MAE 552 Heuristic Optimization

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MAE 552 Heuristic Optimization

• Instructor John Eddy
• Lecture 17
• 3/4/02
• Taguchis Orthogonal Arrays

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S/N Ratio

• Why use the signal / noise ratio?
• Given a product or process with a target
  performance, deviation from that performance can
  typically be expressed in terms of (i.t.o)
  statistics (Taguchi Quality loss).
• Two things you want are for your mean to be on
  target and for your variance to be low. Take
  the example of a machine that throws darts.

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S/N Ratio

• Here are the results of 4 such machines

Good Mean Bad Var.
Good Mean Good Var.
Bad Mean Good Var.
Bad Mean Bad Var.
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S/N Ratio

• Noise can be interpreted as the variance observed
  in the process (since variance is commonly a
  direct result of noise).
• The signal can be interpreted as the desired
  value (the value you would like your mean to
  take).
• The S/N ratio is then

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S/N Ratio

• Another means of calculating the S/N ratio is to
  take -10log(d) where d represents the mean
  squared deviation from the target.
• So in our example, the target is 0 defects, and
  the squared deviation for each count is the
  square of the count itself.

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S/N Ratio

• Should we always use an S/N ratio?
• Taguchi says yes because it accounts for both the
  mean and standard deviation.
• Will it always be a maximize situation?
• Yes it will, assuming that it is always desirable
  to achieve the target value with very little
  deviation.

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Back to our Example
A B C D
1 1 1 1 1
2 1 2 2 2
3 1 3 3 3
4 2 1 2 3
5 2 2 3 1
6 2 3 1 2
7 3 1 3 2
8 3 2 1 3
9 3 3 2 1
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Taguchi Example

• Continuing with our example

Factor Levels Levels Levels
Factor 1 2 3
A Temperature -20 -45 -60
B Pressure -30 -40 -55
C Settling Time -50 -35 -40
D Clean Method -45 -40 -40
All the level averages for our example problem.
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Example

• Using the values for the level averages, we can
  plot the factor effects and see visually which
  factors have the greatest influence on the
  performance of our product or process.

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Example
Overall Mean
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Example

• Considering the effects visualized in the
  previous figure, which levels do you think are
  optimal?
• Recall that we want to maximize our S/N ratio, so
  we choose the levels that cause the greatest
  positive deviation from our mean. Since our
  deviation was
• mfactor, level m

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Example

• We clearly want to choose those levels with the
  largest values because we have a negative overall
  mean and we will therefore be adding that value
  to our level means.
• Consider Factor A, our deviation from the mean
  cause by factor A at level 1 is
• -20 (-41.67) 21.67

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Example

• And our deviation from the mean cause by factors
  2 and three are
• -45 (-41.67) -3.33
• -60 (-41.67) -18.33
• So by comparison, level 1 causes the greatest
  positive deviation from the mean, and will be
  chosen as our optimal setting.

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Example

• Likewise, for the remaining factors, we will
  choose the following levels as optimal
• B) Pressure - Level 1
• C) Settling Time - Level 2
• D) Cleaning Method - Level 2 or 3
• The resulting equivalent experiment is then
• 1 1 2 2/3 - does not appear in our array!!

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Example

• The final step would then be to run a validation
  experiment to see that the optimal solution is
  actually the best thus far.
• If it is not, then you would perhaps do one of
  two things
• Choose the best configuration of any of the
  experiments you did run.
• Refine your system and re-run the experiments.
  (could mean changing level settings to hone in
  on good regions, etc.

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Comparison

• What if we didnt use the Taguchi approach, what
  else might we do to estimate factor effects?
• One approach is a one-at-a-time method in which
  all but one factor are held constant while the
  other is varied throughout its levels. How many
  experiments does this require?

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Comparison

• Answer
• For N factors each having L levels, we would
  need
• L x N experiments
• That would be 12 in our example (and it turns out
  that we would not achieve the same accuracy in
  our estimate of the level means).

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Comparison

• Another approach is the brute-force full
  factorial method. That is, conduct every
  possible experiment for the factors and levels.
• How many experiments would this require?

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Comparison

• Answer
• For N factors each having L levels, we would
  need
• LN experiments
• That would be 81 experiments in our example. The
  only good thing is that we would be guaranteed to
  find the best configuration for our levels.

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ANOM

• The approach we have taken with this problem is
  called the analysis of means.
• This approach relies on 2 assumptions for its
  validity
• Use of an additive model is appropriate
• Use of an orthogonal array is appropriate.

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Additive Model

• What do we mean by additive model?
• First, lets define a short hand for our factor
  effects
• Let ai represent the deviation from the mean
  caused by setting factor a to level i. So i.t.o.
  our previous example,
• b2 mb2 - m

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Additive Model

• We are going to show that the procedure we
  applied to our example is equivalent to applying
  an additive model of the individual factor
  effects to determine performance.
• This means that by using such an additive model,
  we should be able to predict the performance of
  our product or process based on the factor
  effects.

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Additive Model

• Our additive model is stated as follows
• ?(Ai, Bj, Ck, Dl) m ai bj ck dl e
• Where e is an error term that accounts for the
  error incurred by using the additive model and
  the error incurred by any lack of repeatability
  of measuring ? for any given experiment.

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Additive Model

• As in most subject areas, an additive model is
  also sometimes called a superposition model or a
  variables separable model.
• It is possible for our factor effects to be
  non-linear (quadratic, cubic, etc.) but cross
  product terms (between factors) are not allowed.
• Cross correlation would be like The effect of
  factor A at level 2 while factor C is at level 3
  is

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Additive Model

• According to our previous definition of factor
  level effects, we can show that the sum of the
  effects of all the levels of a given factor is
  equal to 0. That is (in our example)
• a1 a2 a3 0
• b1 b2 b3 0
• c1 c2 c3 0
• d1 d2 d3 0
• We will show this for factor a.

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Additive Model

• Recall that ai mAi m

v
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Additive Model

• Now, recall our representation of the observation
  value i.t.o the overall mean, factor effects, and
  error term
• ?(Ai, Bj, Ck, Dl) m ai bj ck dl e
• We can represent factor effects i.t.o this
  equation as shown on the following slide.

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Additive Model
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Additive Model

• So from the previous result, we see that
• mA3 m a3 some error term
• This is sensible enough, recall that our
  representation of ai was originally mAi m.
• So mA3 as we have derived it here, is an estimate
  of m a3 and thus our approach was in fact the
  use of an additive model.

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Additive Model

• A note on the error term.
• We have shown that the error is actually the
  average of 3 error terms (one corresponding to
  each experiment).
• We typically treat each of the individual
  experiment error terms as having zero mean and
  some variance.

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Additive Model

• Define Replication Number nr
• The replication number is the number of times a
  particular factor level is repeated in an
  orthogonal array.
• It can be shown that the error variance of the
  average effect of a factor level is smaller than
  the error variance of a single experiment by a
  factor equal to its nr.

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Additive Model

• So, to obtain the same accuracy in our factor
  level averages using a one-factor-at-a-time
  approach, we would have to conduct 3 experiments
  at each of the 3 levels of each factor for a
  total of 36 experiments.