Post on 06-Apr-2020
transcript
Part 3
Dynamic response designsThe Process
X1 Xn
(Reduce variance in controllable factors)
Control parameters (Xs)
N1 Nn
(Minimise effect of variance of noise factors)
Noise factors
Y
Response
S
Signal
Introduction to Dynamic response designs
o Dynamic designs are applied when the output response is not a static value but varies as per the change in Signal factor
Examples:
• Acceleration of a motor vehicle depends on how much the accelerator pedal (Signal) is pressed upon (Acceleration ∝ Pedal depression)
• Volume increase or decrease depends on how the Volume control button (Signal) of the amplifier is handled
o In general, a linear relationship is desired between the Response & Signal
o To use the Dynamic design, an “Ideal function” between the output response and the signal factor has to be defined [based on system knowledge]
o Hence, the design objective is:
• To estimate the most robust result for the Ideal Function by choosing optimum values for the control factors and minimising the impact of variation triggered by the Noise factors
• This is achieved by optimising the Signal to Noise ratio for the design
[Signal to Noise ratio for Dynamic designs differ from the static designs]
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Dynamic designs: Signal to Noise ratio
The Ideal Function is given by: Y = b . M
§ where Y- response output, M- Signal input, b- Proportionality constant used by the scaling factor to adjust slope between Y and M (Usually b = 1)
§ For M = 0, Y = 0. Hence, 0 acts as a reference point (All lines fit through the Origin)
o Quality loss, in case the optimal slope for Y = b . M is not achieved, is proportional to (𝑌#$−𝑀#)
§ Average Quality Loss is given by: 𝑄* = ,-./∑ ∑ (𝑌#$ − 𝑀#)1/
$23-#23
o Quality Loss in this case can be due to two reasons:
§ Slope characteristics of the Input Signal and Output Response [I/O]:- when Slope b ≠ 1 [i.e. deviation exists from the best fit (linear) line]
§ Linearity characteristics of the Input Signal and Output Response [I/O]:- when there is deviation from linearity [Variance]
o Signal to Noise ratio is given by:
§ n = 10 Log10 [45
65], where 𝜎2 is the variance, which is the mean of the sum of squares of deviations of measured data points from the best fit line
§ Hence, the design objective can be summarised as,
• Relation between the Input Signal and Output Response to be linear and directly proportional
• This linear relationship is to show least deviation from the Ideal Function Line©Harish Haridasan6
Example [Taguchi’s Dynamic Design]
Case:
o Bar code scanners, to be robust, must scan the bar codes accurately & precisely in all weather and lighting conditions. Our objective is to
design a robust scanning technology for such bar code scanners.
o The Output response [Y] is the measured width of bars and spaces between the bars which depends on the actual width of these bars
and spaces [Signal – M]. M has 3 levels (i.e. The bar code has 2 black bars with a white space in between)
o Such scanners can be used any where in the world & hence would be exposed varied conditions of ambient lighting which is a Noise
Factor. We will consider 2 levels of ambient lighting in this experiment
o The critical control factors include the Bar-code distance from the scanner (A), the Mirror radius (B) and the aperture (C). Each factor has
2 levels
o The Factor Levels are:
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Parameters Type Level 1 Level 2 Level 3
Distance [A] Control Factor 1 2
Radius [B] Control Factor 1 2
Aperture [C] Control Factor 1 2
Actual Bar-width [M] Signal 1.25 𝝁 25 𝝁 250 𝝁
Ambient Lighting [N] Noise 50 Lux 150 Lux
Array Experimental runs
Max. # of Factors
Max. # of Factors that can be considered at various Factor Levels
Level 2 Level 3 Level 4 Level 5
L: 4 3 3
L; 8 7 7
L< 9 4 4
L31 12 11 11
Example [Taguchi’s Dynamic Design]
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Factors Levels (S) Degrees of Freedom (Df = S – 1)
[A] 2 1
[B] 2 1
[C] 2 1
Total Degrees of Freedom ∑ [𝑆?−1]?
#233
Systems design
Step 1: Determine the Degrees of Freedom
Step 2:
o Total number of Experimental runs, TExperiments = 1 +
∑ [𝑆?−1]?#23 , would be 1 + (3) = 4
Step 3:
o The suitable Orthogonal Array should be ≥Number of
Experiments [ 4 ]
o Referring to the Standard OA Table, the nearest 2 level
OA is L; [2**7]
Example [Taguchi’s Dynamic Design]
Step 4:
o We visualise the “Required Linear Graph” (RLG)
o Since, we have to study the main factors A, B, C only with no interaction effects, the RLG will look as,
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A B C
Step 5:
o The “Standard Linear Graph” (SLG) to be used in this case, for OA L; is as
shown,
Step 6:
o We will modify this “Standard Linear Graph” (SLG) to match it with the
“Required Linear Graph”
1
2
3
46
5
7
1
2
3
46
5
7 3 5 6
Example [Taguchi’s Dynamic Design]
Step 7:
o In preparation to determine the experimental layout, we need to allocate the Factors to the Columns of OA
o This can be done as shown:
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Factors To be assigned to Column/ Node
A 1
B 2
C 4
1/ A
2/ B 4/ C7 3 5 6
Example [Taguchi’s Dynamic Design]
Step 7:
o We choose the experimental layout referring to the default OA design layout
o The default layout can be depicted as shown below.
