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Blocking and Fractional Factorial Designs

Outline to date...

- definition of experimental design
- experimental design considerations
- two-level factorial designs
- assessment of main, interaction effects
- precision of calculated effects
- estimating inherent (extraneous) noise variance
- In a complete, two-level factorial design

implemented in a - randomized order, we obtain clear indications of

effects - - unambiguous or unaliased - no correlation

between - associated parameter estimates.

Outline

- motivation
- introduction to confounding
- homogeneous experimental conditions
- components of variation
- blocking
- fractional factorial designs
- more on confounding...

An Imperfect Situation

- Suppose we have a 23 design (8 runs) for a

reactor yield experiment - - effects of temperature, pressure and reactant

ratio - however
- we only have enough catalyst to conduct 4 runs
- we prepare another batch of the same catalyst to

conduct the other 4 runs - we decide to conduct all high temperature

experiments with the first batch of catalyst - Check the main effect of temperature...

An Imperfect Situation

- Main effect of temperature -
- (average of yields at high T - average of yields

at low T) - however the same averages can also reflect the

effect of the catalyst batches - Main effect of catalyst preparation -
- (average of yields for cat prep 1-average of

yields for cat prep 2) - which are the same averages as for temperature!
- The effect of temperature is confounded, or

completely aliased, with temperature - ambiguity about which is the effect
- complete correlation of effects of cat prep.,

temperature

Confounding of Effects

- In this instance, the confounding was by

accident - I didnt pay attention to how the runs were being

conducted - unintentional confounding of effects
- In some instances, we may intentionally confound

certain effects in order to reduce the required

number of experimental runs - fractional factorial designs

Confounding of Effects

- What can we do about this dilemma?
- try intentionally confounding an insignificant

effect with the catalyst batch - for example -
- conduct all runs for which TPratio1 with the

old batch - conduct all runs for which TPratio-1 with the

new batch - i.e., intentionally confound cat prep with the

three factor interaction, which is usually

insignificant - randomize all runs conducted for a given catalyst

batch - randomized incomplete block design
- other main effects are now unconfounded with the

catalyst prep - The technique of grouping runs to guard against

unforeseen effects is known as blocking.

Experimental Conditions

- The difficulty in the reactor example arose

because we didnt have homogeneous

experimental conditions - two batches of catalyst were used in the

experimental program - conditions - not to be confused with levels!
- Other examples of non-homogeneous experimental

conditions - - use of more than one unit - e.g., extruder,

analytical instrument - to conduct experimental

program - different individuals conducting experiments
- Experimental conditions refers to the broader

environment of the experimental program

Homogeneous Experimental Conditions

- Homogenous experimental units are as uniform as

possible on all characteristics that could affect

the response. - Blocking is the grouping of experimental runs in

groups, or blocks, so that the runs within each

block are subject to experimental conditions that

are as homogenous as possible - have as much extraneous variation in common with

each other as possible

Components of Variation

- The extraneous variation observed in an

experimental program can arise from a variety of

sources - measurement noise
- variations between lots
- variations between manufacturers

Components of Variation

- Example - Resistivity Measurements on Conductors
- (pages 1-20 to 1-32 - Course Notes)

conductor variation

measurent variation

lot variation

Components of Variation

- Example - Resistivity of Conductors -

lot variation

Lot 1

Lot 10

...

conductor variation

Conductor 1

Conductor 4

Conductor 1

Conductor 4

...

...

Msmt 1

Msmt 2

Msmt 1

Msmt 2

Msmt 1

Msmt 2

Msmt 1

Msmt 2

measurent variation

Total variability in dataset variance of msmts

variance of lots

variance of conductors

Blocking

- Example - Impact of Furnace Position on Oxide

Thickness - (page 1-35, Course Notes)
- study impact of two positions on thickness -

handle, source - issue - each run in furnace may introduce

additional variability - non-homogeneous

conditions - solution - conduct tests of two wafers at a time
- one in the handle position
- one in the source position
- compute differences in thickness for each run
- assess whether mean of differences is

significantly non-zero - use variance of

differences in test - paired comparison of means

Blocking

- Example - Impact of Furnace Position on Oxide

Thickness - The blocks in this example are the furnace tests

- - wafers in same furnace run have more variation in

common - conducting a furnace test for each single

position introduces furnace variability into the

comparison - Read pages 1-19 to 1-47 in the Course Notes

Blocking in 2-Level Factorial Designs

- When dividing factorial experiments into blocks,

try to ensure that number of runs at high level

of each factor is the same as the number of runs

at the low level of each factor - ensures uniform precision
- provides information about primary effects of

interest - Example - reactor yield (the imperfect situation)

