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Blocking and Fractional Factorial Designs

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Resolutions of Fractional Factorial Designs. Resolution IV Design ... saves having to write out the full factorial design initially. CHEE418/801 - Module 3b ... – PowerPoint PPT presentation

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Title: Blocking and Fractional Factorial Designs


1
Blocking and Fractional Factorial Designs

2
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.

3
Outline
  • motivation
  • introduction to confounding
  • homogeneous experimental conditions
  • components of variation
  • blocking
  • fractional factorial designs
  • more on confounding...

4
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...

5
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

6
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

7
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.

8
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

9
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

10
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

11
Components of Variation
  • Example - Resistivity Measurements on Conductors
  • (pages 1-20 to 1-32 - Course Notes)

conductor variation
measurent variation
lot variation
12
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
13
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

14
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

15
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)

16
Outline
  • motivation
  • introduction to confounding
  • homogeneous experimental conditions
  • components of variation
  • blocking
  • fractional factorial designs
  • more on confounding...

17
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

18
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)

19
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.

20
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
21
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...

22
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

23
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)

24
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

25
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?

26
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)

27
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
28
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

29
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

30
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

31
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

32
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!)

33
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

34
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.

35
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

36
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

37
The Home Stretch
  • Building on Fractional Factorial Designs
  • Examining Nonlinear Factor Effects --
    Higher-Order Designs
  • Response Surface Methodology
  • The Taguchi Approach to Quality Improvement

38
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

39
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?

40
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

41
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.

42
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
43
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

44
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

45
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

46
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
47
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

48
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

49
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

50
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.

51
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)

52
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

53
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
54
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

55
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

56
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 ...

57
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.

58
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.

59
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

60
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)

61
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

62
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
63
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

64
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

65
Response Surface Methodology
  • Initial model
  • Gradient

not significant based on confidence limits using
previous external estimate of inherent variance
66
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

67
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

68
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

69
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

70
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

71
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
  • ...

72
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

73
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
74
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
75
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
76
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)

77
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
78
Computing Losses from Measurements
  • Mean Squared Deviation
  • measure of average departure from target
  • yi -- measured value of performance
    characteristic of i-th item

79
Computing Losses from Measurements
  • Off-Centering and Inherent Variation
  • decompose MSD into two components
  • off-centering
  • inherent variation

off-centering
inherent variation
80
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

81
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

82
Signal-to-Noise Ratios
  • 2. Objective - make Y as small as possible
  • 3. Objective - make Y as large as possible

83
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...

84
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
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