Probabilistic Design

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| Probabilistic and Robust Design | The Probabilistic Design Process | Probabilistic Design Course Notes | Research in Probabilistic Design | Mathematica | Contact Details |

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Probabilistic and Robust Design

What are probabilistic and robust design? Probabilistic design is a mathematically based engineering design methodology for producing high quality mass-produced products. Robust design is a methodology for optimizing this quality.

How is probabilistic design done? Probabilistic engineering design makes calculations with the probability distributions of the design parameters, instead of the mean or nominal values only. This allows the designer to design for a specific reliability or for a specific proportion of product to be produced within specification; and hence guarantee safety, quality and economy.

How is robust design done? A robust product is one that is insensitive to variation, be it from material property variation, manufacturing tolerances, or changing environmental conditions. Robust design attempts to determine the values of the design parameters which maximize the reliability of the product without tightening the material, manufacturing or environmental tolerances. Robust design may be done either experimentally or analytically.

What is the difference between experimental and analytical robust design? Robust design has traditionally been done experimentally following Taguchi's method of using orthogonal experimental arrays. However, if you can make mathematical models of the failure modes, you can obtain results which are orders of magnitude faster, more economic, more accurate and more insightful. This is called analytical robust design.

The Probabilistic Design Process

Probabilistic Design involves the following stages:

1.  Identification of the likely critical physical failure modes of the product.
2.  Determination of the quality variables - those variables whose values determine success or failure.
3.  Determination of the quality relations - those relations defining success.
4.  Mathematical modeling of the quality variables as functions of the design parameters.
5.  Determination of the probability distributions of the design parameters.
6.  Computation of the probability distributions of the quality variables.
7.  Computation of the reliability or percent of product within specification by applying the failure relations to the probability distributions of the quality variables.

If the reliability or conformance to specification is unsatisfactory, proceed to the robustification stage:

8. Derive expressions for the variances of the quality variables.
9. Construct a loss function involving these variances.
10. Determine the values of the (means of the) design parameters which minimize the loss whilst satisfying the quality relations and whilst yielding a feasible design.

If this optimum is still unsatisfactory, proceed to the tolerance design stage:

11. Derive a cost function involving the costs of achieving the given tolerances.
12. Determine the tolerances which will minimize this cost.

The design world is indebted to Genichi Taguchi for the concept of robust design. Note however that Taguchi's context is the experimental determination of the optimum, whilst ours is the analytical determination of the optimum from a mathematical model.

Probabilistic Design Course Notes

I taught probabilistic and robust design methods to final year mechanical engineering students for many years in a Design for Quality course. In the course I found Mathematica (see below) to be ideal for formulating and solving the class problems.

If you are interested in probabilistic and robust design you might be interested in looking at my course notes.

Research in Probabilistic Design

Some of the stages in the probabilistic design process still need further research.

Stage 5, the determination of the probability distributions of the design parameters, is not a trivial exercise to do well, since there is often very little information to be had about the probability distributions of the parameters of a design (which most likely has not been built yet). One of the fascinating tools we can apply here is the Principle of Maximum Entropy to give us the least biased distribution for the knowledge we have. Another is the modelling of the variability of manufacturing processes from first principles.

In the Robustification process Stage 10, the minimization is a constrained optimization involving potentially many variables, including the variances of the design paramters.

If you are interested in the spectrum of methods for determining the best probability distribution from scarce information you might like to contact Maxine Nelson who has written a doctoral dissertation on the subject.

If you are interested in the distributions resulting from manufacturing processes, you might like to contact Clint Steele, who, in addition to having written a doctoral dissertation on the subject has written software using genetic algorithms to carry out the robustification process.

Mathematica

Mathematica is a powerful system for doing mathematical modelling, and I have found it particularly useful in doing probabilistic and robust design. It has a statistics package which includes a comprehensive set of probability distributions, and you can easily do the power series expansions needed for the moment method, the differentiations needed to approximate the variances of a function of random variables, or the graphic visualizations of the processes involved.

If you are interested in the way Mathematica can be used in probabilistic design, see the solved problems in my course notes.

You can get more information on Mathematica from wolfram.com

Contact Details

John M Browne
http://grassmannalgebra.info
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