john lupienski ssat 2006 copyright protected 1 statistical process control john lupienski ssat...

John Lupienski SSAT 2006 Copyright protected1

Statistical Process Control

John Lupienski [email protected] 716-830-4620

mailto:[email protected]


Control Chart Basics - Objectives

At the end of this section, you will be able to: Input data into all types of SPC Charts Recognize the various types of statistical process

control (SPC) Select the appropriate MT- 17 control chart for the

given data Interpret control charts and recommend the

appropriate action to be taken on a process Understand the difference between “Common

cause” vs “Special cause” variation


What is Common Cause Variation?

Variation which is natural in the process as it is designed to operate

Seen as the normal, expected, usually random, variation that is inherent in the process factors: Materials, methods, people, machines,

environment, and measurement

If only common cause variation is present The process is stable and therefore,

predictable and controllable


What is Special Cause Variation?

Variation which is not designed to be part of the process The result of undue or extraordinary influence of

factors Seen as abnormal, unexpected, unusual indications – may appear as non-random (patterned) variation

If special cause variation is present, The process is unstable or out of

statistical control and the causes must be investigated


302010

30

25

20

15

_X=21.36

UCL=27.10

LCL=15.62

1

Key Elements of a Control Chart

Courier Delivery TimeData Plotted over Time3 Upper Control Limit

3 Lower Control Limit Centerline


10

20

30

40

50

60

70

80

90

Identify Signals of Special Cause Variation: 3 Primary Rules

One point outside the 3-sigma limit

7 points in a row on one side of centerline

6 points in a row consistently increasing or decreasing

10

20

30

40

50

60

70

80

90

100

10

20

30

40

50

60

70

80

90

Point outside the limit

Run

Trend


Identify Signals of Special Cause Variation:

3 Secondary Rules 15 points in a row

hugging centerline

2 out of 3 points in a row beyond 2 std. deviations from centerline

Cycles (seasonal, time zone close-out)

10

20

30

40

50

60

70

80

90

10

20

30

40

50

60

70

80

90

10

20

30

40

50

60

70

80

90

Hugging

Approaching Control Limits

Cycle


Developing a Control Chart

1. Select the appropriate critical variable to chart 2. Determine data type, frequency, quantity (sub-group

size), and rationale for sampling plan3. Select the type of control chart to use – See

Roadmap 4. Gather and record sufficient data – in the absence of

anything unusual5. Calculate the centerline and control limits using Minitab 6. Interpret the chart – take appropriate action


Recalculating Control Limits

Recalculated control limits when: Trial limits (calculated with less than sufficient data)

are replaced Signals of special cause variation (in original data only)

are explained and removed The process has changed, and the old limits no longer

characterize the new process


Control Chart Roadmap

I, mR Chartnp Chart

Constant sample

size

Variable sample

size

Variable sample

size

Constant sample

size

Defectives Defects

Discrete Data (Attribute) Continuous Data (Variables)

Control Charts

Subgroup size = 2 to 10

p Chart c Chart u Chart

SubgroupSize > 10

Empirical Rule

Subgroup size = 1

Poisson DistributionBinomial Distribution

x, R Chart x, s Chart


Charts for Continuous Data

X Bar & R Chart: Average and Range Chart Multiple observations taken with regular periodicity Each period results in a sub-group of observations The Average chart tracks the sub-group average from the

process and the range chart tracks the range within each sub-group

I-MR Chart: Individual Values and Moving Ranges Chart Used for single observations of continuous data Either infrequent, or insufficient for sub-grouping The Individuals chart tracks individual measurements from

the process and the range chart tracks the moving range between individual measurements

_


Average and Range Chart CalculationsCalculate the statistics ( , R Chart)

Where:x: Average of the subgroup averages, it becomes the centerline of the

control chartXi: Average of each subgroup

k: Number of subgroupsRi : Range of each subgroup (Maximum observation – Minimum

observation)R: The average range of the subgroups, the centerline on the range chartUCLx: Upper control limit on average chart

