doe

INTRODUCTION TO DESIGN OF EXPERIMENTS

A METHOD FOR OPTIMIZATION OF PRODUCTS AND PROCESSES

2007

Bohdan W. Oppenheim Mechanical & Systems Engineering

LMU|LA

©2007 Bohdan W. Oppenheim [email protected]

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DESIGN OF EXPERIMENTS

IntroductionContents

1. Introduction2. Full factorials 2f3. Fractional factorials 2f-i4. Nonlinear methods


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• DEFINITION: DOE = Statistical method for optimization of advanced products and processes

– Self-contained and simple (2 day course)– Practical and useful for most applications– Spreadsheets (EXCEL) are sufficient

• ADVANCED DOE– Statistics more advanced– Variance analysis, multiple regression, math analysis– Special software required

• PRINCIPLES– Determine an empirical equation for the system in the

most efficient way– Determine the factor settings that yield the optimum

product/process• OPTIMUM

– Target value (e.g., dimension) , or maximum (yield) , or minimum (cost)

– Variability (scatter) reduction– Product or process robustness

DESIGN OF EXPERIMENTSIntroduction


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“STATISTICAL METHOD" VERSUS “TAGUCHI METHOD“

• TAGUCHI: > "Guru of quality optimization" - popularized the method in production

> Created the concept of "product life-cycle loss function"

> Created the product robustness concept> These contributions caused a quality revolution

– Basic DOE identical to the rigorous methods– "Signal to noise ratios" and "noise factors" controversial - better methods are

available– Advantages dominate the disadvantages– Rigorous scientific literature comparing the two methods is available

• "STATISTICAL METHOD" - is practiced by most scientists and statisticians

• DOE often yields dramatic improvements: e.g., cost reduction of 30%, quality improvement of 200%, yield improvement of 25%.



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• Sir R.A. Fisher, England, 1920's , agriculture. Routinely used in chemical and biological sciences

• Until 1980, taught as advanced and esoteric statistics

• In Japan, due to Deming and Taguchi, DOE becomes a routine optimization tool, and contributes to the quality revolution in electronics and automotive engineering

• Taguchi moves to Detroit (Am. Suppliers Institute) and popularizes DOE in auto industry

• DOE becomes an integral part of properly presented TQM, together with SPC, DOE and QFD

• DOE still not taught in undergraduate engineering, limited to graduate programs (the leftover of "advanced statistics" mentality)

• Huge list of successes available in literature



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Applications Of DOE

• Traditional: in agriculture, chemical and biological industries

• Ideal for optimization of mass production elements• Ideal for parametric optimization phase of product design

(between "conceptual" and "detailed")• Some successes in "soft" applications of office work and

software development• Least applicable for short-series labor-intensive

processes, crafts and intellectual work



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ONE APPLICATION OF DOE: OPTIMIZATION OF PROTOTYPE• In Japan, DOE is often used in design optimization (optimization of

"stable" prototype)

Traditional Modern

CONCEPTUAL DESIGN

MATURE PROTOTYPE

OPTIMIZATION OF PROTOTYPE

FINAL DESIGN

PRODUCTION OPTIMIZATION

PRODUCTION


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APPLICATIONS OF DOE• IN DESIGN

– Optimization of features, layouts, dimensions, tolerances...– Making the product robust– Parametric design

• IN PRODUCTION– Setting the procedures "on target"– Variability reduction (scatter)– Making the process robust– Maximization of the yield– Minimization of cost

• LIFE CYCLE– Prevention of failures and reduction of warranty costs– Increased customer satisfaction and loyalty– Reduction of life-cycle costs – Increased revenue of the firm



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PRODUCT LIFE-CYCLE LOSS FUNCTION - AFTER TAGUCHI

• Consider a mechanical part with dimension and tolerances given asTarget Value plus or minus Tolerance

Traditional: OK, within specs Taguchi: life-cycle loss function

Loss OK Loss L=k [S2 +(Y -T)2 ]Loss $

LSL Target T USL T Y

• DOE is a tool for– setting the production on target Y ⇒ T– scatter (variability) reduction S ⇒ 0– maximizing product robustness under uncontrollable factors S ⇒ 0

YY

_

_

_


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SUMMARY OF DOE STEPS1. Select one system equation from the available set.

2. Empirically, determine the equation coefficients, using the mosteffective methods (the fewest number of test runs)

3. Based on the equation, set the factors for optimum settings of the output response:1. optimum mean response (minimum, maximum, target value)

2. minimum variance (scatter)

3. maximum robustness (insensitivity to inevitable uncontrollable factors)

4. Repeat steps 1-3 with fewer selected factors using a better equation



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SYSTEM MODELFACTORS

AB Y= Response (single) C

etc.