- - cat prep 1 - TPratio 1
- runs , - -, - -, - -
- cat prep 2 - TPratio -1
- runs - - -, -, - , -
- in each block, we have an equal number of high

and low levels in each factor (2 high, 2 low)

Outline

- motivation
- introduction to confounding
- homogeneous experimental conditions
- components of variation
- blocking
- fractional factorial designs
- more on confounding...

Fractional Factorial Designs

- are a type of screening design
- identify influential factors as efficiently as

possible - Approach
- decide on one or more effects of no interest, and

discard runs which would support estimation of

these effects - reduces total number of runs
- Example - 23 design
- decide that x1x2 x3 is not important
- choose runs for which x1x2 x3 1
- 4 runs - half fraction of a 23 design
- Notation - 23-1 fractional factorial design

Fractional Factorial Designs

- Terminology and Notation
- Blocking variable -
- factor, or combination of factors, whose value is

held constant within block of runs - sometimes referred to as the defining contrast
- 23 example - x1x2 x3 1 is the blocking

variable - 2k-p Fractional Factorial Design
- 2 --gt two levels in each factor
- k --gt number of factors
- p --gt degree of fractionation (e.g., p1 is a

half fraction) - All fractional factorial designs are balanced,

and are orthogonal (effects are uncorrelated)

Fractional Factorial Designs - Aliases

- Aliases are combinations of factors whose levels

are identical to those of other combinations of

factors in the fractional factorial design. - Example - 23 design
- since x1x2 x3 1 in the 23-1 design, x1 and x2

x3 will have identical levels --gt x1 and x2 x3

are aliases (or are said to be aliased) - It is important to understand the alias structure

of a fractional factorial design because the

effect estimated for a given factor or

interaction represents the sum of the effects of

the aliased variables.

Determining the Alias Structure

- Start with the blocking variable (defining

contrast), and apply the following two rules - 1) X1X
- 2) XX1
- Example 23 -1 half fraction -
- x1x2 x3 1
- x1x1x2 x3 x11 gt x1 x2 x3
- x2 x1x2 x3 x2 1 gt x2 x1 x3
- x3 x1x2 x3 x3 1 gtx3 x1x2
- conclusion - all main effects are aliased

(confounded) with two-factor interactions in

this design

Note - aliased confounded

Determining the Alias Structure

- To obtain effects aliased with the main effects -
- multiply the blocking variable expression by the

first factor - multiply the blocking variable expression by the

second factor - To obtain effects aliased with two-factor

interactions - - multiply the blocking variable expression by a

two-factor interaction of interest - repeat this procedure for each two-factor

interaction - and so forth...

Resolution of a 2-Level Fraction Factorial

- The fractional factorial designs can be

classified by their resolution - smallest sum of the orders of aliased effects
- Resolution III Design
- no individual factors aliased with any other

individual factors - at least one individual factor (main effect) is

aliased with a two-factor interaction effect

Resolutions of Fractional Factorial Designs

- Resolution IV Design
- no individual factor is aliased with any other

individual factor or 2-factor interaction - at least one 2-factor interaction is aliased with

another 2-factor interaction - at least one individual factor is aliased with a

3-factor interaction - Resolution V Design
- main effects not aliased with main, 2-factor or

3-factor interaction effects - 2-factor interaction effects not aliased with any

other 2-factor interaction effects - (please see Course Notes)

Example - Half Fraction of 23 Design

- We are looking at a 23 -1 design -
- choose blocking variable as x1x2 x3 1
- we saw that x1 is aliased with x2 x3
- x2 is aliased with x1 x3
- main effects are not aliased with other main

effects - Resolution III design
- notation 23 -1III design

Example - Half Fraction of 24 Design

- We are looking at a 24 -1 design
- choose blocking variable as x1x2 x3 x4 1
- x1 aliased with x2x3 x4
- x2 aliased with x1x3 x4 and so forth
- x1 x2 is aliased with x3 x4
- x1 x3 is aliased with x2 x4 and so forth
- Resolution IV design
- notation 24 -1IV
- What if we used x1x2 x3 1 as a blocking

variable?

