$3.$ Cecil Rice Export is planning on providing incentives for senior personnel to

rotate into other shifts to better mix personnel among the shifts. If the same

control charts were used for all three shifts, what would be the control limits

of these charts?

ChatGPT

To determine the control limits for the three shifts at Cecil Rice Export using the same control charts, we need to calculate the combined control limits based on the data from all shifts. Since the control limits depend on factors such as the sample size and historical data, we'll assume the same sample size and statistical properties for all shifts for simplicity.

Let's break down the steps to calculate the control limits:

Calculate Combined Range Data:

Combine the range data from all three shifts to get a combined dataset.

Find the average range $($R$-$bar$)$ across all samples from the combined dataset.

Calculate Combined Control Limits:

Use the average range $($R$-$bar$)$ to calculate the control limits for the combined data.

Determine the factors needed for the calculations based on the sample size $($n$).$

Let's assume:

Sample size $($n$)=5$ bags per sample $($consistent with previous calculations$).$

Factors:

$$

$3$

D

$3$

$$

for n $=5$ is $0\mathrm{.}37$

$$

$4$

D

$4$

$$

for n $=5$ is $2\mathrm{.}11$

$$

$2$

A

$2$

$$

for n $=5$ is $0\mathrm{.}483$

Given the combined range data, lower control limit $($LCL$),$ center line $($CL$),$ and upper control limit $($UCL$)$ for each sample, we'll proceed with the calculations.

Here's the provided combined data for reference:

markdown

Copy code

Sample $|$ Range Data $|$ LCL $|$ CL $|$ UCL

$----------------------------------------$

$1|4\mathrm{.}9|1\mathrm{.}305|3\mathrm{.}75|6\mathrm{.}195$

$2|5\mathrm{.}7|1\mathrm{.}305|3\mathrm{.}75|6\mathrm{.}195$

$3|5\mathrm{.}6|1\mathrm{.}305|3\mathrm{.}75|6\mathrm{.}195$

$4|2\mathrm{.}3|1\mathrm{.}305|3\mathrm{.}75|6\mathrm{.}195$

$5|2\mathrm{.}3|1\mathrm{.}305|3\mathrm{.}75|6\mathrm{.}195$

$6|2\mathrm{.}6|1\mathrm{.}305|3\mathrm{.}75|6\mathrm{.}195$

$7|3\mathrm{.}8|1\mathrm{.}305|3\mathrm{.}75|6\mathrm{.}195$

$8|2\mathrm{.}8|1\mathrm{.}305|3\mathrm{.}75|6\mathrm{.}195$

Calculate R$-$bar:

Combine the range data:

$$

$-$bar

$=$

$$

Range Data

Number of samples

R$-$bar$=$

Number of samples

$$Range Data

$$

Calculate the average range $($R$-$bar$).$

$$

$-$bar

$=$

$4\mathrm{.}9$

$+$

$5\mathrm{.}7$

$+$

$5\mathrm{.}6$

$+$

$2\mathrm{.}3$

$+$

$2\mathrm{.}3$

$+$

$2\mathrm{.}6$

$+$

$3\mathrm{.}8$

$+$

$2\mathrm{.}8$

$8$

R$-$bar$=$

$8$

$4\mathrm{.}9+5\mathrm{.}7+5\mathrm{.}6+2\mathrm{.}3+2\mathrm{.}3+2\mathrm{.}6+3\mathrm{.}8+2\mathrm{.}8$

$$

$$

$-$bar

$=$

$29$

$8$

$$

$3\mathrm{.}625$

R$-$bar$=$

$8$

$29$

$$

$3\mathrm{.}625$

Calculate Control Limits:

Use the calculated R$-$bar and factors to determine the control limits$.$

UCL for R$-$chart

$=$

$$

$-$bar

$\backslash$times

$$

$4$

$=$

$3\mathrm{.}625$

$\backslash$times

$2\mathrm{.}11$

$$

$7\mathrm{.}64$

UCL for R$-$chart$=$R$-$bar$\backslash$times D

$4$

$$

$=3\mathrm{.}625\backslash$times $2\mathrm{.}117\mathrm{.}64$

LCL for R$-$chart

$=$

$$

$-$bar

$\backslash$times

$$

$3$

$=$

$3\mathrm{.}625$

$\backslash$times

$0\mathrm{.}37$

$$

$1\mathrm{.}34$

LCL for R$-$chart$=$R$-$bar$\backslash$times D

$3$

$$

$=3\mathrm{.}625\backslash$times $0\mathrm{.}371\mathrm{.}34$

UCL for X$-$bar chart

$=$

CL

$+$

$$

$2$

$\backslash$times

R$-$bar

UCL for X$-$bar chart$=$CL$+$A

$2$

$$

$\backslash$times R$-$bar

$=$

$3\mathrm{.}75$

$+$

$0\mathrm{.}483$

$\backslash$times

$3\mathrm{.}625$

$$

$5\mathrm{.}25$

$=3\mathrm{.}75+0\mathrm{.}483\backslash$times $3\mathrm{.}6255\mathrm{.}25$

LCL for X$-$bar chart

$=$

CL

$$

$$

$2$

$\backslash$times

R$-$bar

LCL for X$-$bar chart$=$CL$$A

$2$

$$

$\backslash$times R$-$bar

$=$

$3\mathrm{.}75$

$$

$0\mathrm{.}483$

$\backslash$times

$3\mathrm{.}625$

$$

$2\mathrm{.}25$

$=3\mathrm{.}750\mathrm{.}483\backslash$times $3\mathrm{.}6252\mathrm{.}25$

So$,$ the calculated control limits for the combined data from all three shifts are approximately:

UCL for R$-$chart: $7\mathrm{.}64$

LCL for R$-$chart: $1\mathrm{.}34$

UCL for X$-$bar chart: $5\mathrm{.}25$

LCL for X$-$bar chart: $2\mathrm{.}25$

These control limits can be used to monitor the bagging machine's performance across all shifts and detect any variations that may need attention or further investigation