Taguchi's Designs Ver1.13 Print -...

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]

©Harish Haridasan4

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:

©Harish Haridasan8

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


©Harish Haridasan9

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]


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,

©Harish Haridasan10

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


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:


Factors To be assigned to Column/ Node

A 1

B 2

C 4

1/ A

2/ B 4/ C7 3 5 6


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.


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


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


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


Most Statistical Softwares, even as simple as Minitab, provide for such designs. In Minitab – the above layout can be generated by following steps:



o This provides the design layout


o Experiments are conducted and the response is updated as shown:


Step 9: Analysis of the experiment output

In Minitab, the analysis can be initiated as shown:


Objectives include:

§ Finding the optimum design (Vital Factors)

§ Understanding the effects (Main/ relative) of individual Factors

§ Product/ process performance at the optima


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


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.


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


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


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


Mea

n of

SN

ratio

s

21

23

22

21

20

19

21

21

23

22

21

20

19

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



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


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]


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


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


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.


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)


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


Parameters design

Taguchi’s Design: Approach Summary


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 || [email protected] || +91- 94492 40463 || Bangalore, India

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