8. Assessing Product Reliability
8.1. Introduction
8.1.6. What are the basic lifetime distribution models used for non-repairable populations?

8.1.6.2.

Weibull

• Formulas and Plots
• Uses of the Weibull Distribution Model
• DATAPLOT and EXCEL Functions for the Weibull

Weibull Formulas

Formulas and Plots

The Weibull is a very flexible life distribution model with two parameters. It has CDF and PDF and other key formulas given by: 

with the scale parameter (the Characteristic Life), (gamma) the Shape Parameter, and is the Gamma function with (N) = (N-1)! for integer N.

The Cum Hazard function for the Weibull is the integral of the failure rate or

A more general 3-parameter form of the Weibull includes an additional waiting time parameter µ (sometimes called a shift or location parameter). The formulas for the 3-parameter Weibull are easily obtained from the above formulas by replacing t by (t - µ) wherever t appears. No failure can occur before µ hours, so the time scale starts at µ, and not 0. If a shift parameter µ is known (based, perhaps, on the physics of the failure mode), then all you have to do is subtract µ from all the observed failure times and/or readout times and analyze the resulting shifted data with a 2-parameter Weibull.

NOTE: Various texts and articles in the literature use a variety of different symbols for the same Weibull parameters. For example, the characteristic life is sometimes called c (or = nu or = eta) and the shape parameter is also called m (or = beta). To add to the confusion, EXCEL calls the characteristic life and the shape and some authors even parameterize the density function differently, using a scale parameter

Special Case: When = 1, the Weibull reduces to the Exponential Model, with = 1/ = the mean time to fail (MTTF).

Depending on the value of the shape parameter , the Weibull model can empirically fit a wide range of data histogram shapes. This is shown by the PDF example curves below.

Weibull data 'shapes'

From a failure rate model viewpoint, the Weibull is a natural extension of the constant failure rate exponential model since the Weibull has a polynomial failure rate with exponent { - 1}. This makes all the failure rate curves shown in the following plot possible.

Weibull failure rate 'shapes'

The Weibull is very flexible and also has theoretical justification in many applications

Uses of the Weibull Distribution Model

1. Because of its flexible shape and ability to model a wide range of failure rates, the Weibull has been used successfully in many applications as a purely empirical model.
2. The Weibull model can be derived theoretically as a form of Extreme Value Distribution, governing the time to occurrence of the "weakest link" of many competing failure processes. This may explain why it has been so successful in applications such as capacitor, ball bearing, relay and material strength failures.
3. Another special case of the Weibull occurs when the shape parameter is 2. The distribution is called the Rayleigh Distribution and it turns out to be the theoretical probability model for the magnitude of radial error when the x and y coordinate errors are independent normals with 0 mean and the same standard deviation.

Dataplot and EXCEL functions for the Weibull

DATAPLOT and EXCEL Functions for the Weibull

The following commands in Dataplot will evaluate the PDF and CDF of a Weibull at time T, with shape and characteristic life .

SET MINMAX 1
LET PDF = WEIPDF(T,,0, ),
LET CDF = WEICDF(T,,0,)

For example, if T = 1000, = 1.5 and = 5000, the above commands will produce a PDF of .000123 and a CDF of .08556.

NOTE: Whenever using Dataplot for a Weibull analysis, you must start by setting MINMAX equal to 1.

To generate Weibull random numbers from a Weibull with shape parameter 1.5 and characteristic life 5000, use the following commands:

SET MINMAX 1
LET GAMMA = 1.5
LET SAMPLE = WEIBULL RANDOM NUMBERS FOR I = 1 1 100
LET SAMPLE = 5000*SAMPLE

Next, to see how well these "random Weibull data points" are actually fit by a Weibull, we plot the points on "Weibull" paper to check whether they line up following a straight line. The commands (following the last commands above) are:

X1LABEL LOG TIME
Y1LABEL CUM PROBABILITY
WEIBULL PLOT SAMPLE

The resulting plot is shown below. Note the log scale used is base 10.

Dataplot Weibull Probability Plot

EXCEL also has Weibull CDF and PDF built in functions. EXCEL calls the shape parameter = alpha and the characteristic life = beta. The following command evaluates the Weibull PDF for time 1000 when the shape is 1.5 and the characteristic life is 5000:

WEIBULL(1000,1.5,5000,FALSE)

For the corresponding CDF

WEIBULL(1000,1.5,5000,TRUE)

The returned values (.000123 and .085559, respectively) are the same as calculated by Dataplot.