LCLx: Lower control limit on average chart

UCLR: Upper control limit on range chart

LCLR : Lower control limit range chart

A2, D3, D4: Constants that vary according to the subgroup sample size

k

xx

k

1ii

k

RR

k

ii

RAxUCL 2x RAxLCL 2x

RDUCL 4R RDLCL 3R

Centerline Control Limits

X


Average & Standard Deviation Chart Calculations

Calculate the statistics ( , s Chart)

k

xx

k

1ii

sAxUCL 3x


sAxLCL 3x k

ss

k

1ii

sBUCL 4S sBLCL 3S

Where:x : Average of the subgroup averages, it becomes the centerline of the average chartxi : Average of each subgroup

k : Number of subgroupssi : Standard deviation of each subgroup

s : Average of the subgroup standard deviations, the centerline on the s chartUCLx: Upper control limit on average chart

LCLx: Lower control limit on average chart

UCLs: Upper control limit on s chart

LCLs : Lower control limit s chart

A3, B3, B4: Constants that vary according to the subgroup sample size

X


Constants TableX-bar Chart for sigma R Chart Constants S Chart Constants Constants estimate

Sample Size = n

A2 A3 d2 D3 D4 B3 B4

2 1.880 2.659 1.128 -- 3.267 -- 3.267

3 1.023 1.954 1.693 -- 2.574 -- 2.568

4 0.729 1.628 2.059 -- 2.282 -- 2.266

5 0.577 1.427 2.326 -- 2.114 -- 2.089

6 0.483 1.287 2.534 -- 2.004 0.030 1.970

7 0.419 1.182 2.704 0.076 1.924 0.118 1.882

8 0.373 1.099 2.847 0.136 1.864 0.185 1.815

9 0.337 1.032 2.970 0.184 1.816 0.239 1.761

10 0.308 0.975 3.078 0.223 1.777 0.284 1.716


Basic Control charts - dialogs & menus Menus are better organized by category…

Let go into the Control Chart Menus and look at the various types of information available


Basic Control chart - X Bar & R Chart

Open file: Camshaft.mtw Stats> Control Charts>Var. charts for SubGrp> XBar R chart


Basic Control chart - dialogs & menus

Lets now review Scale & Option

Do you have C5 Time Stamp?


X Bar & R Control Chart file Camshaft.mtw subgroup size = 5 & time dated

Time stamp

Sam

ple

Mean

3/31/20043/21/20043/11/20043/1/20042/20/20042/10/20041/31/20041/21/20041/11/20041/1/2004

602

601

600

599

598

__X=600.072

UCL=601.722

LCL=598.422

Time stamp

Sam

ple

Range

3/31/20043/21/20043/11/20043/1/20042/20/20042/10/20041/31/20041/21/20041/11/20041/1/2004

6.0

4.5

3.0

1.5

0.0

_R=2.860

UCL=6.048

LCL=0

1

Xbar-R Chart of Length


Control Chart Testsstats>control charts>var. charts for sub grp>X Bar R>

Options>Tests


Control Charts: Update Define TestsSelect Tools>Options>Control Charts and Quality Tools>Define TestsChange Test 2 from 9 to 7


Control Chart TestsSelect Tools>Options>Control Charts and Quality Tools> Tests to PerformSelect Perform all eight tests


X Bar & R Control Chart for Camshaft.mtw data subgroup size = 5, with time dated and all tests

clicked

Time stamp

Sam

ple

Mean

3/31/20043/21/20043/11/20043/1/20042/20/20042/10/20041/31/20041/21/20041/11/20041/1/2004

602

601

600

599

598

__X=600.072

UCL=601.722

LCL=598.422

Time stamp

Sam

ple

Range

3/31/20043/21/20043/11/20043/1/20042/20/20042/10/20041/31/20041/21/20041/11/20041/1/2004

6.0

4.5

3.0

1.5

0.0

_R=2.860

UCL=6.048

LCL=0

66

1

Xbar-R Chart of Length


Example- I-MR Chart for the Camshaft.mtw same data

Try to make this chart check options and tests

Observation

Ind

ivid

ua

l V

alu

e

9181716151413121111

604

602

600

598

596

_X=600.072

UCL=603.591

LCL=596.553

Observation

Mo

vin

g R

an

ge

9181716151413121111

4

3

2

1

0

__MR=1.323

UCL=4.323

LCL=0

66

6

22

2

222

6668

6

2

2

22

2

2

22

I-MR Chart of Length


X Bar, R Chart The chart is best used when

Data are available in sufficient quantity and frequency for small samples to be pulled with regularity

Process knowledge is the basis of rationale for meaningful interpretation

Ideally, 30 subgroups (k = 30) have been collected Trial limits can be calculated with less than k = 30

Examples of uses (sampled hourly, by shift, etc.) Time to process transactions, response times Inquiry handling times, time from request to receipt Dollar metrics – value, sales, costs, usages, variances Variable dimensions or environmental readings or

conditions

What Else?