Resolution of a Design

- An alternative way for determining the resolution

is as the length of the blocking variable - for half fraction designs (for smaller fractions,

we must be more careful) - 23 -1 design
- x1x2 x3 1 --gt length is 3 (therefore

Resolution III) - 24 -1 design
- x1x2 x3 x4 1 --gt length is 4 (therefore

Resolution IV)

Smaller Fractions

- A (1/2p)th fraction of a 2k design is a 2k-p

fractional factorial design - e.g., 25 -2 is a1/4 fraction of the 25 design
- To define a (1/2p)th fraction, we need p

blocking variables - Example - 25 -2
- choose runs from the 25 design for which x1x2 x3

x4 1 AND x3 x4 x5 1 - these blocking variables imply a third blocking

variable, x1x2 x3 x4 x3 x4 x5 x1x2 x5 1 - notation
- x1x2 x3 x4 x1x2 x5 ( x1x2 x5 ) 1

( ) denote implied

Smaller Fractions

- The alias structure of these fractions is

determined by looking at the aliases implied by

all of the defining blocking variables (actual

and implied) - take each blocking variable, and go through

sequence of multiplying by single factors,

2-factor interactions, ... - The resolution of the fraction is defined as the

length of the smallest blocking variable

expression - Example - 25 -2
- x1x2 x3 x4 x1x2 x5 ( x1x2 x5 ) 1
- smallest blocking variable definition has length

3 - Resolution III design

Example - 26-2 Design

- Blocking variables -
- select runs for which x1x2 x3 x5 1 AND x1x4 x5

x6 1 - implied blocking variable is x2x3 x4 x6 1

(obtained by multiplying the defining blocking

variables together) - x1x2 x3 x5 x1x4 x5 x6 ( x2x3 x4 x6 ) 1
- smallest blocking variable expression has length

of 4 - Resolution IV design
- A partial catalogue of fractional factorial

designs is available in the Course Notes - pp.

4-20 -- 4-28

Methods for Obtaining the Runs

- Which runs do we need for fractional factorial?
- Approach 1 -
- list all runs for the full factorial design
- select those runs satisfying the blocking

variable(s) - Approach 2 -
- start with a 2m design in m of the factors, where

mk-p - assign levels to the additional factors using the

blocking variable relationships defining the

fraction - Added Factors technique
- saves having to write out the full factorial

design initially

Saturated Designs

- are designs in which the effects of a specified

number of factors are investigated using the

minimum number of runs - focus is on single-factor effects
- give up on interaction effects (will be

confounded with main effects) - Resolution III designs
- example 27 -4III design provides info on 7 main

effects in 8 runs - these are screening designs - provide info on

main effects - no provision for estimating error - could augment

with centre point runs

Plackett-Burman Designs

- Saturated designs in runs that are multiples of 4
- Examples -
- 12 run design - use for 7-11 factors
- 16 run design - use for 12 - 15 factors (215 -11

design!)

Calculating Effects for Fractional Factorial

Designs

- Same procedure as for the full 2-level Fractional

Factorial designs - - formal definitions - main effects, interaction

effects - effects representation - divide by number of

(non-centre point) runs (2k -p) - regression
- The key difference is in the interpretation -
- calculated effect represents the effect of the

factor PLUS the effects of all factors aliased

with that factor - e.g., in half fraction of 23 design, x1 is

aliased with x2 x3 so the main effect calculated

for x1 represents the effect of x1 the effect

of x2 x3

Calculating Effects

- Precision is handled in the same way - obtain

estimate of extraneous variance from - replicate runs (typically, centre-point runs)
- external estimate from previous runs
- insignificant effects - detected using Normal

probability plot - Key distinction is in the number of runs being

used.