System equation:

Y=Y + cAA + cBB + cCC +...+ cABAB + cACAC + cBCBC + ...+ cABCABC+...cA2A2 + ...

where: A,B,C,... = Factors that can be numerical or categorical (e.g., contractor )cA, cB , ...= Coefficients to be determined empirically AB, AC, BC,..., ABC, ....= two-way, three-way ... interactionsA2, B2, A2B,...... = terms with nonlinear powers


SYSTEM (PROCESS)

=


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PROJECT STEPS

1. Administrative: objectives, budget, schedule, instruments, people responsible

2. Select: factors and their ranges, the response, replicates.

If done incorrectly, DOE wasted

3. Select the DOE method (i.e., the system equation)

Trade off between the number of runs and statistical confidence

4. Perform test runs

5. Analysis (usually on computer), compute the equation coefficients, and – Confidence measures, if applicable

– Use spreadsheets for simple cases, special software otherwise

6. Verify the equation validity with one or more control runs

7. Reject unimportant factors. Select the factor settings for optimization.

Often: conflicts between various optimization objectives. Resolved from non-DOE considerations.

8. Further optimization: next DOE using only the selected factors, and a better equation



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TRADITIONAL METHOD: VARY ONE-FACTOR-AT-A-TIME (OFAT)

• Ignores all interactions (simultaneous effects of factor combinations)• Wrong method, routinely taught to engineers• Often leads to incorrect conclusions

EXAMPLE (after Montgomery): Conception of a babyFactors: M= Male (1=present, 0=absent)

F= Female (1=present, 0=absent)Response Y= conception

0,1 1,1 conception

M

0,0 1,0F

• Simultaneous effect of factors M and F decides about the response• OFAT method ignores the MF interaction• Observation of factors M and F alone not sufficient


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DESIGN OF EXPERIMENTSIntroduction- Examples of Interactions


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SUMMARY OF THE OFAT METHOD• No guarantee of finding the optimum• If interactions present - incorrect results• The OFAT method is ineffective with a large number of factors (see

Montgomery)

Number of factors Relative efficiency of the factorial method2 1.53 24 2.55 36 3.5

where Efficiency = the ratio of the number of test runs in the full factorial experiment to the number of runs required by the OFAT method. (Montgomery, "Design of Industrial Experiments")



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MAKING THE PRODUCT OR PROCESS ROBUST

• Definition of robustness– Insensitivity to the inevitable uncontrollable factors (e.g., outside temperature,

pressure, humidity, geographical location where the product is used, tool condition, human factors,...)

• Robustness is best determined from simple consideration of interactions

– Abscissa (horizontal axis) = uncontrollable factor (must be controlled in DOE)– Ordinate (vertical axis) = response Y– Parameter = controllable factor

• Select the controllable factor so as to minimize the variation of Y when the uncontrollable factor varies spontaneously



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EXAMPLE OF ROBUSTNESS ANALYSIS• Use above example, in transponded coordinates

Pressure = controllable factorTemperature = factor that is almost uncontrollable in production

but can be controlled in the experiment

• Yield decreases when pressure = 1.0, but increases when pressure = 0.5• Supposedly, the yield becomes independent of temperature (is robust ) when

pressure is about 0.75 (dashed line). This is a crude but powerful approach. • The yield is then 62 • In DOE optimization, we often have to trade off the best mean (here max. yield),

best robustness, and minimum variability


50

60

70

Y

50 75Temp

5045

8570

P=1

P=0.5

P=0.75


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Full Factorials 2f

"The most informative but also the most expensive"



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DESIGN OF EXPERIMENTSFull Factorials 2f

We explain the method using the following example.Grooves in a PCB are cut by a machine which excites vibrations in the PCB and the tool. Two factors are thought to influence the vibration. The factors and their ranges are listed:

9040B. Speed [RPM]

42A. Cutter size [mm]

HIGH LEVELLOW LEVELFACTOR

PCB, machine tool, cutter, foundation, ...

A (cutter size)

B (speed)Y = vibration amplitude

The “factor domain”plot:

A (cutter size)

4

2

40 Speed 90


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Step 1:Normalize the low and high levels of each factor to [-1,+1] (after Montgomery and Lawson; others use 0-1, 1-2, 0-2)We normalize categorical factor ranges similarly: e.g., contractor P = -1, contractor Q = 1

-1 +1 A normalized

A2

4


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Step 2.List all possible combinations of the factorsTwo factors (f=2) in our example are: A, BPossible combinations of A, B A,B,AB

Other examples: factors A,B, C A,B,C, AB,AC,BC, ABC

factors A,B,C,D A,B,C,D, AB, AC, AD, BC, BD,CD, ABC, ABD, ACD, BCD, ABCD

Class exercise: do it for 5 factors: A,B,C,D,E

What is the formula for the number of combinations?