Summary ofVariable Charts

Chart Type Purpose Subgroup Notes

X & R Average X

Range (R)

Monitor the average of a characteristic over time Monitor the variability of a characteristic / time

n > 1

2 £ n £ 10

Use for sets of measurements

Standard Deviation (S)

Monitor the variability of a characteristic / time

Used with the X charts when the

sample size is > 10

Increased sensitivity due to increased sample size. Use when tight control is necessary or sampling size cost is a factor

Individual & Moving

Range (I - MR)

Monitor the variability of an individual characteristic over time

None

For example use for sales, costs, variances, customer satisfaction score, total

Exponentially

Weighted Moving Average (EWMA)

Monitor small shifts in the process

n ³ 1

Smoothes data to emphasize trends. Uses weights to emphasize importance of recent data

Cumulative Sum

(CUSUM)

Used for specialized applications (similar to EWMA)

n ³ 1

Same sensitivity as EWMA.


Questions?


Discrete Control Chart Roadmap

I, mR Chartnp Chart

Constant sample

size

Variable sample

size

Variable sample

size

Constant sample

size

Defectives Defects


Control Charts



SubgroupSize > 10

Empirical Rule

Subgroup size = 1




Charts for Discrete Data

np Chart: number of defective unitsMust have equal subgroup size

p Chart: proportion of defective unitsCan have unequal subgroup size

c Chart: number of defectsMust have equal subgroup size

u Chart: number of defects per unitCan have unequal subgroup size


np & p Control Chart Roadmap

I, mR Chartnp Chart

Constant sample

size

Variable sample

size

Variable sample

size

Constant sample

size

Defectives Defects


Control Charts



SubgroupSize > 10

Empirical Rule

Subgroup size = 1




np Chart

The np chart will record, and plot, the count of the number of defective (non-conforming) units

As with all discrete charts, larger subgroup size (n = 50 or greater) is desirable – though not always possible

As with all discrete charts, the possibility exists that the computed lower control limit would be negativeIn that case the chart will simply bottom out at zeroOnly an upper control limit will be set and interpreted


np Chart Examples

Examples of applications of an np chart: The number of unresolved trouble calls out of 50

sampled each day at the help desk The number of trades with incorrect broker codes, out

of 100 sampled each day The number of incomplete credit applications, out of 60

sampled each week The number of accounts not balanced out of 200 that

are tracked each billing period

What Else?


Example np Chart

You work in a toy manufacturing company and your job is to inspect the number of defective bicycle tires. You inspect 200 samples in each lot and then decide to create an NP chart to monitor the number of defectives. To make the np chart easier to present at the next staff meeting, you decide to split the chart by every 10 inspection lots.

1 Open the worksheet TOYS.mtw. 2 Choose Stat > Control Charts > Attributes Charts > NP. 3 In Variables, enter Rejects. 4 In Subgroup sizes, enter Inspected. 5 Click NP Chart Options, then click the Display tab. 6 Under Split chart into a series of segments for display purposes,

choose Each segment contains __ subgroups and enter10. 7 Click OK in each dialog box. Review Session window output, What is your interpretations and

conclusion?


Example np Chart for Rejects for Toys.mtw

Interpreting the resultsInspection lots 9 and 20 fall above the upper control limit, indicating that special causes may have affected the number of defectives for these lots. You should investigate what special causes may have influenced the out-of-control number of bicycle tire defectives for inspection lots 9 and 20.

