Control Charts for Monitoring Weibull Distribution

Department of Mathematics wtrnumber2013-1

Control Charts for Monitoring Weibull Distribution Chuanping Sze and Francis Pascual March 2013

Postal address: Department of Mathematics, Washington State University, Pullman, WA 99164-3113 Voice: 509-335-3926 • Fax: 509-335-1188 • Email: [email protected] • URL: http://www.sci.wsu.edu/math

Control Charts for Monitoring Weibull Distribution

Chuanping Sze and Francis Pascual, Department of Mathematics Washington State University, Pullman WA 99164

ABSTRACT Tables of constants are provided for computing the control limits for monitoring processes described by the Weibull distribution. The smallest extreme value distribution (SEV) is used in the simulation process, and we obtain results for Weibull by taking exponentials. We monitor the quantiles of the Weibull distribution and the SEV mean. When monitoring the quantile, we considered the cases when  is known and  is unknown. Monitoring the SEV mean is done using the 0.4296 quantile estimate or the sample mean of SEV samples. Tables and control limits equations are given for each case. Average run length (ARL) charts are constructed for the problem of detecting the shift of a percentile of a Weibull distribution. An illustrative example is presented concerning the tensile strength of carbon fibers from Nichols and Padgett (2005).

1. Introduction In this paper, we provide tables of constants (quantiles of pivotal quantities) that practitioners can use for control charts with the Weibull distribution. We consider the two-parameter Weibull distribution with the probability density function   x f ( x)      

 1

  x   exp     ,     

x>0(1.1)

where is the scale parameter and  is the shape parameter. Because of its flexible shape and ability to model a wide range of failure rates, Weibull has been used successfully in many applications as a purely empirical model, for example, in biomedical applications and in modeling ball bearing, relay and material strength failures. The Weibull distribution includes as special cases known distributions. If X has a Weibull distribution with , it is identical to the exponential distribution. When , it is the Rayleigh distribution. If X is Weibull, then Y= ln(X) has an SEV distribution. This relationship between the Weibull and SEV is similar to the relationship between normal and lognormal. Hence, any statistical results we get from the Weibull distribution can be applied easily to SEV and vice versa. The Weibull distribution is equivalent to the SEV with parameter log(and where log is natural logarithm. The probability density function of SEV is written in the form f ( y) 

y  y     exp  exp         1

-∞ < y < ∞

(1.2)

1

The mean and variance of SEV are, respectively, E(Y)    

V(Y) 

  2

(1.3)

2

where =0.5772 is Euler’s constant. Since the parameters of the SEV are location-scale parameters, it is more convenient to develop control chart techniques with SEV. The 100pth percentile or p quantile of the SEV distribution is y p     log[  log(1  p)] . The 100pth percentile of the Weibull distribution is exp (yp). Note that when p=0.4296, yp = E (Y), that is, the SEV mean is also its 0.4296 quantile. 6

Limits for control charts are obtained from the distributions of pivotal quantities. In Section 2, we introduce the basic definition of control limits and ARL. We consider two cases when simulating the tables of the percentiles of pivotal quantities. In Section 3, we monitor the quantile of the SEV when  is known and  is unknown. In Section 4, with  known, we monitor the mean of SEV using the maximum likelihood estimate (MLE) of the 0.4296 quantile or, alternatively, the sample mean. We discuss chart performance using ARL in Section 5. We also provide one-sided and two-sided charts to see how the ARL changes with varying sample size and how fast a shift is detected. In Section 6, we use the carbon fiber tensile strength data from Nichols and Padgett (2005) and construct one and two-sided control charts for monitoring the 0.01 quantile. We also compare the performance between charts using y and the MLE of E(Y) when  is known. Ramalhoto and Morais (1999) study the design and performance of the Shewhart control charts for monitoring the scale parameter  of the three-parameter Weibull distribution. They assume that the Weibull shape and threshold parameters are known and use a pivotal quantity for  to compute control limits. They also perform an ARL study to assess how fast the charts detect changes in . Nichols and Pagett (2005) also study control charts for the Weibull distribution by simulation. However, they fail to take the advantage of relevant pivotal quantity that they have to perform the simulation process every time data are gathered to generate the control limits. In addition, Ryan (2000), Jobe and Vardeman (1998), Montgomery (2004), and Wetherill and Brown (1991) discuss control charts in quality control almost exclusively for the normal distribution. 2. Probability limits and Average Run Length 2.1 Probability Limit Suppose a quantity  is being monitored using a sample statistic  . Assume that the process is ^

stable and let     1 , where 0