1 control charts for moving averages and r charts track the performance of processes that have long...

1

Control Charts for Moving Averages

• and R charts track the performance of processes that have long production runs or repeated services.

• Sometimes, there may be insufficient number of sample measurements to create a traditional and R chart.

• For example, only one sample may be taken from a process.

• Rather than plotting each individual reading, it may be more appropriate to use moving average and moving range charts to combine n number of individual values to create an average.

X

X

2


• When a new individual reading is taken, the oldest value forming the previous average is discarded.

• The new reading is combined with the remaining values from the previous average to form a new average.

• This is quite common in continuous process chemical industry, where only one reading is possible at a time.

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• By combining individual values produced over time, moving averages smooth out short term variations and provide the trends in the data.

• For this reason, moving average charts are frequently used for seasonal products.

4


• Interpretation:– a point outside control limits

• interpretation is same as before - process is out of control

– runs above or below the central line or control limits

• interpretation is not the same as before - the successive points are not independent of one another

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Example: Eighteen successive heats of a steel alloy are tested for RC hardness. The resulting data are shown below. Set up control limits for the moving-average and moving-range chart for a sample size of n=3.

Heat Hardness Average Range Heat Hardness Average Range

1 0.806 10 0.809

2 0.814 11 0.808

3 0.810 12 0.810

4 0.820 13 0.812

5 0.819 14 0.810

6 0.815 15 0.809

7 0.817 16 0.807

8 0.810 17 0.807

9 0.811 18 0.800

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Example: Eighteen successive heats of a steel alloy are tested for RC hardness. The resulting data are shown below. Set up control limits for the moving-average and moving-range chart for a sample size of n=3.

Heat Hardness Average Range Heat Hardness Average Range

1 0.806 10 0.809 0.810 0.002

2 0.814 11 0.808 0.809 0.003

3 0.810 0.810 0.008 12 0.810 0.809 0.002

4 0.820 0.815 0.010 13 0.812 0.810 0.004

5 0.819 0.816 0.010 14 0.810 0.811 0.002

6 0.815 0.818 0.005 15 0.809 0.810 0.003

7 0.817 0.817 0.004 16 0.807 0.809 0.003

8 0.810 0.814 0.007 17 0.807 0.808 0.002

9 0.811 0.813 0.007 18 0.800 0.805 0.007

7000500

01300050572

806000500218110

816000500218110

005016

0070010001000080

811016

8050816081508100

3

4

2

2

).)((LCL

.).)(.(UCL

.).)(.(.LCL

.).)(.(.UCL

readings of number where

.....

.....

RD

RD

RAX

RAX

Xg

g

RR

g

XX

R

R

x

x

i

8

Moving Average Chart

0.8

0.805

0.81

0.815

0.82

0 5 10 15 20

Heat

Ha

rdn

es

s UCLxbar

Average

Xbarbar

LCLxbar

Moving Range Chart

0

0.005

0.01

0.015

0 5 10 15 20

Heat

Ha

rdn

es

s R

an

ge

UCLr

Range

Rbar

LCLr

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Exponentially Weighted Moving Average (EWMA)

• The EWMA values are obtained as follows:

• Control limits are set at

Where

)( 101 1 ttt YXY

yX 30

2/y

42 c

s

d

R or

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Chart with a Linear Trend X

• As the tool or die wears– a gradual change in the average is expected and

considered to be normal– the measurement gradually increases– the R chart is likely to remain in control - the

estimate of may not change.• The difference between upper and lower

specifications limits is usually set substantially greater than 6 , to provide some margin of safety against the production of defective products

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Step 1: A trend line is obtained for the chart. A simplified formula is available if– there are an odd number of subgroups– subgroups are taken at a regular interval and– the origin is assumed at the middle subgroup

X

subgroup middle the for

number subgroup the is where,

, ,

0

2

h

h

bhaXh

XhbXa

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Step 2: For each subgroup a separate pair of control limits is obtained above and below the trend line (so, the control limits are sloping lines parallel to the trend line)

RA2

RAbha

RAbha

x

x

2

2

LCL

UCL

13


Step 3: Estimate . For k = 3, 4 etc. the initial aimed-at mean value, is set k above the lower specification limit and the process is stopped for readjustment (a new setup is made, tool/die is changed) when the observed mean value reaches k below the upper specification limit.

2dR /0X

X

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Text Problem 10.25: A certain manufacturing process has exhibited a linear increasing trend. Sample averages and ranges for the past 15 subgroups, taken every 15 minute in subgroup of 5 items, are given in the following table.

Fit the linear trend line to these data, and plot a trended control chart with 3-sigma limits.

X

Subgroup # Average Range Subgroup # Average Range1 198.8 7 9 209.2 72 197.6 2 10 207.8 163 204.6 10 11 210 94 203.8 12 12 214.4 85 205.6 17 13 211.8 166 204.8 9 14 211.8 67 205.4 10 15 213.8 88 210.6 9

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Text Problem 10.26: Specifications on the process in Problem 10.25 are 20030. The process may be stopped at any time and readjusted. If on readjustment the mean is to be set exactly 4 above the lower specification and the process is to be stopped for readjustment when the mean reaches a level exactly 4 below the upper specification:

(a) Calculate the aimed-at starting and stopping values of

(b) Estimate the duration of a run between adjustments

0X

16

X Bar Chart with Linear Trend

150

200

250

-10 -5 0 5 10Subgroup Number

Sub

grou

p M

ean

XbarUSLUCLxbarTrend lineLCLxbarLSL

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Reading and Exercises

• Chapter 10 (moving average and linear trend): – pp. 382-391 (Sections 10.6-7)– 10.24, 10.27

1 control charts for moving averages and r charts track the performance of processes that have long...

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