(365d) New and Efficient Interval Sampling Method for Use of P-Boxes in Off-Line Quality Control

Parameter design is an off-line quality-control method for designing products and manufacturing processes that are robust in the face of uncontrollable variations. The goal of parameter design is to identify design settings that make the product's performance less sensitive to manufacturing and environmental variations and deterioration. The parameter design problem can be formulated as optimization under uncertainty problem, where variance in the output variable of interest is minimized in the face of uncertainty.

Risk analysts recognize two fundamentally distinct forms of uncertainty. The first is the variability that arises from environmental stochasticity, inhomogeneity of materials, fluctuations in time, variation in space, or heterogeneity or other differences among components or individuals. Variability is sometimes called Type I uncertainty, or less cryptically, aleatory uncertainty to emphasize its relation to the randomness in gambling and games of chance. It is also sometimes called irreducible uncertainty because, in principle, it cannot be reduced by further empirical study (although it may be better characterized). The second kind of uncertainty is the incertitude of scientific ignorance, measurement uncertainty, unobservability, censoring, or other lack of knowledge. This is sometimes called Type II uncertainty or simply epistemic uncertainty. In contrast with aleatory uncertainty, epistemic uncertainty is sometimes called reducible uncertainty because an additional empirical effort can generally reduce it. For situations in which the uncertainty about quantities is purely aleatory, probability theory is usually preferred. However, when strictly epistemic uncertainties, or both aleatory and epistemic uncertainties, are present several researchers have converged on essentially the same idea: that one can work with bounds on probability for this purpose. In the 1990s, researchers introduced interval-type bounds on cumulative distribution functions, which have been called "probability boxes" or "p-boxes" for short [2,3,8]. Uncertainty propagation for uncertainties described by probability distributions is generally carried out using sampling techniques. For p-box uncertainties, researchers use interval Monte Carlo for propagation [1,6,7]. However, this method is computationally intensive. Recently, we developed and evaluated five new efficient interval sampling techniques for P-boxes. These sampling techniques are derived from quasi-Monte Carlo sequences like Hammersley and Sobol sequences, uniformity theory, and interval mathematics. These sampling techniques are evaluated using various functions of two to 800 uncertain variables defined by p-box uncertainties. Heuristics for selecting an efficient sampling technique based on the number of uncertainties is presented [5].

In this paper, we present a real-world example of quality control of a CSTR using one of the efficient sampling techniques presented earlier and is applicable to this problem [6]. The example illustrates how a change in estimates of uncertainty can affect the optimal quality control designs.

References

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