BUS606: Operations and Supply Chain Management

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.