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Process capability analysis is used to determine the process performance as capable or incapable within a specified tolerance. Basic indices C_{p}, C_{pk}, C_{pm}, C_{pmk} initially developed for normally distributed processes showed inappropriate for processes with non-normal distributions. A number of authors worked on non-normal distributions which were most notably those of Clements, Pearn and Chen, Montgomery and Johnson-Kotz-Pearn (JKP). Obtaining PCIs based on the parameters of non-normal distributions are completely disregarded and ignored. However parameters of some non-normal distributions have significance for knowing the status of process as capable or incapable. In this article we intend to work on the shape parameter of Weibull distribution to calculate PCIs. We work on two data sets for verification and validation purpose. Efficacy of the technique is assessed by bootstrapping the results of estimate and standard error of shape parameter.

Process capability indices (PCIs) reflect the potential of a process performance by a unit-less measure. These indices have been used to check a statistical controlled process capability of producing items that meets the quality condition predetermined by the product designer, basic indices as C_{p} by Juran [_{pk} by Kane [_{pm} by Chan et al. [_{pmk} by Pearn et al. [

This article is organized as follows. In Section 2, we describe few special features of Weibull distribution; propose procedure to obtain process parameters of Weibull shape parameter to estimate PCIs from earlier proposed superstructure forms. In Section 3, we analyze the two data sets and observe the effect of shape parameter on PCIs. In Section 4, we conclude the proposed procedure and in Section 5, we discuss some applications of Weibull distribution.

The Weibull distribution is first applied by Swedish Physicist W. Weibull in the 1930s in studying material strength in tension and fatigue. This distribution is extensively used in reliability and lifetime modeling fatigue and survival analysis and to model an extensive range of failure rates because of its flexible shape.

The two parameter Weibull distribution has density function as;

where

The cumulative distribution function of Weibull distribution is as

Equation (2) becomes the cumulative distribution function of exponential distribution for

If

The different

Aleksander and Thierry [

We may thus conclude that the variation in shape parameter is sensitive for measuring the process capability.

There are many methods proposed for estimation of parameters of known theoretical distribution as maximum likelihood estimation method, method of moments, regression methods etc. For our capability index based on Weibull shape parameter we prefer regression method. This method is easy and straight forward even for obtaining the standard errors of the parameters without involving iterative procedures as required for other methods of estimation see Ahmed and Safdar [

The simple linear regression model with only one independent variable (regressor) is

This model has two regression parameters

Suppose that

Taking

For plotting position

mator) is used for an efficient approximation of complete samples. Here i is the rank of the data in ascending order, n is number of items in data and

Comparing Equation (8) with Equation (4) we have

One may note the estimator of

Now we have

The estimate and standard error for Weibull shape parameter using the above method are estimated using the nls function based on delta method in R-console version 3.0.1 [

The sampling distribution of regression coefficients of a linear regression model follows t-distribution. The histogram and the box plot for the estimated parameter for different size (or bootstrap) showed the symmetrical behavior (see

processes but for parameters (shape and scale) we do not have any set specification limits. Based on the sampling distribution of regression parameters we may estimate the limits by

We check the normality by histogram with density curve and box plot. We drew sample of size 50 to obtain the sampling distribution of shape parameter

We have used the following superstructures earlier proposed by numerous authors for normal and non-normal populations to estimate PCIs for Weibull shape parameter