Answer: 2f-1


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Step 3.Prepare the following table, using EXCEL, by hand, or standard software

Columns• N = run designator (will be explained later)• I = the column contains only pluses (i.e., the values of +1 for all runs)• 2f-1 combinations of the letters, as above (A,B,AB in this example)• R columns for measurements, R>=1.

R is called the replicates (repetitions of the runs)R > 1 is recommended. R > 1 is required for variability (scatter) analysis.

• R columns for residuals • Average (average of R measurements) • Standard deviation S (this is possible only if R>1)

R3 R4R2R1Y4Y3Y2Y1ABBA

SResidualsMeasurementsCombinationsIN Y=

Y=


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Step 4: generate the table rows (runs):

• 2f runs (all combinations of signs for f factors )In our example: f=2 factors, 2f=22=4 runs, so we need 4 rows.

• The order of the runs is arbitrary.• Each column of signs must be balanced, i.e., it must contain the

same number of "+" and "-". In other words, the sum of all terms in the column vector must equal 0.

• If all columns of signs are balanced, the design is called orthogonal. • The orthogonality is necessary for avoiding bias towards the low or

the high level.


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• We use the following short hand notation for labeling the runs:(1) = all factors are set at low levels (-1).a = factor A alone is set at high levelb = factor B alone is at high levelab = factors A and B are set at high levels

++ab

+-b

-+a

--(1)

BARUNS

Another example:In the design with factors A,B,C,D,E, acd denotes the fact that factorsA,C,D are "high" (+1) and the remaining factors B and E are "low" (-1).


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Step 5. Enter the signs into the combination columns• We enter the signs into all interaction columns (AB in our example, and

generally AB, AC, BC, ABC,...) where the signs of any combination vector result from multiplying the component vectors element by element

• Example: signs of AB = the signs of vector A times the signs of vector B• The columns (vectors) of interactions must also be balanced.• Non-orthogonal designs exist in theoretical literature but are difficult to use

and require special software, and for that reason are rarely used.

+

-

-

+

AB

++ab

+-b

-+a

--(1)

BARUNS


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Step 6: the last two rows in our table:Δ= “Effect” of the given combination of factors (explained below)ci = Δi/2 = directional coefficients in the system equation

• Exception: the “I” column, where ci = the grand average

Y

Δ++++ab-+-+b--++a

c

+--+(1)R3 SR4R2R1Y4Y3Y2Y1ABBA

ResidualsMeasurementsCombinationsIN

Y=

Y=


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• Example of three factors: 23, factors: A, B, C.• The table of factor settings:

++++++++abc

---+-+++ab

--+-+-++ac

++----++a

-+--++-+bc

+-+--+-+b

+--++--+c

=+++---+(1)

ABCBCACABCBAIRUN

Note:• The number of runs = 2f. Such designs are called full factorials, in contrast to fractional

factorials that have 2f-i runs, where i > 1.• The number of factor and interaction columns = 2f-1 = all combinations of factors, up to the product of

all factors. Including the I column, we have 2f columns of signs.


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CLASS EXERCISE

Prepare a complete table for a two-level full factorial design with four factors 24, A,B,C,D.


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Step 7 (optional). Randomization• It is often desired to reorder the runs randomly, using a table of random

numbers or a computer-generated sequence of random numbers. • Random order is desirable in order to reduce the effects of an unknown

systematic variation such as ambient temperature (e.g., the shoptemperature which increases in the morning and decreases in the evening and which could introduce a trend into the measurements that would be difficult to explain on our model). This step is called randomization.

• In effect, we convert an environmental factor into noise • We change the original (arbitrary) order of runs from (1), a, b, ab, to a

random order, using the table on the following page:(1,2,3,4) = ("1",a,b,ab) reordered randomly become (3,1,2,4)=(b,"1",a,ab)

• (Other random orders - how many ? - are possible and are just as good).• In subsequent work in these notes, we ignore randomization.


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Step 8: Execute the runs and record the measurements.

Δ

39.936.343.941.0++++ab

14.215.114.515.9-+-+b

22.522.424.027.2--++a

c

14.412.918.918.2+--+(1)

R3 SR4R2R1Y4Y3Y2Y1ABBAResidualsMeasurementsCombinationsIN

Always use a "sanity check" on the numbers entered.