Precision of Calculated Effects

- Fractional factorial designs are still balanced
- equal number of high and low levels of factors
- Variance of calculated effect can be developed by
- using formal defn -gt avg. of the responses at

high level (half of the runs) - avg. of the

responses at low level - there are 2k-p/2 runs at each level, and

variances are additive - using regression approach
- effect is 2estimated parameter
- variance is 4variance of estimated parameter
- End result - variance of calculated effects in a

2k-p fractional factorial design is

Example - Wafer Thickness Uniformity

- Described on page 5-15, Course Notes.
- What type of design is this?
- What is the alias structure of this design?
- What are the main and two-factor interaction

effects? - Assess significance of main and two-factor

interaction effects using - confidence intervals
- normal probability plot

The Home Stretch

- Building on Fractional Factorial Designs
- Examining Nonlinear Factor Effects --

Higher-Order Designs - Response Surface Methodology
- The Taguchi Approach to Quality Improvement

Building on Fractional Factorial Designs

- After an initial fractional factorial design has

been implemented, we can obtain better resolution

by conducting additional designs. - Two approaches -
- repeat design with levels of ONE factor reversed

- keep same pattern for other factors - reverse patterns of levels for ALL factors -

Foldover Design

Reversing the Levels of a Single Factor

- Example - 26 -3 Resolution III design
- x1x2 x4 x1x3 x5 x2x3 x6 ( x2x3x4x5

x1x3x4x6 x1x2x5x6 x4x5x6 ) 1 - phase 1 - conduct this experiment - now for Phase

2?

Reversing the Levels of a Single Factor

- Phase 2 - reverse the levels of factor 1
- impact - changes defining relations involving x1
- other factor levels are fixed, x1 is reversed
- x1x2 x4 -1 , x1x3 x5 -1

Reversing Levels of a Single Factor

- Calculated main effect for x1
- represents (x1 x2x4 x3x5 ) for Phase 1design
- represents (x1 - x2x4 - x3x5 ) for Phase 2 design
- assuming higher-order interaction effects are

negligible - Averaging the calculated main effects for x1 over

both experimental phases yields - 1/2 (x1 x2x4 x3x5 ) (x1 - x2x4 - x3x5 )

x1 - Combining the runs from both experimental phases

yields eliminates terms aliased with x1 - finer

resolution. - Aliasing of other main effects with two-factor

interactions involving x1 is also eliminated.

Reversing the Levels of a Single Factor

- Impact on interaction effects involving x1
- Consider
- 1/2 (x1 x2x4 x3x5 ) - (x1 - x2x4 - x3x5 )

(x2x4 x3x5 ) - 1/2 (x2 x1x4 x3x6 ) - (x2 - x1x4 x3x6 )

x1x4 - Conclusion - we can obtain unaliased estimates of

the two-factor interactions involving x1, and

eliminate aliasing of other interaction terms by

x1xj terms

sign doesnt change since this is from x2x3x6

blocking relation

Reversing the Levels of a Single Factor

- General Result - a sequence of two 2k -pIII

designs in which the second design differs from

the first only in the reversal of the pattern of

levels for one factor yields unaliased estimates

of the main effect of that factor, and all

2-factor interaction effects involving that

factor - pages 5-26 to 5-34 -- Course Notes
- can be extended to designs of other resolutions,

however primary incentive is for Resolution III

designs - improve resolution of a single factor (and

2-factor interactions of that factor) of

particular interest - zooming in

Foldover Designs

- Example - 27 -4 Resolution III design -

saturated, in fact! - x1x2x5 x2x3x7 x1x3x6 x1x2x3x4
- ( x2x3x5x6 x3x5x7 x1x2x6x7 x3x4x5

x1x4x7 x2x4x6 ) 1 - Phase 1 experimental programme

Foldover Designs

- Phase 2 - reverse levels of ALL factors
- Impact - the defining relations consisting of 3

factors become - x1x2x5 x2x3x7 x1x3x6 ( x3x5x7 x3x4x5

x1x4x7 x2x4x6 ) -1 - the defining relations consisting of 4 factors

remain 1

Foldover Designs

- Impact on Calculated Main Effects
- averaging over both designs results in

cancellation of aliasing of main effects by

2-factor interactions - example - calculated main effect of x1
- (ignoring 3-factor interactions or greater)
- 1/2 (x1 x2x5 x3x6 x4x7 ) (x1 - x2x5 -

x3x6 - x4x7 ) x1 - The same result is obtained for EACH main factor.
- Result - a Resolution III design is transformed

into a Resolution IV design.

