Introduction

The manufacturing process was allowed to run for 23 hours of the day. Each hour active had a sample size of 5 (\(n=5\)). The processes for each day has a mean of 25 (\(\mu=25\)) and a standard deviation of 1.75 (\(\sigma=1.75\)). Using the sampling standard deviation formula below we can calculate the sample standard deviation (\(\sigma_{\bar{x}}\)).

\(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\)

\(\sigma_{\bar{x}}=0.7826238\)

The upper control limit (UCL) is 3 standard deviations above the mean and the lower control limit (LCL) is 3 standard deviations below the mean. Since the data is a sample, we will use the sample standard deviation (\(\sigma_{\bar{x}}\)) for our calculations.

\(UCL=\mu+3(\sigma_{\bar{x}})\)

\(UCL=27.3478714\)

\(LCL=\mu-3(\sigma_{\bar{x}})\)

\(LCL=22.6521286\)

In the control charts below, there are green and blue dotted lines. The green line shows \(2\sigma_{\bar{x}}\) above and below the mean and the blue line shows \(1\sigma_{\bar{x}}\) above and below the mean. To determine if a process is out of control, the data will be compared to the 8 Nelson Rules.

Day 1 Sample Control Chart

Starting on hour 6 of the day, the process begins to become out of control. According to the third Nelson rule, if 4 out of 5 consecutive points are above \(\mu+\sigma_{\bar{x}}\) or below \(\mu-\sigma_{\bar{x}}\), then the process should be halted. On hours 6-10, the process produced products above the mean with hours 6-9 above \(\mu+\sigma_{\bar{x}}\). The process should have been stopped on hour 10.

Day 2 Sample Control Chart

The process on day 2 showed very early signs of malfunction. Nelson rule 2 states that 2 out of 3 consecutive points above or below the center line and above \(\mu+2\sigma_{\bar{x}}\) or below \(\mu-2\sigma_{\bar{x}}\), then the process should be halted. Hours 2-4 were above the mean with hours 2 and 4 above \(\mu+2\sigma_{\bar{x}}\). The process should have been halted at hour 4.

Day 3 Sample Control Chart

Day 3 did not break any of the Nelson rules. The process was working well and did not need to be halted at any point.

References

The website https://ximera.osu.edu/qcstats/QC_stats/STAT_QC-0250/main was used as a reference for Nelson rules to determine when a process was out of control.