Sample

Sample Count 10987654321

20

10

0

__NP=10.6

UCL=20.10

LCL=1.10

20191817161514131211

20

10

0

__NP=10.6

UCL=20.10

LCL=1.10

30292827262524232221

20

10

0

__NP=10.6

UCL=20.10

LCL=1.10

1

1

NP Chart of Rejects


p Chart The p chart will record the number of defectives and the

sample size - but calculate and plot the proportion of defectives

As with all discrete charts, larger subgroup size (n=>50) is desirable – though not always possible

As with all discrete charts, the possibility exists that the computed lower control limit would be negativeIn that case the chart will simply bottom out at zeroOnly an upper control limit will be set and interpreted


p Chart – Variable Subgroup Size

The control limits vary depending on the subgroup size

UCL

LCL


p Chart – 25% Rule

(1.25)nn(0.75)n i

n

)p-(1p3 p LCL UCL, Then use

If

Wherek

n n i

In other words, if the sample size of any given sample (ni) is within 25% of the mean of all your sample sizes

(n), then just use the sample size mean (n) to calculate your control limits.


p Chart – 25% Rule

The control limits can be constant when the 25% rule is used


P Chart Example - with unequal subgroup sizes

Suppose you work in a plant that manufactures picture tubes for televisions. For each lot, you pull some of the tubes and do a visual inspection. If a tube has scratches on the inside, you reject it. If a lot has too many rejects, you do a 100% inspection on that lot. A P chart can define when you need to inspect the whole lot.

1 Open the worksheet EXH_QC.mtw. 2 Choose Stat > Control Charts >Attributes Charts > P. 3 In Variables, enter Rejects. 4 In Subgroup sizes, enter Sampled. Click OK. Review Session window output and P Chart Graph.

What is your interpretation and conclusion?


Example : P Chart for Rejects from Exh_qc.mtw Data

Interpreting the resultsSample 6 is outside the upper control limit. Consider inspecting the lot.

Sample

Pro

port

ion

191715131197531

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

_P=0.1685

UCL=0.3324

LCL=0.0047

2

2

1

P Chart of Rejects

Tests performed with unequal sample sizes


C & U Control Chart Roadmap

I, mR Chartnp Chart

Constant sample

size

Variable sample

size

Variable sample

size

Constant sample

size

Defectives Defects


Control Charts



SubgroupSize > 10

Empirical Rule

Subgroup size = 1




c ChartUsed to track number of defects

given a constant area of opportunity (sample size)

The c chart is based on the Poisson distribution which stipulates that the potential for a defect is essentially infinite, while the relative occurrence is rare given the opportunity

The number of defects will be plotted and recorded on the c chart


c Chart Calculations

Calculate the C Chart statistics

Where:ci: Number of nonconformities for any given sample of units

c: Average number of nonconformities per unitk: Number of subgroupsLCLc: Lower control limit on u chart.

UCLc: Upper control limit on u chart.

k

cc

k

1ii

c3cUCLc


c3cLCLc


Example c Chart Data

Suppose you work for a linen manufacturer. Each 100 square yards of fabric can contain a certain number of blemishes before it is rejected. You want to track the number of blemishes per 100 square yards over a period of several days, to see if your process is stable & normal. You also want the control chart to show control limits at 2, and 3 standard deviations above and below the center line.

1 Open the worksheet EXH_QC.MTW. 2 Choose Stat > Control Charts > Attributes Charts > C. 3 In Variables, enter Blemish. 4 Click C Chart Options, then click the S Limits tab. 5 Under Display control limits at, enter 2 3 in These multiples of the

standard deviation. 6 Under Place bounds on control limits, check Lower standard

deviation limit bound and enter 0. 7 Click OK in each dialog box. Graph window output


Example c Chart for Blemishes/100 sq. ft.

Are the Blemishes stable and in control?

Sample

Sam

ple

Count

37332925211713951

8

7

6

5

4

3

2

1

0

_C=2.725

+3SL=7.677

LB=0

+2SL=6.027

LB=0

C Chart of Blemish

Interpreting the resultsBecause the points fall in a random pattern, within the bounds of the 3s and 2s control limits, you conclude the process is behaving predictably and is in control.


u Chart The u chart is an attribute control chart that is used

with non-conformities or defects per unit The u chart is based on the Poisson distribution Like the p chart, the u chart is a proportion and can

handle a varying subgroup size; the u chart is to the c chart as the p chart is to the np chart

The u chart is typically applied to more complex processes such as continuous processes


u Chart The u chart is an attribute control chart that is used with

nonconformities or defects per unit Based on the Poisson distribution Used for unequal sample sizes

Like the p chart, the u chart is a proportion and can handle a varying subgroup size; the u chart is to the c chart as the p chart is to the np chart

Typically applied to more complex processes such as continuous processes


u Chart Calculations

Calculate the u Chart statistics

Where:u: Average number of nonconformities per unit.ni: Number inspected in each subgroupLCLu: Lower control limit on u chart.