Phase 1

Phase 2

Foldover Designs

- Impact on Calculated 2-Factor Interaction

Effects - examine difference between calculated main effect

for x1 for Phases 1 and 2 - 1/2 (x1 x2x5 x3x6 x4x7 ) - (x1 - x2x5 -

x3x6 - x4x7 ) - x2x5 x3x6 x4x7
- aliasing of the combination of 2-factor

interaction terms with main effects is eliminated - aliasing between 2-factor interaction terms is

NOT eliminated

Foldover Designs

- General Statement - Foldover Designs can be used

to improve initial Resolution III designs to

Resolution IV designs - key point - need an ODD number of terms in the

smallest defining relation for the necessary

cancellation during averaging over both phases to

occur - Foldover Designs CANNOT be used to improve

Resolution IV designs to Resolution V designs - Foldover Designs CANNOT be used to improve

Resolution III designs to Resolution V designs - pages 5-35-5-49, Course Notes

A Last Look at Blocking

- Blocking of Full Factorial Designs
- Scenario - we have non-homogenous conditions, and

we have to run a full 2k factorial design in two

blocks - - Think of the conditions as an extra factor
- k1 factors to be run in 2k runs
- treat 2k design as a half fraction
- blocking variable is x1x2 ...xk xk1 1
- execute xk1 1 runs for block 1
- execute xk1 -1 runs for block 2
- half fraction concept allows identification of

effects aliased with blocking

Example - Blocking of 24 Design

- We can only do 8 runs per day - need 2 blocks
- think of 24 design as 25 -1 half fraction (5th

factor is day) - blocking variable - gt x1x2 x3 x4x5 1
- Day 1 runs -gt x5 1
- Day 2 runs -gt x5 -1
- check alias structure - what is aliased with

day? - This approach can be generalized to deal with

blocking into more than two blocks, and blocking

of fractional factorial designs.

Looking at Aliasing WITHIN a Block

- Given a 2k design implemented in two blocks,

each block represents a 2k-1 design in the k

factors - Example - 24 design in two blocks
- block 1 -- x1x2 x3 x4 1
- block 2 -- x1x2 x3 x4 -1
- We may elect to examine effects within a given

block, and if so, we should recognize this fact. - This approach can also be used to determine

blocking strategies for fractional factorial

designs. - Note
- block 1 -- x1x2 x3 x4 1 (x5 1)
- block 2 -- x1x2 x3 x4 -1 (x5 -1)

Higher-Order Designs

- How should we design experiment when we want to

estimate a full second-order model in k factors? - quadratics --gt require more than two levels in

each factor - Options
- 3-level factorial design -- 3k
- large numbers of runs
- all combinations of high, middle and low factor

levels for all factors - Central Composite Design
- Face-Centred Central Composite Design
- Box-Behnken Design

Central Composite Designs

- Start with a 2k design -
- add centre-point runs
- add star points - runs with a given factor

level A greater than /-1, and all other factors

at their centre-points - star point distance A (2k)1/4
- selected to provide rotatability - uniform

precision for responses an equal DISTANCE from

the centre point

Total number of points 2k 2k m for k

factors, m runs at centre point

Face-Centred Central Composite

- Star points can take us outside our desired

experimental region. - Solution - clamp star points at /-1
- Cost - our design is no longer rotatable
- Number of Points - same as Central Composite

Design

Box-Behnken Designs

- offer another alternative for keeping the runs

within the high/low boundaries - resulting design is rotatable, or nearly

rotatable - design is formed by combining 2-level factorial

designs with incomplete block designs - available for 3 factors or more

Estimating Second-Order Models

- Use the standard multiple linear regression

approach - models are still linear in the parameters
- Full second-order model includes
- intercept
- main effects
- two-factor interactions
- quadratics in each factor
- Quadratic terms will be correlated with each

other, and with the intercept term. The main and

interaction effects will be uncorrelated with all

terms. - Second-order models provide empirical

representaiton of curvature in the response

surface, and are the basis for ...

