A New Generalized Range Control Chart for the Weibull Distribution

Arif, Osama H.; Aslam, Muhammad

doi:https://doi.org/10.1155/2018/9453589

Complexity

On this page

Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 9453589 | https://doi.org/10.1155/2018/9453589

A New Generalized Range Control Chart for the Weibull Distribution

Osama H. Arif^1,2and Muhammad Aslam²

Academic Editor: Lucia Valentina Gambuzza

Received03 Mar 2018

Revised02 Aug 2018

Accepted30 Aug 2018

Published05 Nov 2018

Abstract

In this study, a generalized range control chart is designed for the Weibull distribution using generally weighted moving average statistics. The proposed chart is based on minimum generally weighted moving average statistic and maximum generally weighted moving average statistics. We utilize the inverse erf function to transform the Weibull data to normal data. The necessary measures are given to assess the performance of the proposed control chart. The comparison study shows that the proposed control chart outperforms the existing control charts based on exponentially weighted moving average statistic in terms of the average run length. A real example is given for applying the proposed chart in the industry.

1. Introduction

The control charts are an integral part of statistical process control which is used to monitor and improve the manufacturing process in the industry. Among the control charts, the Shewhart control charts are preferred due to ease in assumptions and use in the industry. These charts are usually designed under the assumption that the quality of interest follows the normal distribution. But the data obtained from the chemical, semiconductor, cutting tool wear, and failure time processes are often skewed [1, 2]. So, monitoring the skewed data using the Shewhart control chart may mislead to observe the manufacturing process.

The Weibull distribution has been widely used to characterize material breaking strength, describe the lifetime of electronic items, and monitor the reliability data obtained from life test [3, 4]. For the skewed data, it is not a proper approach to monitor the process using Shewhart X-bar, standard deviation, and range charts. According to [5], these charts are not effective to detect a shift in the process in low population percentiles of these distributions and increase the false signal out-of-control. Several people worked on the designing of control chart for the Weibull distribution. Nelson [6] designed various control charts for the Weibull process. Bai and Choi [1] designed X-bar and range charts to monitor the skewed data. Hawkins and Olwell [7] proposed cumulative sum (CUSUM) control chart to monitor the scale parameter of the Weibull distribution. Ramalhoto and Morais [8] focused on the development of Shewhart for the Weibull distribution with fixed and variable sampling intervals. Chang and Bai [9] designed various control chart including exponentially weighted moving average (EWMA) chart using weighted standard deviations for the skewed distribution. More details about these charts can be seen in [10–12].

Transforming the Weibull distributed data to normal data may make ease for industrial engineering. Therefore, several transformation methods including power transformation, inverse erf function, and Box–Cox transformation are available in the literature. Batson et al. [13] designed moving average control chart using the power transformation. Pascual [14] studied moving average chart using natural logarithm of the Weibull distribution. More details about control chart using transformation can be read in [3, 15–17] and in [18].

Recently, Wang [19] proposed the MaxEWMA control chart for the Weibull distribution using three different transformations. Wang [19] recommended that inverse erf function performs better than power transformation and Box–Cox transformation when shape parameter of the Weibull distribution is greater than one. Woodall and Montgomery [20] pointed that development of control chart for the Weibull distribution has a potential for future research.

In this study, a new generalized range control chart is designed for the Weibull distribution using inverse erf function which has been commonly used to transform the Weibull distributed data to normal data. Some tables for practical use are also given. The comparison study shows that the proposed control chart outperforms the existing control chart in terms of average run length. A real example is also given to apply the proposed chart in the industry.

2. Design of Proposed Control Chart

The Weibull distribution with shape parameter and scale parameter has the following probability density function:

Usually, industrial engineers have the information about the parameters and . These parameters can be estimated from the monitoring data if unknown in the practice. This density function is skewed, so symmetric type of control limits may not work for a Weibull distribution. But there are several methods of transforming Weibull distributed data to a normal distributed data. Wang [19] discussed and analyzed three transformation methods and recommended the method in [17] when .

Faraz et al. [17] suggested the use of erf function to convert the Weibull distributed random variable to standard normal variable. When is a Weibull distributed data, the following transformed data were shown to follow a standard normal data: where .

As suggested by Wang [19], we utilized another statistic from a sequence of above transformed data ’s:

Note here that presents the inverse cumulative standard normal distribution and cumulative chi-square distribution with one degree of freedom.

