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5. Process Improvement
5.6. Case Studies
5.6.1. Eddy Current Probe Sensitivity Case Study

Using the Fitted Model

Model Provides Additional Insight Although deriving the fitted model was not the primary purpose of the study, it does have two benefits in terms of additional insight:
  1. Global prediction
  2. Global determination of best settings
Global Prediction How does one predict the response at points other than those used in the experiment? The prediction equation yields good results at the eight combinations of coded -1 and +1 values for the three factors:
  1. X1 = Number of turns = 90 and 180
  2. X2 = Winding distance = 0.38 and 1.14
  3. X3 = Wire gauge = 40 and 48
What, however, would one expect the detector to yield at target settings of, say,
  1. Number of turns = 150
  2. Winding distance = 0.50
  3. Wire gauge = 46
Based on the fitted equation, we first translate the target values into coded target values as follows:
    coded target = -1 + 2*(target-low)/(high-low)
Hence the coded target values are
  1. X1 = -1 + 2*(150-90)/(180-90) = 0.333333
  2. X2 = -1 + 2*(0.50-0.38)/(1.14-0.38) = -0.684211
  3. X3 = -1 + 2*(46-40)/(48-40) = 0.5000
Thus the raw data
    (Number of turns, Winding distance, Wire gauge) = (150, 0.50, 46)
translates into the coded
    (X1, X2, X3) = (0.333333, -0.684211, 0.50000)
on the -1 to +1 scale.

Inserting these coded values into the fitted equation yields, as desired, a predicted value of

    \( \hat{Y} \) = 2.65875 + 1.55125(0.333333) - 0.43375(-0.684211) = 3.47261
The above procedure can be carried out for any values of turns, distance, and gauge. This is subject to the usual cautions that equations that are good near the data point vertices may not necessarily be good everywhere in the factor space. Interpolation is a bit safer than extrapolation, but it is not guaranteed to provide good results, of course. One would feel more comfortable about interpolation (as in our example) if additional data had been collected at the center point and the center point data turned out to be in good agreement with predicted values at the center point based on the fitted model. In our case, we had no such data and so the sobering truth is that the user of the equation is assuming something in which the data set as given is not capable of suggesting one way or the other. Given that assumption, we have demonstrated how one may cautiously but insightfully generate predicted values that go well beyond our limited original data set of eight points.
Global Determination of Best Settings In order to determine the best settings for the factors, we can use a DOE contour plot. The DOE contour plot is generated for the two most significant factors and shows the value of the response variable at the vertices (i.e, the -1 and +1 settings for the factor variables) and indicates the direction that maximizes (or minimizes) the response variable. If you have more than two significant factors, you can generate a series of DOE contour plots with each one using two of the important factors.
DOE Contour Plot The following is the DOE contour plot of the number of turns and the winding distance.

The DOE contour plot identifies X1=-1 and X2=1 as the optimal

The maximum value of the response variable (eddy current) corresponds to X1 (number of turns) equal to -1 and X2 (winding distance) equal to +1. The lower right corner of the contour plot corresponds to the direction that maximizes the response variable. This information can be used in planning the next phase of the experiment.

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