Y=


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Step 9: calculate the mean, residuals, and standard deviation for each run:• E.g., the mean in the first row: (18.2+18.9+12.9+14.4)/4=16.1• Residual = (replication) - (the mean of R replications)• E.g., in the first row

18.2-16.1 = 2.1, 18.9-16.1=2.80, 12.9-16.1=-3.20, 14.4-16.1=-1.70• Standard deviation for each run (we divide by R-1 rather than R since this is sample std.

deviation)SQRT { [ 2.102 +2.802 +(-3.20)2 +(-1.70)2 ] / 3} = 2.91

Δ

3.1440.275-.375-3.9753.6250.72539.936.343.941.0++++ab

0.7514.925-.7250.175-.4250.97514.215.114.515.9-+-+b

2.2424.025-1.525-1.625-.0253.17522.522.424.027.2--++a

c

2.9116.100-1.700-3.2002.8002.10014.412.918.918.2+--+(1)R3 R4R2R1Y4Y3Y2Y1ABBA

SResidualsMeasurementsCombinationsIN Y


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Step 10. Calculate the effects Δ in columns I, A, B, AB• Multiply the given column by the Y-bar column. For example, for column A:

- 1 16.100 - 16.100+1 x 24.025 = 24.025- 1 14.925 - 14.925+1 40.275 40.275

• Next, sum up the resultant column and divide by the number of pluses in that column, (by 2). We always divide by the number of positive contributors to the sum.

(-16.1+24.025-14.925+40.275) / 2=33.275 / 2=16.6375 • Column I has four pluses, so we divide by 4

(+1*16.1+1*24.025+1*14.925+1*40.275)/4=23.83• This result represents the grand mean of the system, which is the first term in the system equation.


Y

8.717.5416.6423.83Δ

3.1440.275-.375-3.9753.6250.72539.936.343.941.0++++ab

0.7514.925-.7250.175-.4250.97514.215.114.515.9-+-+b

2.2424.025-1.525-1.625-.0253.17522.522.424.027.2--++a

c

2.9116.100-1.700-3.2002.8002.10014.412.918.918.2+--+(1)R3 R4R2R1Y4Y3Y2Y1ABBA

SResidualsMeasurementsCombinationsIN


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Step 11. Calculate coefficients ci in the system equation for all factors and all interactions

• ci = Δi/2 , except in the I column. For example, cA=16.64/2=8.32 etc. • The reason for dividing ΔA by 2 that in the experiment, normalized A changes by two units, from -

1 to +1. For example, for A cA = directional coefficient of Y w/r/t A

cA = Δ/2 = 16.64 / 2 = 8.32


8.717.5416.6423.83Δ

3.1440.275-.375-3.9753.6250.72539.936.343.941.0++++ab

0.7514.925-.7250.175-.4250.97514.215.114.515.9-+-+b

2.2424.025-1.525-1.625-.0253.17522.522.424.027.2--++a

4.363.778.3223.83c

2.9116.100-1.700-3.2002.8002.10014.412.918.918.2+--+(1)

R3 R4R2R1Y4Y3Y2Y1ABBA

SResidualsMeasurementsCombinationsIN

Y

Δ

-1 1A

2


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Step 12. Complete the system equationY = + cAA + CBB +CABAB + ERROR = 23.83 + 8.32 A + 3.77 B + 4.36 AB + ERROR

NOTES:• The system equation has the mathematical form of the selected model and is only as good as the

model. The coefficients in the equation are empirically derived for the given model, using simple regression that we performed in the above table.

• Grand mean = 23.83 is the intercept of the surface Y=Y(A,B) with the Y axis (where A=B=0). This is a direct analogy to the straight line equation


Y

xn

m

Y= m x + n

X


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• Mathematically speaking, the term AB is nonlinear. Here, it represents the AB interaction, that is the simultaneous effect of factors A and B on response Y. Notice that Y=Y(A,B) is a curved surface rather than a plane (which would be described by Y=+ΣckK). Because the measurements are taken in the corners of the factor domain (A= -1, +1 and B= -1, +1), the values given by the system equation in the corners should reproduce the tabulated values exactly (this is a good check of the algebra). In the interior of the domain, the state equation serves as an interpolating function.

• Note: in our simple example, the curved surface is obtained from straight lines along both A and B axes. This straight-line feature will be absent in more complex models.


AB

• The system equation is valid for normalized factor values only. The ranges of validity (subject to subsequent empirical verification) are -1 < A < 1 and -1 < B < 1.


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NOTES:• The ERROR term is never known. It exists due to a combination of

the model fit limitations, measurement errors, the presence of unknown nuisance factors (such as air temperature and humidity, the worker state of mind at a given time, etc.), and, perhaps (and we hope not !), the fact that certain active factors have been ignored in the model.

