The following experiment was performed on a process for making leaf springs. We are interested in optimizing the process using this data. The goal will be to hit a target mean output while minimizing variance and production time.

The output measures are the sample mean and variance of the rest height of the spring after hot forming. The factors are x₁: Temperature, x₂: Heating Time,x₃: Transfer Time, andx₄: Hold Down Time

X1	X2	X3	X4		S2
-1	-1	-1	-1	7.372	0.038
1	-1	-1	1	7.660	0.017
-1	1	-1	1	7.670	0.091
1	1	-1	-1	7.785	0.053
-1	-1	1	1	7.520	0.001
1	-1	1	-1	7.640	0.008
-1	1	1	-1	7.540	0.090
1	1	1	1	7.902	0.071

1.2 Noticing that the column x₄= x₁x₂x₃, what is the defining relation and resolution for this design? 1.4 By looking at the aliases we decide to fit a linear model of the following form:

= ₀+ ₁x₁+ ₂x₂+ ₃x₃+ ₄x₄+ ₁₂x₁x₂+ ₁₃x₁x₃+ ₂₃x₂x₁₃

Fill in the coefficients for the model for ands². 1.5 Use normal probability plots of these coefficients to figure out which factors are significant for both models. Hint: assume that₀is significant in both models and don't add it to the plot. If you use Matlab, ignore the line fit provided by the code.

1.6 We are given the specification target value of 7.45, and want to find the best combination of inputs to get that value. Given the models derived above, and the desire to also minimize variance and production time, what are the best settings for this process?x_1, x_2, x_3, x₄ (Assume that all inputs must be in the range -1 to 1.)

1.7 Instead we now want to meet the target with minimum variance, but wish to minimize energy use. What would be the new operating point assuming lower temperatures mean less energy use?x_1, x_2, x₄