Response Surface Methodology

- Response Surface - characterizes behaviour of the

response as a function of several factors - graph of response vs. factors
- empirical representation of behaviour
- developed from designed experiments in a region
- LOCAL representation of behaviour
- types of surfaces - refer to pages 9-6 to 9-10,

Course Notes - surface plot
- contour plot
- Response Surface Methodology (RSM) uses a

sequence of planned experiments and estimated

response surfaces to optimize process performance.

Direction of Steepest Ascent

- Given a function yf(x1, , xk), the direction of

steepest ascent of y is the gradient of the

function f with respect to the factors x1, , xk - We can obtain this gradient from our empirical

model of the response surface, and move the

process in this direction in order to increase

the response y. - Note - gradient at a point is perpendicular to

the contours of the response function.

Response Surface Methodology

- Phases
- Pre-Screening - use screening design (fractional

factorial) to identify which factors have

significant impact on response. - Phase 1 - conduct a 2-level factorial (possibly

fractional factorial) centred at current

operating point - estimate main and 2-factor interaction effects
- assess significance of terms
- if interaction terms are insignificant, calculate

gradient from model without interaction terms

Response Surface Methodology

- Phase 2 - conduct number of experiments along

direction of steepest ascent (gradient) of linear

model - continue experiments until response DECREASES
- weve gone to far
- return to previous best run
- repeat Phase 1 about this new centre point
- repeat Phase 1, Phase 2 until interaction terms

and/or main effect terms become insignificant - were near a possible optimum (stationary point)

Response Surface Methodology

- Phase 3 - explore the stationary region with a

higher-order design and full second-order model - augment last 2-level factorial design with

additional runs (centre plus star points for

Central Composite design) - estimate second-order model, and use it to

determine whether location is a maximum, minimum

or saddle point

A Note on Significant Effects

- A term may be insignificant because
- 1) we have straddled a local maximum or minimum
- 2) poor signal to noise
- 3) term has no significant effect on response

y

x

-1

1

Direction of Steepest Ascent

- is calculated as gradient of first-order model,

which is estimated in CODED form - must transform back to original units for

implementation - multiply each element by 1/2 range of respective

factor - normalize direction in original factor units
- move a certain portion in normalized direction
- stepsize - typically choose the range of one

factor - Normalizing the direction
- divide by magnitude of vector

Response Surface Methodology

- Example - maximizing yield of a component in a

chemical reactor - nominal starting point - yield of 49.4 at

initial centre point - 3 of 5 factors significant based on pre-screening
- factors - temperature, feed concentration,

residence time - coded form

Response Surface Methodology

- Initial model
- Gradient

not significant based on confidence limits using

previous external estimate of inherent variance

Response Surface Methodology

- Planned Runs
- Now conduct a 2-level factorial design centred at

the point corresponding to T135 - yields a model with significant interaction terms
- augment to a central composite design

Response Surface Methodology

- Central Composite Design - in three factors
- yields following model
- check second derivatives to determine whether

this is a maximum, minimum or saddle point - check whether 2nd derivative matrix (Hessian) is

positive or negative definite

Identifying a Max/Min/Saddle Point

- Check the second derivative matrix - Hessian
- Here, H is

- det 1x1 -5.68 lt 0
- det 2x2 2.0 gt 0
- det 3x3 -0.2 lt 0
- negative definite
- MAXIMUM

Identifying a Max/Min/Saddle Point

- To determine whether Hessian is positive or

negative definite, look at determinants of

sub-matrices - 1 x 1 ( the 1,1 element)
- 2 x 2 ( the upper left 2x2 matrix)
- 3 x 3 ( the upper left 3x3 matrix)
- if all determinants are positive --gt positive

definite, and the stationary point is a minimum - if determinants alternate, -ve, ve, -ve, --gt

negative definite, and the stationary point is a

maximum - all other cases --gt saddle point

What can we do with a Saddle Point?