The generally weighted moving average (GWMA) statistic is the generalization of traditional EWMA statistic. The GWMA statistic empowers the decision about the state of process by utilizing the weight assigned to current and past information; for more details, see [21–24]. Suppose that denotes the number of samples in the sequence of independent samples until the 1^st occurrence of event since the previous occurrence of event [25]. Suppose that are assigned weight for current and previous sample such that shows the weight to target of the process.

Two GWMA statistic to monitor the mean and variability of the process are defined.

The GWMA statistics applied to two transformed data are given by where and the starting values of and are usually set at 0.

The MaxEWMA control chart has the ability to detect a small shift in the mean or variability than the traditional EWMA control charts [25]. The range chart () which is computed from the difference of maximum quantity, say maxGWMA, and minimum quantity, say minGWMA, has been commonly used to monitor the system variability.

We define the maxGWMA statistic as

The minGWMA statistic is defined as follows

Following [26], Sheu et al. [25] suggested that when and are independently normally distributed, the quantity can be evaluated as follows for the design parameter and adjustment parameter . where , , and .

Now, the plotting statistic can be defined as follows

The proposed control chart is stated as follows.

Step 1. Obtain the Weibull data ’s at each subgroup of size and transform it to the standard normal data using and compute .

Step 2. Obtain GWMA statistics and . Determine and and compute statistic .

Step 3. Declare the process out-of-control if or .
The proposed control chart is a generalization of several control charts. The proposed control chart reduces the chart in [27] when and , where is smoothing constant in EWMA statistic. The proposed control chart becomes the chart in [19] when and .

The operational process of the proposed control chart is based on two control limits which are defined as where is the average and SD is the standard deviation of statistic , respectively. Also, is the control chart coefficient which can be determined by closed-form solution, the Markov chain approach, and Monte Carlo simulation (MCS). These are alternative methods for obtaining the properties of the run-length distribution. In this paper, the control chart coefficient will be determined through the following MCS. (1)Specify the shape parameter and scale parameter of the Weibull distribution. Generate a random sample of size from the Weibull distribution by assuming in-control state(2)Compute statistic and plot it on the control limits. Note that the first of out-of-control is called run length (RL). Repeat this process 10,000 times(3)Compute average run length (ARL) and standard deviation (SD) using these RL. Determine for which , where is specified ARL(4)Repeat Steps 1-3 to find the ARL for the shifted process, say when scale parameter is shifted to , where is a constant

Using the above MCS, the values of and SD are reported for various control chart parameters in Tables 1–5 as well as the mean and standard deviation when =0.95, =5, and various values of , , and . The , , and SD for various values of shape and scale parameters of the Weibull distribution and shift constant are reported in Tables 2–4.

From Tables 1–5, we note the following trend in the control chart parameters. (1)For same values of all other parameters, the values of and SD decrease as values decrease. For example, when and , the value of from Table 2 is 347, and it is 346 when (2)For the same values of all other parameters, the values of and SD decrease for and decrease when (3)For the same values of all other parameters and , , the values of and SD increase when increases and decreases

3. Comparison and Simulation Study

In this section, we compare the performance of the proposed control chart with chart based on EWMA statistic and the chart in [19].

3.1. Proposed Chart vs EWMA Chart

As mentioned earlier, the proposed control chart is the extensions of several charts. The proposed control chart reduces to control chart based on EWMA statistic when and . The performance of the proposed control chart will be compared with chart based on EWMA statistic proposed by [27] using the simulation data. In this study, it is assumed that , , and . Suppose that the process is declared to be and . The first 20 observations of subgroup size are generated from the Weibull distribution with parameters and . The next 30 observations of subgroup size are generated from the Weibull distribution with parameters and shift in scale parameter that is . The simulation data, and , of the proposed control chart are shown in Table 6.

The calculations of statistic is also shown in Table 6. The values of statistic are plotted when ; , 1.10; (exponential distribution) and ; ([27] statistic) are plotted on control charts given in Figure 1. The UCL and LCL of the three charts are also shown in Figure 1. From Figure 1, it can be noted that the proposed chart using GWMA statistic detects shift in the process at around 45^th values while the proposed control charts using the statistic in [27] and exponential distributions are unable to detect shift in the process. From Figure 1, it can be concluded that the proposed control chart using GWMA statistic performs better than the control chart using the statistic in [27] in detecting earlier shift in the process.

