chapter 8b qualtiy control

8/9/2019 Chapter 8B Qualtiy Control

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ncs3x

n

c

s

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becomelimitscontrolthat theSoc

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:isdeviationstandardprocesstheofestimateOur

n3x

:centerlinethefromxoferror

standardthetimesaway threetakingandaddingby

foundarelimitscontrolThe.xaverages,subgrouptheofaveragetheischartxfor thecenterlineThe

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s!

s

!W

Ws


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sAx)xLCL(

sAx)xUCL(

nc

3

ALetting

3

3

43

!

!

!

Where the value for A3 depends on subgroup size.


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nc

s2xBandAzonesupperbetweenBoundary

ncs2xCandBzonesupperbetweenBoundary

nc

s-xCandBzoneslowerbetweenBoundary

nc

s2-xBandAzoneslowerbetweenBoundary

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An Example

y

In a converting operation, a plastic film is combined withpaper coming off a spooled reel.

y As the two come together, they form a moving sheet thatpasses as a web over a series of rollers.

y

The operation runs in a continuous feed, and thethickness of the plastic coating is an important productcharacteristic.

y Coating thickness is monitored by a highly automated

piece of equipment that uses 10 heads to take 10measurements across the web at half-hour intervals.


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Head # 8:30 9:00

1 .08 .1

.26 2.02

3 2.13 2.1

1.9 1.9

5 2.30 2.30

6 2.15 2.08

7 2.07 1.9

8 2.02 2.12

9 2.22 2.15

10 2.18 2.36

ve. 2.1 2.120.111 0.137

17:00 17:30 18:00

1.98 2.08 2.22

2.30 2.12 2.05

2.31 2.11 1.93

2.12 2.22 2.08

2.08 2.00 2.15

2.10 1.95 2.272.15 2.15 1.95

2.35 2.1 2.11

2.12 2.28 2.12

2.26 2.31 2.10

2.18 2.1 2.10

0.121 0.113 0.106


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There are no indications of a lack of control, so the process can

be considered to be stable and the output predictable withrespect to time, as long as conditions remain the same.


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y It is now appropriate to use some of the methods that will be

described in Chapter 10 (such as check sheets, Paretoanalysis, or brainstorming) to attempt to reduce the common

causes of variation in the never-ending quest to decrease the

difference between process performance and customer

needs.


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Individuals and Moving Range Charts

It is not uncommon to encounter a situation where only asingle variable value can be periodically observed forcontrol charting.

Perhaps measurements must be taken at relatively long

intervals, or the measurements are destructive and/orexpensive; or perhaps they represent a single batchwhere only one measurement is appropriate, such as thetotal yield of a homogenous chemical batch process.

Whatever the case, there are circumstances when datamust be taken as individual units that cannotconveniently be divided into subgroups.


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Individuals and moving range charts have two parts:

y

One charting the process variabilityy One charting the process average

The two parts are used in tandem much as the

and R chart. Stability must first be established in

the portion charting the variability, because theestimate of process variability provides the basis for

control limits of the portion charting the process

average.

x


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Single measurements of variables are considered a subgroup ofsize one.

Hence, there is no variability within the subgroups themselves,and an estimate of the process variability must be made insome other way.

An estimate of variability is based on the point-to-point

variation in the sequence of single values, measured by themoving range (the absolute value of the difference betweeneach data point and the one that immediately preceded it):


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1-ii R !

An avera eofthe moving ranges isusedas

the centerlineforthe moving rangeportion

ofthe chart andasabasisofan estimateof

theoverallprocessvariation:

1-k

RRRange)(MovingCenterline

!!

Where k is the numberofsingle measurements.

As it is impossible to calculate to moving rangeforthefirst subgroupbecause noneprecede it, thereareonly k-1

range measurements;so thesum ofthe Rvalues isdivided

by k-1.


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8.6Individuals and Moving Range

Charts

For the individuals portion of the control chart, thecenterline is the average of the single measurements. Wefind control limits by adding and subtracting three timesthe standard deviation of the single measurements,

estimated by Rbar/d2:

Centerline(x) = xbar = x/k (8.54)

UCL(x) = xbar + 3(Rbar/d2) (8.55)

Using the factor E2 to represent 3/d2, the expression forthe upper control limit becomes

UCL(x) = xbar + E2Rbar (8.56)

Where E2 depends on subgroup size.


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Forthe individualsportionofthecontrolchart,the

centerline istheaverageofthesinglemeasurements.

