BGH 12 BE BFG BDFH ACE ACFG ACDFH ACDEGH BDEGH 13 BF BEG BDEH ACF ACEG ACDEH ACDFGH BDGH 14 DE DFG FH ABCDE ABCDFG ABCFH ABCEGH EGH 15 DF DEG EH ABCDF ABCDEG ABCEH ABCFGH FGH 16 EF G DH ABCEF ABCG ABCDH ABCDEFGH DEFGH 17 ABD ABDEFG ABEFH CD CDEFG CEFH CGH ABGH 18 ABE ABFG ABDFH CE CFG CDFH CDEGH ABDEGH 19 ABF ABEG ABDEH CF CEG CDEH CDFGH ABDFGH 20 ADE ADFG AFH BCDE BCDFG BCFH BCEGH AEGH 21 ADF ADEG AEH BCDF BCDEG BCEH BCFGH AFGH 22 AEF AG ADH BCEF BCG BCDH BCDEFGH ADEFGH 23 BDE BDFG BFH ACDE ACDFG ACFH ACEGH BEGH 24 BDF BDEG BEH ACDF ACDEG ACEH ACFGH BFGH 25 BEF BG BDH ACEF ACG ACDH ACDEFGH BDEFGH 26 DEF DG H ABCDEF ABCDG ABCH ABCEFGH EFGH 27 ABDE ABDFG ABFH CDE CDFG CFH CEGH ABEGH 28 ABDF ABDEG ABEH CDF CDEG CEH CFGH ABFGH 29 ABEF ABG ABDH CEF CG CDH CDEFGH ABDEFGH 30 ADEF ADG AH BCDEF BCDG BCH BCEFGH AEFGH 31 BDEF BDG BH ACDEF ACDG ACH ACEFGH BEFGH 32 ABDEF ABDG ABH CDEF CDG CH CEFGH ABFEGH
Confounding pattern
Chalmers University of Technology
1 I 2 A -49,0625 3 B 143,3125 4 D 46,0625 5 E 13,0625 6 F -23,3125 7 AB -54,8125 8 AD -130,313 9 AE -0,4375 10 AF 10,8125 11 BD -26,6875 12 BE 7,8125 13 BF 30,9375 14 DE 38,9375 15 DF -2,8125 16 EF 8,3125 17 ABD 14,5625 18 ABE -34,0625 19 ABF -6,9375 20 ADE 4,8125 21 ADF 10,0625 22 AEF -17,4375 23 BDE -8,0625 24 BDF 31,9375 25 BEF 30,1875 26 DEF 92,5625 27 ABDE -4,8125 28 ABDF -10,5625 29 ABEF 17,3125 30 ADEF -6,1875 31 BDEF -2,0625 32 ABDEF -25,8125
Contrasts!
Chalmers University of Technology
Analysis of the experiment
-3
-2
-1
0
1
2
3
-150 -100 -50 0 50 100 150 200
G
contrasts
BBG
Chalmers University of Technology
1 I EFG DEFH ABC ABCEFG ABCDEFH ABCDGH DGH 2 A AEFG ADEFH BC BCEFG BCDEFH BCDGH ADGH 3 B BEFG BDEFH AC ACEFG ACDEFH ACDGH BDGH 4 D DEFG EFH ABCD ABCDEFG ABCEFH ABCGH GH 5 E FG DFH ABCE ABCFG ABCDFH ABCDEGH DEGH 6 F EG DEH ABCF ABCEG ABCDEH ABCDFGH DFGH 7 AB ABEFG ABDEFH C CEFG CDEFH CDGH ABDGH 8 AD ADEFG AEFH BCD BCDEFG BCEFH BCGH AGH 9 AE AFG ADFH BCE BCFG BCDFH BCDEGH ADEGH 10 AF AEG ADEH BCF BCEG BCDEH BCDFGH ADFGH 11 BD BDEFG BEFH ACD ACDEFG ACEFH ACGH BGH 12 BE BFG BDFH ACE ACFG ACDFH ACDEGH BDEGH 13 BF BEG BDEH ACF ACEG ACDEH ACDFGH BDGH 14 DE DFG FH ABCDE ABCDFG ABCFH ABCEGH EGH 15 DF DEG EH ABCDF ABCDEG ABCEH ABCFGH FGH 16 EF G DH ABCEF ABCG ABCDH ABCDEFGH DEFGH 17 ABD ABDEFG ABEFH CD CDEFG CEFH CGH ABGH 18 ABE ABFG ABDFH CE CFG CDFH CDEGH ABDEGH 19 ABF ABEG ABDEH CF CEG CDEH CDFGH ABDFGH 20 ADE ADFG AFH BCDE BCDFG BCFH BCEGH AEGH 21 ADF ADEG AEH BCDF BCDEG BCEH BCFGH AFGH 22 AEF AG ADH BCEF BCG BCDH BCDEFGH ADEFGH 23 BDE BDFG BFH ACDE ACDFG ACFH ACEGH BEGH 24 BDF BDEG BEH ACDF ACDEG ACEH ACFGH BFGH 25 BEF BG BDH ACEF ACG ACDH ACDEFGH BDEFGH 26 DEF DG H ABCDEF ABCDG ABCH ABCEFGH EFGH 27 ABDE ABDFG ABFH CDE CDFG CFH CEGH ABEGH 28 ABDF ABDEG ABEH CDF CDEG CEH CFGH ABFGH 29 ABEF ABG ABDH CEF CG CDH CDEFGH ABDEFGH 30 ADEF ADG AH BCDEF BCDG BCH BCEFGH AEFGH 31 BDEF BDG BH ACDEF ACDG ACH ACEFGH BEFGH 32 ABDEF ABDG ABH CDEF CDG CH CEFGH ABFEGH
Confounding pattern