(a)

(b)

(c)

3.2. Proposed Chart vs the Chart in [19]

The performance of the proposed chart in terms of ARL and SD will be compared with the MaxEWMA control chart proposed by [19] for the same values of specified control chart parameters. To save the space, the comparison will be given when , , , and . The ARLs of both control charts are depicted in Figure 2.

From Figure 2, it can be noted that the proposed control chart has lower curve of ARLs as compared to the chart proposed by [19]. From Figure 2, it can be observed that the proposed chart has smaller values of ARLs than the chart in [19] for small-shift level that is . For example, when , the ARL and SD from the proposed control charts are 338 and 332, respectively, while the ARL and SD from the chart in [19] are 344 and 336, respectively. The two control charts have the same performance when shift is large that is when . So, by comparison, it is concluded that the proposed control chart is more sensitive to detect small shift in the process.

4. Case Study

Monitoring the breaking strength fibrous composite is very important in the industries which guarantee the safety of material used in the aero industry and construction of bridges. The application of the proposed control chart is given on the breaking strength of carbon fiber data which is modeled by the Weibull distribution with and . The data were originally reported by [5] and used by [17] in the area of the control chart. Let and the control chart coefficient for this real data is 4.27. The data for the breaking strength of carbon fibers for are reported in Table 7 whereas the first 10 samples are recorded when the process is in-control. The values of statistic are also reported in Table 7.

The values of are also plotted in Figure 3. From Figure 3, it can be noted that 13^th values are out-of-control which lead the experimenter to take action to bring back the process to the in-control state.

5. Concluding Remarks

In this study, a new generalized range control chart is designed for the Weibull distribution using inverse erf function which has been commonly used to transform the Weibull distributed data to normal data. Some tables for practical use are also given. From comparison study, it is concluded that the proposed control chart is more efficient than the existing charts in detecting quick shift at small-shift levels. The proposed control chart can be applied in the industry for monitoring of the process that follows the Weibull distribution. The proposed control chart using some other sampling scheme and cost models can be considered as future research.

Data Availability

The data are given in the paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality of this manuscript. This work was supported by the Deanship of Scientific Research (DSR), Jouf University, Sakaka, under Grant no. (39/828). The authors, therefore, gratefully acknowledge the DSR technical and financial support.