Wefindcontrollimitsbyaddingandsubtractingthree

timesthestandarddeviationofthesinglemeasurements,

estimatedby:d

R

si e.subgoupondependsEWhere

RE)(U L

dforEfactorthengUsi

d

R)(U L

k( )enterline

2

2

2

2

2


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In this case the subgroup size is 2, as we use two

observations to calculate each moving range value.

Hence, E2 = 2.66, and

R2.66x(x):usingfoundisli itcontrollowertheSi ilarly

R2.66x(x)

!

!


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y A chemical company produces 2,000-gallon batches of a chemicalproduct, A-744, once every two days.

y The product is a combination of six raw materials, of which threeare liquids and three are powdered solids.

y Production takes place in a single tank, agitated as the ingredientsare added, and for several hours thereafter.

y Shipments of A-744 to the customer are made in bins as single lotswhen the batches are finished.

y The chemical company is concerned with the density of thefinished product, which it measures in grams per cubic centimeter.

y As batches are constantly stirred during production, the density is

assumed to be relatively uniform throughout each batch.y Therefore, management decides that density will be measured by

only one reading per batch. During a 60-day period, 30 batches ofA-744 are produced.


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Date Density Moving

Range

Date Density Moving

Range

5/6 1.2 2 - 6/10 1.253 0.018

5/8 1.289 0.0 7 6/12 1.257 0.00

5/10 1.186 0.103 6/1 1.275 0.018

5/13 1.197 0.011 6/17 1.232 0.0 3

5/15 1.252 0.055 6/19 1.201 0.031

5/17 1.221 0.031 6/21 1.281 0.080

5/20 1.229 0.078 6/2 1.27 0.007

5/22 1.323 0.02 6/26 1.23 0.0 0

5/2 1.323 0.000 6/28 1.187 0.0 7

5/27 1.31 0.009 7/1 1.196 0.009

5/29 1.299 0.015 7/3 1.282 0.0865/31 1.225 0.07 7/5 1.322 0.0 0

6/3 1.185 0.0 0 7/8 1.258 0.06

6/5 1.19 0.009 7/9 1.261 0.003

6/7 1.235 0.0 1 7/11 1.201 0.060

Totals 37. 98 1.087


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The process appears to be in a state of statistical control, since

there are no points beyond the control limits and no other

signs of any trends or patterns in the data.


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Special Characteristics ofIndividuals

and Moving Range Charts

Because each subgroup consists of only one value, andprocess variation is estimated on the basis of observation-to-observation changes, individuals and moving range

charts have certain unique characteristics that distinguishthem from other control charts.

For example, for an individuals and moving range chart tobe reliable, it is best to have at least 100 subgroups,

whereas 25 will suffice for most other control chartforms.


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Correlation in the Moving Range.

The moving ranges tend to be correlated.

Large moving range values tend to be followed by other largemoving range values tend to be followed by other largemoving range values, and small moving range values tend to

be followed by other small moving range values.

Because of this, users must be cautious in applying rules for alack of control dealing with patterns in the data.


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It is usually best to be conservative when applying the rules

concerning patterns in the data other than points beyond the

control limits that indicate a lack of control in moving range

charts.

For example, instead of 8 consecutive values above or below

the centerline indicating a lack of control, we might require

10 or 12.

Knowledge and experience are the best guides in establishing

policy in this case.


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Inflated Control Limits.

The control limits for individuals and moving range charts arecomputed from individual measurements.

One indication that the control limits are inflated is the

occurrence of at least of the data points below the

centerline of the moving range portion of the control chart.


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When this happens, control limits should be based on

the median of the moving range values, rather than

those based on the average of the moving Range

values. is calculated using the following

procedure:

1. Arrange the subgroup ranges from low to high.

2. If there are an odd number of subgroup ranges, select the middle

subgroup range as the median range.

3. If there are an even number of subgroup ranges, select the two

middle most subgroup ranges and compute their average. Thisaverage is the median range.

ReM

ReM


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Ifinflated control limits are suspected, control limits

based on the median of the moving ranges should becalculated and compared to those based on the averagemoving range; the narrower of the two sets should beused. Control limits for the moving range portion,based on the median, can be computed using:


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0.000range)movingLCL(median

M3.865range)movingUCL(median e

!

!

(Again, recallthatthe assumption ofnormality is notrequired to interpretthe controllimits;the Empirical

ule maybe used instead).


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Controllimitsforthesinglemeasurements

portionarecreatedusingthefollowing

equations:

e

e

M.CL(x)

Similarl

M.xCL(x)

chapter 8b qualtiy control

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