Comparing and Weighting Two Weibull Models

The Weibull model is widely used for estimating the reliability of engineering materials. More reliable parts tend to last longer (i.e. have a large time before failure). Estimates of Weibull model parameters can be achieved by using a least squares method to fit a straight line on a logarithmic plot. This Demonstration describes a method for comparing two Weibull models for reliability via weight measures. The PDFs or CDFs are plotted.
The weight functions measure a combination of failure time and failure probability. For a chosen weight function, larger weight values indicate higher reliability of the material whose failure time follows that Weibull distribution.


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Snapshots 1 and 2: two cases where the more reliable model could be easily determined from comparing PDF plots based on the single crossing point
Snapshots 3 and 4: cases where there are two crossing points and it is not as clear which model is for a more reliable material
This raises the questions: How can one determine which Weibull model is for a more reliable material? Can high probability of early failure be compensated by high probability of later failure when compared to another model?
This Demonstration is an attempt to answer this question by comparing the models via a weighted measure.
Weight function is a combination of two variables, the failure probability and failure time. Larger weight values indicate higher reliability of the material whose failure time follows that Weibull distribution.
Weight function 1 is used for a practical, approximate engineering calculation. It is the sum of the products of the probability density function (PDF) and failure times: , where is the weight result, is the failure time, and is the PDF of the Weibull model. Specify a suitable time range and sample rate to simplify the calculation.
Weight function 2 is the corresponding theoretical mathematical expression: . The integral weight function precisely calculates over the entire Weibull distribution.
Left sliders: generate two Weibull models with different parameters and , plot option and control parameters for weight function 1.
Top-right panel: indicate parameters and weight results for the two Weibull models.
Bottom-right graphic: both models are shown using either their PDF or CDF plots. The color function illustrates the weight function concept. The earlier failure probability is represented in warm colors to indicate a warning. The later failure probability is represented in cool colors to indicate safety.
Reference: W. Weibull, "Fitting of Curves to Observations," Fatigue Testing and Analysis of Results, New York: Pergamon Press, 1961 pp. 201–203.
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