• It is possible to compute the model goodness of fit, expressed as confidence levels for each term in the equation, using ANOVA=the analysis of variance. Various DOE software include such computations. We ignore them here.

• The system equation is next used for optimization of our vibration problem.



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We obtained: Y = 23.83 + 8.32 A + 3.77 B + 4.36 ABY=vibration amplitudeA=normalized cutter sizeB=normalized cutter speed

By inspection of the equation terms (and also from the table), we deduce that:Minimum amplitude of the vibrations occurs when:

A= -1 ( 2 mm cutter), B= +1 ( RPM of 90)then Y=14.925

In more complex models, we can always find the factor settings required for a given level of Y, by trial and error, by varying one factor at a time and computing the corresponding Y from the system equation. For example, at:

A= -1, -0.95, -0.9, -0.85., ... 0.9, 0.95, 1B= -1, -0.95, -0.9, -0.85., ... 0.9, 0.95, 1etc., in all possible combinations.

Spreadsheets are ideally suited for this purpose.

DESIGN OF EXPERIMENTSFull Factorials 2fInterpretation and Optimization


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• We can show the results from the Y column in the table graphically:

• Conclusions (which confirm the results from the system equation):> Smaller cutter reduces the vibrations> When the smaller cutter is used, the speed is not important, so we select the

speed based on some other considerations, e.g.: - Fast speed to maximize the yield- Small speed to reduce the cutter wear- Medium speed for the smoothest cutting surface- etc.



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WHEN CAN WE ELIMINATE SOME FACTORS ?• Generally, the first experiment has a masking character. One of its objectives is to

examine all potential factors (often 8, 10 or more), in order to see which are important and which are not.

• In the present example, none of the terms is significantly smaller than others, therefore none can be eliminated.

• In practice, any factor which has small both the effects and all interactions which contain the factor, can be ignored.

• The criteria for ignoring a factor can be subjective or objective. • Subjective:

- We set some factor limits "by eve"- We make a normal probability plot (shown later) on which the

"important" points are located off the fit line, while those unimportant are close to the fit line.

• Objective:- We calculate the probability that a given factor is important using

ANOVA (analysis of variance) available in most DOE software.



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Interactions• In our example, the lines on the AB interaction plot intersect. This indicates a strong

AB interaction. Parallel lines would indicate no or weak interaction. The crossing angle is not a good quantitative measure of the interaction strength because the vertical scale is arbitrary. However, it is a qualitative indication whether a given interaction is strong or weak.

• Recall: even if the effect of a given factor is weak, we should not eliminate the factor unless all interactions involving the factor are also weak.

• In practice, it is very rare that three-way or higher-order interactions are important or strong. For well-behaved functions, the higher the interaction order, the weaker is the interaction. This is analogous to the Taylor series expansion where we expect that each next term is weaker than the previous term. Thus, in practice, we often limit ourselves to two-way interactions only. These interactions are easy to plot, analyze and interpret.

• In order to perform the robustness analysis, we must include in our model both controllable and uncontrollable factors. Both are treated the same way in the experiment. The factors which are uncontrollable in real life must be controlled in the experiment.



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Interactions averaging over one or more factors


39++++abc

60-+++ab

40+-++ac

44--++a

55++-+bc

35-+-+b

32+--+c

22---+(1)

ABCBCACABCBAIN

In our simplest example with only one interaction AB, making the interaction plot was easy: we plotted Y versus A for all sign combinations: A-B-, A-B+, A+B-, A+B+ and then connected the dots for constant B+ and constant B- since B was the parameter.

In the models with more than two factors, we have more than one value of at each combination of two letters. Consider the following experiment 23 with factors A, B and C. Now we have three two-way interactions: AB, AC and BC.

Suppose we wish to plot the AB interaction, that is plotting four values of Y that correspond to A-B-, A-B+, A+B- and A+B+. Now we have two data points for each combination of the AB signs. For example, for A+B+ we have Y = 60 at C- and Y = 39 at C+.


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Interactions averaging over one or more factorsWith a larger number of factors, the number of data points increases. In 24 we have four data points at each AB sign combination, in 25 we have eight data points, etc.

It is clear that in such cases, the number of data points can become too large to plot in any informative way.

Therefore, it is customary to average over such data points. In the 23 example above, we would average (60+39)/2 = 49.5 and plot this value for A+B+. Then we would average similarly for the remaining AB sign combinations. The result is shown below.


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Optimization of Variability (Scatter)

• The analysis of the mean Y indicated that the small cutter reduces vibrations and the speed is unimportant.