References

D.-S. Bai and I. Choi, “ and R control charts for skewed populations,” Journal of Quality Technology, vol. 27, no. 2, pp. 120–131, 1995.
View at: Publisher Site | Google Scholar
K. Derya and H. Canan, “Control charts for skewed distributions: Weibull, gamma, and lognormal,” Metodološki zvezki, vol. 9, no. 2, 2012.
View at: Google Scholar
F. Pascual and S. Li, “Monitoring the Weibull shape parameter by control charts for the sample range of type II censored data,” Quality and Reliability Engineering International, vol. 28, no. 2, pp. 233–246, 2012.
View at: Publisher Site | Google Scholar
W. Weibull, A Statistical Theory of the Strength of Materials, Ingeniors Vetenskaps Akademien, 1939.
W. Padgett and J. D. Spurrier, “Shewhart-type charts for percentiles of strength distributions,” Journal of Quality Technology, vol. 22, no. 4, pp. 283–288, 1990.
View at: Publisher Site | Google Scholar
P. R. Nelson, “Control charts for Weibull processes with standards given,” IEEE Transactions on Reliability, vol. R-28, no. 4, pp. 283–288, 1979.
View at: Publisher Site | Google Scholar
D. M. Hawkins and D. H. Olwell, Cumulative Sum Charts and Charting for Quality Improvement, Springer, New York, NY, USA, 1998.
View at: Publisher Site
M. Ramalhoto and M. Morais, “Shewhart control charts for the scale parameter of a Weibull control variable with fixed and variable sampling intervals,” Journal of Applied Statistics, vol. 26, no. 1, pp. 129–160, 1999.
View at: Publisher Site | Google Scholar
Y. S. Chang and D. S. Bai, “Control charts for positively-skewed populations with weighted standard deviations,” Quality and Reliability Engineering International, vol. 17, no. 5, pp. 397–406, 2001.
View at: Publisher Site | Google Scholar
M. Aslam, N. Khan, M. Azam, and C.-H. Jun, “Designing of a new monitoring t-chart using repetitive sampling,” Information Sciences, vol. 269, pp. 210–216, 2014.
View at: Publisher Site | Google Scholar
L. K. Chan and H. J. Cui, “Skewness correction and R charts for skewed distributions,” Naval Research Logistics, vol. 50, no. 6, pp. 555–573, 2003.
View at: Publisher Site | Google Scholar
E. Santiago and J. Smith, “Control charts based on the exponential distribution: adapting runs rules for the t chart,” Quality Engineering, vol. 25, no. 2, pp. 85–96, 2013.
View at: Publisher Site | Google Scholar
R. G. Batson, Y. Jeong, D. J. Fonseca, and P. S. Ray, “Control charts for monitoring field failure data,” Quality and Reliability Engineering International, vol. 22, no. 7, pp. 733–755, 2006.
View at: Publisher Site | Google Scholar
F. Pascual, “EWMA charts for the Weibull shape parameter,” Journal of Quality Technology, vol. 42, no. 4, pp. 400–416, 2010.
View at: Publisher Site | Google Scholar
M. Aslam, “A mixed EWMA–CUSUM control chart for Weibull-distributed quality characteristics,” Quality and Reliability Engineering International, vol. 32, no. 8, pp. 2987–2994, 2016.
View at: Publisher Site | Google Scholar
M. Aslam, M. Mohsin, and C.-H. Jun, “A new t-chart using process capability index,” Communications in Statistics - Simulation and Computation, vol. 46, no. 7, pp. 5141–5150, 2017.
View at: Publisher Site | Google Scholar
A. Faraz, E. M. Saniga, and C. Heuchenne, “Shewhart control charts for monitoring reliability with Weibull lifetimes,” Quality and Reliability Engineering International, vol. 31, no. 8, pp. 1565–1574, 2015.
View at: Publisher Site | Google Scholar
G. Chen, S. W. Cheng, and H. Xie, “Monitoring process mean and variability with one EWMA chart,” Journal of Quality Technology, vol. 33, no. 2, pp. 223–233, 2001.
View at: Publisher Site | Google Scholar
F. K. Wang, “MaxEWMA control chart for a Weibull process with individual measurements,” Quality and Reliability Engineering International, vol. 33, no. 2, pp. 369–379, 2017.
View at: Publisher Site | Google Scholar
W. H. Woodall and D. C. Montgomery, “Some current directions in the theory and application of statistical process monitoring,” Journal of Quality Technology, vol. 46, no. 1, pp. 78–94, 2014.
View at: Publisher Site | Google Scholar
N. Chakraborty, S. Chakraborti, S. W. Human, and N. Balakrishnan, “A generally weighted moving average signed-rank control chart,” Quality and Reliability Engineering International, vol. 32, no. 8, pp. 2835–2845, 2016.
View at: Publisher Site | Google Scholar
N. Chakraborty, S. W. Human, and N. Balakrishnan, “A generally weighted moving average exceedance chart,” Journal of Statistical Computation and Simulation, vol. 88, no. 9, pp. 1759–1781, 2018.
View at: Publisher Site | Google Scholar
S.-L. Lu, “Non parametric double generally weighted moving average sign charts based on process proportion,” Communications in Statistics-Theory and Methods, vol. 47, no. 11, pp. 2684–2700, 2018.
View at: Publisher Site | Google Scholar
M. Mohsin, M. Aslam, and C.-H. Jun, “A new generally weighted moving average control chart based on Taguchi’s loss function to monitor process mean and dispersion,” Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, vol. 230, no. 8, pp. 1537–1547, 2016.
View at: Publisher Site | Google Scholar
S.-H. Sheu, C.-J. Huang, and T.-S. Hsu, “Extended maximum generally weighted moving average control chart for monitoring process mean and variability,” Computers & Industrial Engineering, vol. 62, no. 1, pp. 216–225, 2012.
View at: Publisher Site | Google Scholar
T. Nakagawa and S. Osaki, “The discrete Weibull distribution,” IEEE Transactions on Reliability, vol. R-24, no. 5, pp. 300-301, 1975.
View at: Publisher Site | Google Scholar
S. Roberts, “Control chart tests based on geometric moving averages,” Technometrics, vol. 1, no. 3, pp. 239–250, 1959.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2018 Osama H. Arif and Muhammad Aslam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies