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It is quite common in industrial applications to have two or more responses to be optimized. Look at this example from Peter Tkaczuk involving an adhesive. He had two factors and four responses. He fitted the interaction model for each response from data at the four corners of the region of interest. He made four contour plots, one for each response. (See his paper or request our item, "Groping in the Dark?" to see the graphics.)
Peel Looptack Shear Cost
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x1
For the first three on the left, he had a target level for his product. The OK area (on the full graphic) shows where the model is predicting performance that meets the goal.
The fourth response is cost, which is to be minimized. It's OK to the left of the curve.
When all four plots were overlaid, Peter found a triangular sweet spot about 1/3 down on the left.
How will you find the sweet spot in your application? List your responses and your goals for them: to minimize or maximize or hold to target. Choose factors which have a reasonable chance of changing the responses. Choose a model, hopefully you will have the budget to afford a quadratic because it performs so well so often. (Peter, above, got good results from the simpler interaction model: Y = (b0) + (b1)(x1) + (b2)(x2) + (b12)(x1)(x2).) Choose a design. If there is none in a textbook or on your software to suit your model, then call us at 800-732-7381; we will prepare a design to suit your model. Or, use our software, STRATEGY(tm).
Then compute the b's for each response. If you have only two factors, as Peter did, then overlaying the contour plots works very well. But if you have three or more factors, there is a simpler way than making contour plots: use the GridSearch feature of our software STRATEGY(tm) to let the computer do the looking through the contour plots. You tell it your goals; GridSearch saves and lists the combinations of factor levels that meet your goals. GridSearch not only saves you time and money, it also saves trees by printing out less paper.
Then, when you have one or several sweet spots, consider making a few contour plots to show how sensitive the response is to changes in the factors in the vicinity of the sweet spot. A "robust" sweet spot is one around which the process may vary somewhat and still give good results. Factor settings in your plant might be hard to hold exactly. A sweet spot is more attractive the more the x factors can be allowed to slip away from their settings yet still give good results.