Robust Designing

Robust parameter designs, introduced by Taguchi, are experimental designs used to exploit the interaction between control and uncontrollable noise variables by robustification -- finding the settings of the control factors that minimize response variation from uncontrollable factors.^[1] Control variables are variables of which the experimenter has full control. Noise variables lie on the other side of the spectrum, and while these variables may be easily controlled in an experimental setting, outside of the experimental world they are very hard, if not impossible, to control. Robust parameter designs use a naming convention similar to that of FFDs. A 2^{(m1+m2)-(p1-p2)} is a 2-level design where m1 is the number of control factors, m2 is the number of noise factors, p1 is the level of fractionation for control factors, and p2 is the level of fractionation for noise factors.

Effect Sparsity. Interactions can only significantly effect the response if at least one of the parent factors has an effect on the response.

Consider an RPD cake baking example from Montgomery (2005), where an experimenter wants to improve the quality of cake.^[2] While the cake manufacturer can control the amount of flour, amount of sugar, amount of baking powder, and coloring content of the cake; other factors are uncontrollable, such as oven temperature and bake time. The manufacturer can print instructions for a bake time of 20 minutes but in the real world has no control over consumer baking habits. Variations in the quality of the cake can arise from baking at 325^o instead of 350^o or from leaving the cake in the oven for a slightly too short or too long period of time. Robust parameter designs seek to minimize the effects of noise factors on quality. For this example, the manufacturer hopes to minimize the effects in fluctuation of bake time on cake quality, and in doing this the optimal settings for the control factors is required.

RPDs are primarily used in a simulation setting where uncontrollable noise variables are easily controlled. Whereas in the real world noise factors are hard to control, in an experimental setting control over these factors is easily maintained. For the cake-baking example, the experimenter can fluctuate bake time and oven temperature to understand the effects of such fluctuation that may occur when control is no longer in his hands.

Robust parameter designs very similar to fractional factorial designs (FFDs) in that the optimal design can be found using Hadamard matrices, principles of effect hierarchy and factor sparsity are maintained, and aliasing is present when full RPDs are fractionated. Much like FFDs, RPDs are screening designs and can provide a linear model of the system at hand. What is meant by effect hierarchy for FFDs is that higher-order interactions tend to have a negligible effect on the response.^[3] As stated in Carraway, main effects are most likely to have an effect on the response, then two-factor interactions, then three-factor interactions, and so on.^[4] The concept of effect sparsity is that not all factors will have an effect on the response. These principles are the foundation for fractionating Hadamard matrices. By fractionating, experimenters can form conclusions in fewer runs and with fewer resources. Oftentimes, RPDs are used at the early stages of an experiment. Because two-level RPDs assume linearity among factor effects, other methods may be used to model curvature after the number of factors has been reduced.

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