Clements [

(PC) Pearn and Chan [

(PKJ) Pearn and Kotz [

B | Methods | C_{p} | C_{pk} | C_{pm} | C_{pmk} |
---|---|---|---|---|---|

20 | Normal | 1.000 | 1.000 | 1.000 | 1.000 |

Clement | 1.607 | 1.509 | 1.542 | 1.447 | |

PC | 1.607 | 1.495 | 1.524 | 1.418 | |

P-K-J | 1.607 | 1.396 | 1.524 | 1.347 | |

30 | Normal | 1.000 | 1.000 | 1.000 | 1.000 |

Clement | 1.543 | 1.495 | 1.527 | 1.479 | |

Pearn & | 1.543 | 1.514 | 1.537 | 1.508 | |

P-K-J | 1.543 | 1.311 | 1.537 | 1.307 | |

40 | Normal | 1.000 | 1.000 | 1.000 | 1.000 |

Clement | 1.448 | 1.443 | 1.448 | 1.443 | |

Pearn | 1.448 | 1.437 | 1.447 | 1.437 | |

P-K-J | 1.448 | 1.370 | 1.447 | 1.369 | |

50 | Normal | 1.000 | 1.000 | 1.000 | 1.000 |

Clement | 1.221 | 1.196 | 1.218 | 1.193 | |

Pearn | 1.221 | 1.214 | 1.220 | 1.213 | |

P-K-J | 1.221 | 1.128 | 1.220 | 1.128 | |

100 | Normal | 1.000 | 1.000 | 1.000 | 1.000 |

Clement | 1.119 | 1.063 | 1.104 | 1.048 | |

Pearn | 1.119 | 1.062 | 1.103 | 1.046 | |

P-K-J | 1.119 | 0.932 | 1.103 | 0.923 |

B | Methods | C_{p} | C_{pk} | C_{pm} | C_{pmk} |
---|---|---|---|---|---|

20 | Normal | 1.00 | 1.00 | 1.00 | 1.00 |

Clement | 1.67 | 1.57 | 1.60 | 1.50 | |

PC | 1.67 | 1.59 | 1.62 | 1.54 | |

P-K-J | 1.67 | 1.50 | 1.62 | 1.47 | |

30 | Normal | 1.00 | 1.00 | 1.00 | 1.00 |

Clement | 1.71 | 1.60 | 1.63 | 1.53 | |

Pearn & | 1.71 | 1.60 | 1.62 | 1.52 | |

P-K-J | 1.71 | 1.59 | 1.62 | 1.53 | |

40 | Normal | 1.00 | 1.00 | 1.00 | 1.00 |

Clement | 1.41 | 1.36 | 1.39 | 1.34 | |

Pearn | 1.41 | 1.37 | 1.40 | 1.36 | |

P-K-J | 1.41 | 1.04 | 1.40 | 1.03 | |

50 | Normal | 1.00 | 1.00 | 1.00 | 1.00 |

Clement | 1.37 | 1.34 | 1.36 | 1.34 | |

Pearn | 1.37 | 1.34 | 1.36 | 1.34 | |

P-K-J | 1.37 | 1.26 | 1.36 | 1.26 | |

100 | Normal | 1.00 | 1.00 | 1.00 | 1.00 |

Clement | 1.29 | 1.26 | 1.29 | 1.26 | |

Pearn | 1.29 | 1.25 | 1.28 | 1.24 | |

P-K-J | 1.29 | 1.19 | 1.28 | 1.18 |

To illustrate how to calculate the PCIs based on shape parameter of Weibull distribution, we used two data sets and obtain the results of indices given in Equations (11a)-(11d).

The data consisting of 22 annual minimum mean daily flows, given in m^{3}/s, were recorded at the proposed diversion site in the Hasdo subcatchment of the Mahanadi basin in India pp-236 [

Assuming annual minimum flow follows two parameter Weibull distributions with shape parameter 2.29 and scale parameter 2.67. These parameters are estimated from the mean and variance of the given data sets.

The reason for estimating PCIs for annual minimum daily flow is the notice on the variation when we vary the magnitude of shape parameter in either direction. It is observed that the probability that an annual minimum low flow will be less than 2 m^{3}/s over a two-year period is 0.40. This probability decreases by increasing the magnitude of the shape parameter from 2.29 and increases by decreasing the magnitude. So the stability of annual minimum low flow may analyzed by obtaining PCIs based on the shape parameter of the Weibull distribution.

We may observe from the analysis a single conclusion that Weibull shape parameter may use in the place of measurements coming from Weibull distribution. So in any industrial or mechanical production of process follows Weibull distribution with known parameters, capability indices using the proposed procedures can be obtained in a very straightforward way along with statistical process control tools.

Pearn and Chan used rubber edge measurements to exemplify his proposed generalization of basic indices to cover non-normal distributions. Rubber edge of a speaker woofer driver is one of the key components which reflect the sound quality of the driver units, such as musical image and clarity of the sound. One characteristic of the rubber edge is weight which is considered here as a quality characteristic. Since the sound quality is time dependent for any life-time system so use of this data as Weibull distribution is justified. We used a sample of size 80 from Pearn and Chen measurements [

The shape and scale parameter based on mean and standard deviation are found 259.1926 and 8.681957. The probability that process produce the weight of rubber edge on target (T = 8.7) at these parameters is found to be 0.819, if beta is increased by 10 the probability becomes 0.826 and 0.813 by decreasing the beta to 10. These variations in probabilities justified the use of PCIs for the shape parameter of the Weibull distribution.

For the data set of Pearn and edge measurements if shape parameter which should be the function of weight selected by process designer increases the chance that process producer the rubber edge of speaker woofer driver on target might be increases for this case.

In this research paper we are the first who discussed the PCIs based on shape parameter of Weibull distribution that have been disregarded always. The importance of shape parameter cannot be ignored in making a process behavior symmetrical or skewed. Some other non-normal distributions as Gumbel (EVI) or fatigue reliability distribution can also be assessed using this technique. In studying of estimating the PCIs for the shape parameter of Weibull distribution for the data sets (discussed in Section 3.1 and 3.2), we noted that the process stability depended on the parameters of the distribution. One can show that the annual minimum low flow as stable or instable and the quality of woofer speaker as good or bad might be the functions of the magnitude of the shape parameter. We also observed that the size of the sample has no one-to-one influence on capability of a process. We used the three popular generalizations by Clements [

In studying of estimating the PCIs for the shape parameter of Weibull distribution for two data sets, we noted that the process stability also depended on shape parameters of the distribution. We also observed that the size of the sample has no significant effect on capability of a process.

This distribution provides the close approximation to the probability laws of many natural phenomena as estimation of low flows, time to failure for an electrical and mechanical system(s), earthquakes or floods breaking strengths in censored or truncated situations and many more. Few examples are Lieblein and Zelen [

The authors thank the Editor-in-Chief Prof. Qihua Wang (Chinese Academy of Sciences, China) and the anonymous referees for their careful reading of the paper and their suggestions which improved the paper.