• The column of standard deviations S in the table indicates that the smallest scatter of the results (scatter of the vibration amplitudes) is obtained with the following factor settings:

A = -1 (small cutter, 4 mm),B = +1 (high speed, 90 RPM)

• We are lucky, the vibration amplitude is then also the smallest. This luck does not occur very often. Usually, there is a conflict in the factor setting between the optimum mean and the minimum scatter or highest robustness. Such conflicts require a trade off, as follows.



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• The final choice of the factor settings is outside the domain of DOE. It must be based on external engineering considerations.

• Examples of external engineering considerations:– Small vibrations are more important than the vibration scatter (or vice versa)– High speed is important for the yield– Small speed reduces the cutter wear, and cutters are important– Perhaps a compromise setting would be the best ?, e.g., A=0 and B=0

(medium values)• A bar plot of mean values next to the std. dev. Values is often useful in making

judgments about their relative importance. • Taguchi uses here the signal to noise ratios (the ratios of to or similar), to

obtain a “happy optimum”. Statisticians question this method since it ignores the underlying process physics, such as that above, and, in some circumstances (not known a priori) yields incorrect results. A video will be shown later with Dr. Montgomery discussing this.



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Two levels versus three levels

• Most experiments are limited to two levels, in the expectation that the world is "mostly linear".

• This is a correct approach for the masking experiments where we are forced to begin with a large number of potential factors and, among others, have to determine which are and which are not important.

• In practice, only a few factors are truly important, but we rarely know which ones before hand.


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• After eliminating the weak factors, we can (and should) perform a gross check for nonlinearity by adding a special run with all factors set at the center point of the domain. We can then plot the response at the center point of the domain, or at the 0 value of each factor or interaction (-1, 0, +1).

• This is not always possible. For example, when one factor is only categorical with two levels.

• In this case, we perform such a check for each level of the categorical factor.

• If no strong nonlinearlity is evident, we accept the experiment results as final. If a strong nonlinearity is detected, we must change the model to a more advanced one, typically using three levels but with a reduced number of factors.



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Homework 1An automatic welding process has three factors, as follows:A = weld type (2 categorical types), B = belt speed (5, 7.5 m/s), C = welding temperatureOur goal is to reduce the number of bad welds. The following has been measured.

RUN NUMBER OF REJECTED WELDS(1) 22 31 25c 32 43 29b 35 34 50bc 55 47 46a 44 45 38ac 40 37 36ab 60 50 54abc 39 41 47

1) Derive the process equation. 2) Select the factor settings that reduce the number of rejects. 3) Sketch all effects and all two-way interactions (vertical axis = response, horizontal axis = earlier

letter, parameter = later letter of the alphabet).4) Sketch the factor domain as a cube, label the corners and axes with normalized values of A,B,C,

and show the mean responses in each corner. 5) Discuss: Can any factor be ignored ? Which and why ? Explain.6) Plot both the mean and the std. dev. values as bar graphs on the same bars. Discuss which

standard deviation is the best? For what factor setting is the variability the smallest? Are these settings in conflict with the settings required for the best mean values? If there is a conflict, briefly describe how you would resolve it ?



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Homework 2FACTORS: A=aperture (type 1, type 2)

B=time the aperture is open (-20%, +20% off nominal)C=exposure time (30, 45 seconds)D=mask type (type 1, type 2)E=immersion time (14.5, 15.5 seconds)

RESPONSES (indicating the number of rejects)(1)=7 e=8 d=8 de=6a=9 ae=12 ad=10 ade=10b=34 be=35 bd=32 bde=30ab=55 abe=52 abd=50 abde=53c=16 ce=15 cd=18 cde=15ac=20 ace=22 acd=21 acde=20bc=40 bce=45 bcd=44 bcde=41abc=60abce=65 abcd=61 abcde=63

1) Derive the process equation. 2) Select the settings that reduce the number of rejects. 3) Sketch all two-way interactions.4) Can any factor be ignored ? Explain.



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FRACTIONAL FACTORIAL

DESIGNSMore economical, less informative



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When we deal with many factors, the full factorial experiments can require a huge number of runs (and measurements).

NUMBER OF FACTORS MODEL NUMBER OF RUNS2 22 43 23 84 24 165 25 326 26 647 27 1288 28 2569 29 51210 210 1024

In order to reduce the number of runs, we use fractional factorial experiments.