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Runs Column-1 Column-2 Column-3 Column-4 Column-5 Column-6 Column-7
1 1 1 1 1 1 1 1
2 1 1 1 2 2 2 2
3 1 2 2 1 1 2 2
4 1 2 2 2 2 1 1
5 2 1 2 1 2 1 2
6 2 1 2 2 1 2 1
7 2 2 1 1 2 2 1
8 2 2 1 2 1 1 2
o Based on our factor allotment of Aà1, Bà2, Cà4, we choose the 1st, 2nd and 4th columns from the default array
Example [Taguchi’s Dynamic Design]
Parameter design: Conducting experiments to identify
the optimal settings of design parameters that improve
the performance characteristics and reduce the
sensitivity of engineering design to sources of variation
or noise
Step 8: The Experimental Layout (as shown à)
o Since we have one Noise factor at 2 levels, the outer
array will have 2 Columns where we will collect
Output response data twice, once for each Noise level
o We have a Signal Factor at 3 levels. Hence, each run
(for a specific combination of Factor levels) will be
repeated thrice, i.e. for each Signal Level
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Distance [A]
Radius [B]
Aperture [C]
Bar width [Signal – M]
Output ResponseLighting (N-1) Lighting (N-2)
1 1 1 1.25 1.303 1.3121 1 1 25 25.057 25.0721 1 1 250 250.207 250.4411 1 2 1.25 1.286 1.2921 1 2 25 25.048 25.0751 1 2 250 250.257 250.0391 2 1 1.25 1.286 1.2911 2 1 25 25.047 25.0751 2 1 250 250.213 250.0131 2 2 1.25 1.296 1.2961 2 2 25 25.057 25.0861 2 2 250 250.219 250.352 1 1 1.25 1.287 1.292 1 1 25 25.049 25.0612 1 1 250 250.217 250.62 1 2 1.25 1.285 1.2862 1 2 25 25.044 25.0652 1 2 250 250.2 250.5382 2 1 1.25 1.288 1.2882 2 1 25 25.049 25.0692 2 1 250 250.217 250.5852 2 2 1.25 1.283 1.2882 2 2 25 25.044 25.0552 2 2 250 250.2 250.553
Example [Taguchi’s Dynamic Design]
Most Statistical Softwares, even as simple as Minitab, provide for such designs. In Minitab – the above layout can be generated by following steps:
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Example [Taguchi’s Dynamic Design]
o This provides the design layout
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o Experiments are conducted and the response is updated as shown:
Example [Taguchi’s Dynamic Design]
Step 9: Analysis of the experiment output
In Minitab, the analysis can be initiated as shown:
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Objectives include:
§ Finding the optimum design (Vital Factors)
§ Understanding the effects (Main/ relative) of individual Factors
§ Product/ process performance at the optima
Example [Taguchi’s Dynamic Design]
Interpretation:
o Check the Estimated Model Coefficients
§ The order of the coefficients by absolute value indicates the relative importance of each factor to the response (Factor with highest coefficient has the greatest impact)
o At 10% a, the Factor with p value < 0.1 is statistically significant
o In the ANOVA table also, the order of Sum of squares & adjusted sum of squares indicate the relative importance of each factor to the response
In this case, the Object distance from scanner has the greatest impact & is statistically significant
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o Check the Response table:
§ This table summarises the average of each response characteristic (like S/N ratio) for each factor level
§ The level average for each factor determines the level which provides the best output response
§ The delta statistic is the highest minus lowest average for each factor
§ Rank indicates the relative importance of each factor to the response. Factor with highest delta value is assigned Rank-1
o Here, Object distance is again ranked the most critical.