Chalmers University of Technology
Error structure of a Strip-Block ExperimentD E F G H-1 -1 -1 1 -1 20751 -1 -1 1 1 2117-1 1 -1 -1 1 22211 1 -1 -1 -1 2227
-1 -1 1 -1 1 22011 -1 1 -1 -1 2179-1 1 1 1 -1 19881 1 1 1 1 1858-1 -1 -1 1 -1 18291 -1 -1 1 1 1978-1 1 -1 -1 1 21111 1 -1 -1 -1 2205
-1 -1 1 -1 1 2127
A B C 1 -1 1 -1 -1 2106-1 -1 1 -1 1 1 1 -1 1870
1 -1 -1 1 1 1 1 1 1879-1 1 -1 -1 -1 -1 1 -1 22451 1 1 1 -1 -1 1 1 2242
-1 1 -1 -1 1 22451 1 -1 -1 -1 2258
-1 -1 1 -1 1 22061 -1 1 -1 -1 2207-1 1 1 1 -1 20531 1 1 1 1 2188-1 -1 -1 1 -1 22191 -1 -1 1 1 2145-1 1 -1 -1 1 21741 1 -1 -1 -1 2265
-1 -1 1 -1 1 22411 -1 1 -1 -1 2187-1 1 1 1 -1 22081 1 1 1 1 2181
εsεs1
εs2
εw1
εw2
εw3
εw4
εw
ε32
ε1
ε
Chalmers University of Technology
A B D AD error -1 -1 -1 1 εw1+ εs1+ ε1 -1 -1 1 -1 εw1+ εs2+ ε2 -1 -1 -1 1 εw1+ εs3+ ε3 -1 -1 1 -1 εw1+ εs4+ ε4 -1 -1 -1 1 εw1+ εs5+ ε5 -1 -1 1 -1 εw1+ εs6+ ε6 -1 -1 -1 1 εw1+ εs7+ ε7 -1 -1 1 -1 εw1+ εs8+ ε8 1 -1 -1 -1 εw2+ εs1+ ε9 1 -1 1 1 εw2+ εs2+ ε10 1 -1 -1 -1 εw2+ εs3+ ε11 1 -1 1 1 εw2+ εs4+ ε12 1 -1 -1 -1 εw2+ εs5+ ε13 1 -1 1 1 εw2+ εs6+ ε14 1 -1 -1 -1 εw2+ εs7+ ε15 1 -1 1 1 εw2+ εs8+ ε16 -1 1 -1 1 εw3+ εs1+ ε17 -1 1 1 -1 εw3+ εs2+ ε18 -1 1 -1 1 εw3+ εs3+ ε19 -1 1 1 -1 εw3+ εs4+ ε20 -1 1 -1 1 εw3+ εs5+ ε21 -1 1 1 -1 εw3+ εs6+ ε22 -1 1 -1 1 εw3+ εs7+ ε23 -1 1 1 -1 εw3+ εs8+ ε24 1 1 -1 -1 εw4+ εs1+ ε25 1 1 1 1 εw4+ εs2+ ε26 1 1 -1 -1 εw4+ εs3+ ε27 1 1 1 1 εw4+ εs4+ ε28 1 1 -1 -1 εw4+ εs5+ ε29 1 1 1 1 εw4+ εs6+ ε30 1 1 -1 -1 εw4+ εs7+ ε31 1 1 1 1 εw4+ εs8+ ε32
Chalmers University of Technology
Variances of the contrasts
8
1
32
1
4
1
32
1
32
1
416
1
816
1
16
1
isi
jifactorsmaterial
iwi
jifactorsprocess
iinsinteractiomaterialprocess
contrasts
contrasts
contrasts
22
22
2
8
1
2
18
18
1
sfactorsmaterial
wfactorsprocess
nsinteractiomaterialprocess
contrastsVar
contrastsVar
contrastsVar
Chalmers University of Technology
-3
-2
-1
0
1
2
3
-150 -100 -50 0 50 100 150
Identification of location effects
•B, G and BG was determined to be active based on engineering knowledge and the normal plots
Process factors Factors and interactionsassociated with incoming material
Interactions between ”process factors”and ”incoming material factors”
Chalmers University of Technology
Model
ˆ( , ) 2132 72 65 46
2132 72 46 65
y B G B G BG
B B G
B ≈ 1.4
Chalmers University of Technology
Conclusions
• The storage time of the incoming material (G) is causing variation in the bending strength of the composite material.
• If the pressure (B) is set at high level the bending strength is made insensitive to the storage time.
Chalmers University of Technology
Randomisation and split-plot• View randomisation as an insurance against
unknown factors - buy as much as you can afford
• It is not always advisable to reset all factor levels between each experiment!– Can be very time consuming and expensive
– Split-plot designs allow some contrasts of interest to be estimated with great precision. This characteristic can, for example, be useful in robust design experiments