DESIGN OF EXPERIMENTSFractional Factorial Designs


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• Fractional factorial is a subset of a full factorial. Ideal for screening (masking) experiments with many factors

• The benefit: fewer runs are required. The compromise: some interactions are confounded with other interactions

• The subset size = the number of runs of the corresponding full factorial / Li, L=number of levels (2 or 3)

• That is, the subset size = 1/2, 1/4, 1/8, or 1/3, 1/9, 1/27 fraction. Hence the name "fractional"

• Example: f=7 factors and L=2 levelsFull factorial requires 2f=27=128 runsFractional factorial requires 27-i runs

i=1 27-1=64 runsi=2 27-2=32 runsi=3 27-3=16 runsi=4 27-4=8 runs



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+++++++

-+--++-

--+-+-+

+--++--

---+-++

+-+--+-

++----+

-+++ ---

ABCBCACABCBA

+++++++

-+--++-

--+-+-+

+--++--

---+-++

+-+--+-

++----+

-+++ ---

D=ABCBCACABCBA

Suppose we must investigate four factors and we have time/budget for only 8 runs. This would be fractional factorial 24-1=23=8. 8 is the number of runs of a full factorial 23, without one factor, say, the last factor D. We begin with a full factorial for factors A,B,C. (Any other set of three factors could also be used).

Full factorial 23 in A,B,C Fractional factorial 23-1 in A,B,C,D

The ignored factor D has been inserted into column ABC. If we label this column D, we will not be able to separate the effects of D from the effects of ABC. D is said to be aliased or confounded with ABC. It is as if two pipes D and ABC were joined together - observing the ouflowing fluid D+ABC, we cannot tell how much of it came from pipe D and how much from pipe ABC. The result is a fractional factorial 24-1 design.


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DESIGN OF EXPERIMENTSFractional Factorial Designs-Confounding relations

Confounding Relations

abcd

bc

ac

cd

ab

bd

ad

(1)

N

+

-

+

-

+

-

+

-

BCD

A

+

+

-

-

+

+

-

-

ACD

B

+

+

+

+

-

-

-

-

ABD

C

+

-

-

+

-

+

+

-

ABC

D

+

-

-

+

+

-

-

+

CD

AB

+

-

+

-

-

+

-

+

BD

AC

+

+

-

-

-

-

+

+

AD

BC Confounding generator:D=ABC

Multiply by D to obtain "I" vector on the LHS.

The result is the so-called "generating equation":

I=DD=ABCD

The resultant confounding relations are A=BCD AC=BDAB=CD C=ABDB=ACD AD=BC

The consequence of the single confounding D=ABC:• Factor effects become confounded with three-way interactions• Two-way interactions become confounded with other two-way interactions. Interactions

not hidden, only confounded

System equation: Y= Y + cAA+ cBB+ cCC + cDD+ cABAB + cACAC + cADAD


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Procedure for fractional design for f factors

• Select the number of runs N based on the available time, budget, etc. N=power of 2

• Write 2f-i = N and solve for i• Create a table for N runs and f-i factors. Arbitrary decision which i factors

to ignore.• Confound the highest-order interactions in the table with the i ignored

factors • Determine the confounding relations for each column (typically we limit

ourselves to those which are of the lowest order; otherwise their number could be too large for practical considerations), and write the confounded pairs into the table, as above

• The remaining steps are identical to the full factorial method.



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Class exerciseCreate the design matrix for 25-1 fractional factorial. Use E=ABCD.



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Resolution• We usually classify experiment designs by their resolution. The resolution tells the experimenter

something about the nature of the alias structure.

Resolution number = the smallest number of letters in any of the generating equations, except letter I.

• For example, I=ABCD has four letters (A,B,C and D) besides I, so the resolution is IV.• Resolution III = fractional factorial design where the main effects and two factor interactions are

aliased. "Cheap" designs. Used for early design stages where we wish to identify the potentially large main effects.

• Resolution IV = fractional factorial design where the main effects and two-way interactions are clear of each other but the two-factor interactions are aliased with each other. Tells of sometwo-way interaction activity, better for screening purpose than resolution III.

• Resolution V = fractional factorial design where the main effects and two factor interactions are clear of each other. The main effects are confounded with four-way interactions and two-way interactions are confounded with three-way interactions. Excellent tools for estimating main effects and two-factor interactions.

• Notation: Lf-iR

L = number of levels, f = number of factors, i = exponent reduction number for fractional factorials, R = (roman numeral) resolution



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Designs Available in Software "Design Ease"


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Homework Problem 3Verify the computed effects for the given design 25-1

V with factors A,B,C,D,E, and with confounding E=ABCD. Show the complete computational table. Find one error.


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Homework Problem 4Re-create the entire computational table. Add one row below the factors, and enter into that row all lowest-order confounded items (effects or interactions) that apply.


