Robustification: Difference between revisions

Revision as of 02:40, 3 October 2022

Revision as of 04:40, 3 October 2022

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===Non-linearities===

Consider the [[Graph (mathematics)|graph]] below of a relationship between an input variable ''x'' and the output ''Y'', for which it is desired that a value of 7 is taken, of a system of interest. It can be seen that there are two possible values that ''x'' can take, 5 and 30. If the tolerance for ''x'' is independent of the nominal value, then it can also be seen that when ''x'' is set equal to 30, the expected variation of ''Y'' is less than if ''x'' were set equal to 5. The reason is that the gradient at ''x'' = 30 is less than at ''x'' = 5, and the random variability in ''x'' is suppressed as it flows to ''Y''.{{Citation needed|date=October 2022}}

[[File:Robustification.JPG]]

This basic principle underlies all robustification, but in practice there are typically a number of inputs and it is the suitable point with the lowest gradient on a multi-dimensional surface that must be found.{{Citation needed|date=October 2022}}

===Non-constant variability===

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''Z'' = ''x y''

For any target value of ''Z'' there is an infinite number of combinations for the nominal values of ''x'' and ''y'' that will be suitable. However, if the standard deviation of ''x'' was proportional to the nominal value and the standard deviation of ''y'' was constant, then ''x'' would be reduced (to limit the random variability that will flow from the right hand side of the equation to the left hand side) and ''y'' would be increased (with no expected increase random variability because the standard deviation is constant) to bring the value of ''Z'' to the target value. By doing this, ''Z'' would have the desired nominal value and it would be expected that its standard deviation would be at a minimum: robustified.{{Citation needed|date=October 2022}}

By taking advantage of the two principles covered above, one is able to optimise a system so that the nominal value of a systems output is kept at its desired level while also minimising the likelihood of any deviation from that nominal value. This is despite the presence of random variability within the input variables.{{Citation needed|date=October 2022}}

==Methods==

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===Experimental===

The experimental approach is probably the most widely known. It involves the identification of those variables that can be adjusted and those variables that are treated as [[Statistical noise|noises]]. An experiment is then designed to investigate how changes to the nominal value of the adjustable variables can limit the transfer of noise from the noise variables to the output. This approach is attributed to [[Genichi Taguchi|Taguchi]] and is often associated with [[Taguchi methods]]. While many have found the approach to provide impressive results, the techniques have also been criticised for being statistically erroneous and inefficient. Also, the time and effort required can be significant.{{Citation needed|date=October 2022}}

Another experimental method that was used for robustification is the Operating Window. It was developed in the [[United States]] before the wave of quality methods from [[Japan]] came to the [[Western world|West]], but still remains unknown to many.<ref name="Clausing">See Clausing (2004) reference for more details</ref> In this approach, the noise of the inputs is continually increased as the system is modified to reduce sensitivity to that noise. This increases robustness, but also provides a clearer measure of the variability that is flowing through the system. After optimisation, the random variability of the inputs is controlled and reduced, and the system exhibits improved quality.{{Citation needed|date=October 2022}}

===Analytical===

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===Numerical===

In the numerical approach a model is run a number of times as part of a [[Monte Carlo method|Monte Carlo simulation]] or a numerical propagation of errors to predict the variability of the outputs. Numerical optimisation methods such as hill climbing or evolutionary algorithms are then used to find the optimum nominal values for the inputs. This approach typically requires less human time and effort than the other two, but it can be very demanding on computational resources during simulation and optimization.{{Citation needed|date=October 2022}}

==See also==