- Canonical Analysis
- place model into a diagonal form using the

eigenvectors of the Hessian matrix - to increase response (yield) further, consider

moving process in direction associated with

largest positive eigenvalue - conduct sequence of experiments (Phase 1, Phase

2), and fit a second-order model when interaction

terms dominate

Taguchi Approach to Continuous Improvement

- Quality - loss imparted to society from the time

a product is shipped - Important Characteristics -
- product performance
- product cost
- product life
- product reliability
- environmental impact
- customer satisfaction
- ...

Loss Functions

- Loss - measure of the cost for poor quality
- Two types of loss functions are considered
- Goalpost loss function
- quadratic loss function
- Goalpost Loss Function
- premise - all values of performance

characteristic lying between lower and upper spec

limits are equally acceptable --gt all such

products are of equal quality - traditional view

Goalpost Loss Function

- Problems
- given uncertainty in determining spec limits, and

sampling to measure performance characteristic,

there is less confidence about products with

performance lying close to the spec limits - tolerance buildup - if a product barely lies

within the spec range at each step of the

manufacturing process, will the final product

satisfy the overall performance specs?

performance measure

upper spec limit

lower spec limit

target

Quadratic Loss Function

- penalizes deviations from target continuously
- implies that target is the preferred value of

performance characteristic

Loss

performance measure

upper spec limit

lower spec limit

target

Quadratic Loss Function

- Potential Difficulties
- is a quadratic function appropriate?
- e.g., asymmetric loss - over-spec more serious

than under-spec - Loss function L(Y) k(Y-m)2

Loss L(Y)

performance measure

upper spec limit

lower spec limit

target m

Determining k in the Loss Function

- Use a benchmark to place the loss at a certain

distance from the target. - Two common choices
- Manufacturers Tolerance
- maximum departure of performance characteristic

from target that is acceptable to manufacturer - Customers Tolerance
- maximum departure of performance characteristic

from target that is acceptable to consumer - determine k from knowledge of loss
- loss incurred by manufacturer when manufacturers

tolerance is exceeded (loss A) - loss incurred by manufacturer and consumer when

consumers tolerance is exceeded (loss A0)

Determining k

- Graphically,

A0

Loss L(Y)

A

m

Manufacturers Tolerance

Consumers Tolerance

Performance Measure Y

Note - this implies relationship between

manufacturers and consumers tolerance

Computing Losses from Measurements

- Mean Squared Deviation
- measure of average departure from target
- yi -- measured value of performance

characteristic of i-th item

Computing Losses from Measurements

- Off-Centering and Inherent Variation
- decompose MSD into two components
- off-centering
- inherent variation

off-centering

inherent variation

Computing Losses from Measurements

- The Average Loss
- using measured characteristics, compute the

average of the loss function computed for each

measurement - Decompose into two components -
- component due to off-centering
- component due to inherent variation
- What can we do with this?
- relative impact of average operation off-target

vs. variability - think of process capability Cp and Cpk

Signal-to-Noise Ratios

- are intended to provide a performance

characteristic accounting for both off-centering

and inherent variation - Goal - achieve as large an SNR as possible
- Types of SNRs
- 1. Objective - maintain nominal target

Signal-to-Noise Ratios

- 2. Objective - make Y as small as possible
- 3. Objective - make Y as large as possible

Concerns about SNRs

- The SNRs proposed by Taguchi
- do not take into account the distribution of the

inherent variation - may not always be

appropriate - arent functions of the target value -- ???!!!
- can be arbitrarily increased by increasing Y to

create larger average, even though target might

be very small - however they are in use, so try to use them

judiciously...

Going Further -- Taguchi Experimental Design

- orthogonal arrays
- concepts similar to the exptl. design studied to

date - 2-level designs, 3-level designs
- interaction tables, linear graphs
- design stages - system/parameter/tolerance
- control factors vs. noise variables
- control factors - factors we can adjust
- noise variables - factors that vary but cant be

controlled - inner and outer arrays
- inner - focus on control factors
- outer - focus on potential noise variables