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Design Folding by a Factor• Frequently, upon completing an experiment, we notice that a factor is not important.

We can then fold the design by half by ignoring the factor. • The resultant number of runs is a half of the original number, and we gain two replicates

for each original run measurement. This permits to perform scatter analysis in the experiments that had none, or to increase the statistical confidence into the results.

• For example, in the following example, a 23 design, with 8 runs and 3 factors, A, B and C, factor A turned out to be negligible.

• We fold the design by ignoring factor A. As a result, we have 4 runs, each with two replicates that correspond to the original runs with A+ and A+, respectively.

• In the new design, the ignored factor (A in this example) should not appear anywhere in the table or in the system equation.

Exercise

• Carry out the described folding by ignoring factor D in Problem 3. Determine the system equation. Re-do the table and determine the new prediction equation. Comment on the values of the coefficients in the original and the reconstructed equation.



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Plackett-Burnam (P-B) Designs


• All above factorial designs require the number of runs which is a power of 2.

• This is sometimes inconvenient. E.g., with 8 variables we cannot use 8 runs, so the next higher number is 16 runs. This may be prohibitive economically.

• P-B developed a series of two-level fractional factorial designs that require that the number of runs be a multiple of 4.

• Their designs with 12, 20, 24, 28 and 36 runs are very useful in practice.

• All designs are orthogonal and are of resolution III.

• It is easy to generate a P-B design, as shown on the following page.


65



66

Class exercise• Generate the sign table for the P-B design

with 12 runs and 11 factors.



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NONLINEAR DESIGNS

For those nasty nonlinear problems, or for more accurate next iteration



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• When a strong nonlinearity is suspected in a design, usually as a result of the lack of verification of an earlier two-level design, we must use a nonlinear model.

• We have the following choices, all of which require special DOE software.– Mixed-level Taguchi Designs

Some factors are at two levels and others at three levels.

– Central Composite Designs (a.k.a. Box-Wilson designs)Ideal for salvaging results from a two-level design by adding a few runs.

– Box-Behnken Designs

– Full three-level design 3f-i involving full factorial or fractional factorial.

– Response Surface Methodology

• We describe these designs in turn.

NONLINEAR DESIGNSComparison of Nonlinear Designs


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• Mixed-level Taguchi designs have some factors at two levels and others at three levels. Also known as "orthogonal Taguchi arrays". The designs are not exactly new; they are derived from the known "Latin Squares"

• Taguchi uses 1, 2, 3 to denote the low, medium and high levels.• The designs and fractional, orthogonal and usually heavily confounded• Taguchi publications contain complete tables for selected numbers of factors and

levels. Special DOE software is needed.

NONLINEAR DESIGNSMixed-level Taguchi Designs


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• These designs are perfect for salvaging the results of an earlier two-level design which was determined to contain a nonlinearity. All we need to do is add a few runs.

• These designs permit terms in the system equation with factors raised to the second power . That is, the following terms are permitted: Y, A, A2, B, B2, ..., AB,...

• These designs require special DOE software.• Example with three factors A, B, C• We begin with a full factorial two-level design (the corners of the cube)• We add 6 runs denoted by asterisks, located at a distance D away from the center.

D depends on the number of factors. For three factors, D= SQRT(2), i.e., about 1.41 units.

• We add Q runs in the center of the cube, where Q depends on the number of factors. Q=5 for three factors.

• Source: Basic Experimental Strategies and Data Analysis"John Lawson and John Erjavec,Bringham Young University, Provo, Utah, USA

NONLINEAR DESIGNSCentral Composite (a.k.a. Box-Wilson) Designs


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• The designs require fewer runs than the Central Composite• Must be planned from the beginning. We cannot salvage a prior two-level design by

adding runs.• The terms in the system (prediction) equation can contain factors squared.• The designs require special software

• Example for three factors:

NONLINEAR DESIGNSBox-Behnken's Designs


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• The full three-level designs can be applied to full factorial or fractional factorial designs (See Montgomery)

• Special DOE software is required.• The levels are typically denoted 1 = low, 0 = center, 1 = high• The possible terms in the system (prediction) equation include linear terms,

interactions, and factors and interactions raised to the powers of 2 and 3, that is Y, A, A2, A3, AB, A2B, AB2, etc.

• Expensive designs in terms of the number of runs. Central composite and Box-Behnken are more efficient.

• Example of a full factorial three level design 33:

NONLINEAR DESIGNSFull Three-Level Designs 3f-i

Class exercise: some design points are missing from the sketch. Which?


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NONLINEAR DESIGNSResponse Surface Methodology

SPC

doe

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