Example [Taguchi’s Dynamic Design]
Interpretation:
o Response tables for both S/N ratio and Standard Deviation show that Object distance
from the scanner is the most critical factor
o Since our objective is to reduce variance in the way the bars (in bar code) are measured,
we would aim to reduce the standard deviation and maximise the S/N ratio
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o Table shows that Level 1 of Factor- Distance, and Level 2 of Factors- Radius, Aperture produce the highest S/N ratio and the lowest standard
deviation
o Optimal Factor Levels: A1, B2, C2
Example [Taguchi’s Dynamic Design]
Interpretation:
o Main Effect Plot for S/N ratio also shows the same Factor Levels to be optimum
§ Level 1 of Factor- Distance, and Level 2 of Factors- Radius, Aperture maximises the S/ N ratio
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Mea
n of
SN
ratio
s
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21
23
22
21
20
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Distance-A Radius-B
Aperture-C
Main Effects Plot (data means) for SN ratios
Dynamic Response: Signal reference 0 Response reference 0
Mea
n of
Sta
ndar
d De
viat
ions
21
0.12
0.11
0.10
0.09
0.08
21
21
0.12
0.11
0.10
0.09
0.08
Distance-A Radius-B
Aperture-C
Main Effects Plot (data means) for Standard Deviations
Dynamic Response: Signal reference 0 Response reference 0
o Main Effect Plot for Standard Deviation also shows the same Factor Levels to be optimum
§ Level 1 of Factor- Distance, and Level 2 of Factors- Radius, Aperture minimises the variance
o Hence, the ideal setting for the Factors is:
§ Object Distance [A] – Level 1
§ Radius [B] – Level 2
§ Aperture [C] – Level 2
Example [Taguchi’s Dynamic Design]
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Step 10: Confirmatory runs
o Once the Optima is arrived at, it is important to verify the results by running a confirmatory experiment at the optimal factor levels
In case we use Nominal – The Best Signal to Noise ratio in Dynamic design, we use a 2-step optimisation process:
1. Reduce the sensitivity of design parameters to sources of variation or noise
o Identify the Factors which have the greatest effect on variance
o Identify the levels for these Factors which minimise the variance
2. Shift the Mean to Target
o Identify a Scaling Factor (out of the Control Factors) that has significant influence on Mean but relatively lesser impact on S/ N ratio [Mean &
Standard deviation scale together]
o Factor can be identified from the Response tables and the Main Effects Plot for S/N ratio & Mean/ Slope
Example [Taguchi’s Dynamic Design]
Step 11: Tolerance Design
o Tolerance design becomes significant when the target response quality
has not been achieved during the parameter design
o The objective is to determine tighter tolerances around the optimal
settings identified during the parameter design process
o It leads to a robust design in which the designed process delivers at the
target and is least impacted by the variance or noise factors
o Approach guide:
§ Variance in Response [Y/ CTQ] is transferred to the Control Factors
[X] as we can control the Xs
§ This tightens the tolerance/ specifications for Xs as defined by the
function Y = f(x)
§ Also helps to identifies which design level is more robust for a given
variability in X [Factor]
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Transfer FunctionY = f(x)
f(A)
f(B)
A B
For the same variability in X, Design level B has lower CTQ variability than A
Example [Taguchi’s Dynamic Design]
Tolerance Design methods:
How do we derive the variation in Response [Y] from the model of Y?
o Partial derivative method
§ Useful for analytical problems when function is non linear and may contain
higher order terms
§ Taylor series expansion is used
• If Y = f(X1, X2, ...Xn), then
• 𝜎C1 ≈ 𝑓′. 𝜎GH1 + 𝑓′. 𝜎G5
1 + ⋯ 𝑓′. 𝜎GK1where f’ represents the first
order differential term NONGP
,from the Taylor series
§ This calculated 𝜎C is then used statistically to recalculate the tolerances
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o Monte Carlo method
§ Useful with complex models of Y
§ Transfer function Y = f(X1, X2, ...Xn) along with
the variability in Xs is used to simulate the 𝜎C in
a simulator like Crystal Ball.
Example [Taguchi’s Dynamic Design]
Tolerance Design example:
o Lets say the design model on Y is given by Y = 10 + 2x. So x = CR3S1
o Specification of Y is defined [USL = 100, LSL = 50]. Hence, corresponding X-values would be XL = 20 and XU = 45 (derived from transfer function)
o Model / Process variance 𝜎C , as calculated in tolerance design, is 3
o There is a measurement system variance 𝜎T of = 2, which also need to be accounted in tolerance calculation
o Hence, total variance 𝜎U is:
§ 𝜎U1 = 𝜎C1 + 𝜎T1 = 31 +21 = 13à 𝜎U =√13 =3.6
o Normally a buffer of 3𝜎 (to 99.73% of data when short term standard deviation is considered) is taken at both USL and LSL
§ USL = 100 - (3 x 3.6) = 89.2 and LSL = (3 x 3.6) + 50 = 60.8
o Hence, the tightened tolerance on X will be X’L = 25.4 and X’U = 39.6 (derived from transfer function)
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Transfer FunctionY = 10 + 2x
XXUXL
Y
USL (100): YU
LSL (50): YL
X’UX’L
Y’U [89.2]
Y’L [60.8]
[45][20] [39.6][25.4]
o Design objectives & S/N ratio
o Systems design incl. Experiment layout
o Parameter design including analysis [ANOVA, Responsetables, Main Effects plot]
o Determining Optima & the confirmatory runs
o 2-Step optimisation
o Tolerance design, when required
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Parameters design
Taguchi’s Design: Approach Summary
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Problem definition
Factors & interactions
Levels for Factors & interactions
Select Orthogonal Arrays (OA)
Factor allocation using Linear graphs
Select Orthogonal Arrays (OA)
Determine the Noise factors, Levels
Determine the Signal Factors
Finalise the Experiment layout
Randomise & Run the experiments
Record the Responses
Determine the S/N ratio
Conduct ANOVA
Determine the critical design factors
Determine optimum factor levels
Use Scaling Factor to shift mean, if reqd.
Determine Confidence intervals
Verify with confirmatory runs
Design tighter tolerances
Draw Conclusions
Is Quality Loss in Response satisfactory?
Systems design Tolerance design
Yes
No
Thank you !!!
Harish Haridasan || connect@sigmaquotient.com || +91- 94492 40463 || Bangalore, India