Statistical Process Control

Assume that in the manufacture of 1 kg Mischmetal ingots, the product weight varies with the batch. Below are a number of subsets taken at normal operating conditions (subsets 1-7), with the weight values given in kg. Construct the \(X\)-Bar, \(R\)-charts, and \(S\)-charts for the experimental data (subsets 8-11). Measurements are taken sequentially in increasing subset number.

Solution:

First, the average, range, and standard deviation are calculated for each subset.

Next, the grand average \(X_{G A}\), average range \(R_{A}\), and average standard deviation \(S_{A}\) are computed for the subsets taken under normal operating conditions, and thus the centerlines are known. Here \(n=4\).

\(\begin{aligned}

&X_{G A}=1.0004 \\

&R_{A}=0.05428 \\

&S_{A}=0.023948

\end{aligned}\)

\(X\)-Bar limits are computed (using \(R_{A}\)).

\(\begin{aligned}

&\mathrm{UCL}=X_{G A}+A_{2} R_{A}=1.0004+0.729(0.05428)=1.04 \\

&\mathrm{LCL}=X_{G A}-A_{2} R_{A}=1.0004-0.729(0.05428)=0.96

\end{aligned}\)

\(X\)-Bar limits are computed (using \(S_{A}\)).

\(\begin{aligned}

&\mathrm{UCL}=X_{G A}+A_{3} S_{A}=1.0004+1.628(0.023948)=1.04 \\

&\mathrm{LCL}=X_{G A}-A_{3} S_{A}=1.0004-1.628(0.023948)=0.96

\end{aligned}\)

Note: Since \(n=4\) (a relatively small subset size), both \(R_{A}\) and \(S_{A}\) can be used to accurately calculate the UCL and LCL.

\(R\)-chart limits are computed.

\(\begin{gathered}

\mathrm{UCL}=D_{4} R_{A}=2.282(0.05428)=0.12 \\

\mathrm{LCL}=D_{3} R_{A}=0(0.05428)=0

\end{gathered}\)

\(S\)-chart limits are computed.

\(\begin{gathered}

\mathrm{UCL}=B_{4} S_{A}=2.266(0.023948)=0.054266 \\

\mathrm{LCL}=B_{3} S_{A}=0(0.023948)=0

\end{gathered}\)

The individual points in subsets 8-11 are plotted below to demonstrate how they vary with in comparison with the control limits.

Figure E-1: Chart of individual points in subsets 8-11.

The subgroup averages are shown in the following \(X\)-Bar chart:

Figure E-2: \(X\)-Bar chart for subsets 8-11.

The \(R\)-chart is shown below:

Figure E-3: \(R\)-chart for subsets 8-11.

The \(S\)-chart is shown below:

Figure E-4: \(S\)-chart for subsets 8-11.

The experimental data is shown to be in control, as it obeys all of the rules given above.