UCLu: Upper control limit on u chart.

Size Subgroup

Found Defectsu

iu n

u3uUCL


iu n

u3uLCL


Example u Chart with variable Subgroup size

The number of defects will be recorded, but due to the varying sample size, a ratio of defects per unit must be calculated and plotted on the u chart

As production manager of a toy manufacturing company, you want to monitor the number of defects per unit of motorized toy cars. You inspect 20 units of toys and create a U chart to examine the number of defects in each unit of toys. You want the U chart to feature straight control limits, so you fix a subgroup size of 100 (the average number of toy cars sampled).

1 Open the worksheet TOYS.MTW. 2 Choose Stat > Control Charts > Attributes Charts > U. 3 In Variables, enter Defects. In Subgroup sizes, enter Sample. 5 Look at the chart. Click U Chart Options, then click the S Limits tab. 6 Under When subgroup sizes are unequal, calculate control limits, Recalculate Assuming all subgroups have size then enter 100. 7 Click OK in each dialog box. See session window for OOC pts.


u Chart – Toy defects with Variable Subgroup Size

Now under u chart option click Tests tab and click all tests.

What do you find now? Sample

Sam

ple

Count

Per

Unit

191715131197531

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

_U=0.0546

+3SL=0.1241

-3SL=0

+2SL=0.1009

-2SL=0.00832

2

2

2

2

11

U Chart of Defects

Tests performed with unequal sample sizes


u Chart –Toy defects with the 25% Rule of Subgroup size Average n=100

Interpreting the resultsUnits 5 and 6 are still above the upper control limit line, indicating that special causes may have affected the number of defects in these units. You should investigate what special causes may have influenced the out-of-control toy car defects for these units.After clicking all tests you find a good trend that should also be investigated-WHY

Sample

Sam

ple

Count Per Unit

191715131197531

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

_U=0.0555

+3SL=0.1262

-3SL=0

+2SL=0.1026

-2SL=0.00842

2

2

2

2

1

1

U Chart of Defects


u Chart –Toy defects with Modified Stage limits

Now let’s assume after point # 6 we found the root cause and fixed the problem and you want to compute new Limits starting with point # 7.

We must first go to worksheet: Toys and create a new column “C8 Variable” and set the first 6 rows equal to “1” and the remaining “14” rows down to row 20 equal to “2”. Under “U” Chart Options click on “Stages” then enter “ C8 Variable”.

Now look at the chart and what conclusions can you draw?

Sample

Sample Count Per Unit

191715131197531

0.20

0.15

0.10

0.05

0.00

_U=0.04

UCL=0.1

LCL=0

1 2

U Chart of Defects by Variable


Summary ofAttribute Charts

Chart Type Purpose Subgroup Notes

NP Monitor the number of nonconforming units to a subgroup (units rejected)

Must be fixed

Only use this chart when n is roughly constant, fractions are based on counts not measurements

P

Monitor the fraction of nonconforming units in a subgroup Units rejected Units passed

Variable subgroup size, but must be

large enough for a high probability of

at least one nonconformance

in the sample

Often used at quality control points where one or more attributes are inspected. Fractions are based on counts

C

Monitor the number of nonconformances per inspection unit

Area of

opportunity is constant size

Size = 1 inspection unit

Always plot data in order if there is a natural chronological sequence. Note c is the count of occurrence, c is the average

U

Monitor the number of nonconformances per inspection unit

Size of the area of opportunity may

vary – will be described in a

certain number of inspection units (definition of

inspection unit stays the same)

Use when the area of opportunity varies, that is, if the size of the area or population is at risk. For example reorganization doubles the number of employees in one area, doubling the risk.


Control Charts Summary

At this point, you should be able to: Recognize the elements of statistical process

control (SPC) Understand the basis of process stability

studies Select the appropriate control chart for a given

process metric Interpret control charts and recommend the

appropriate action to be taken on a process


Questions?

john lupienski ssat 2006 copyright protected 1 statistical process control john lupienski ssat...

Documents

common cause variation

statistical control

observed variation

patterned variation

process factors

data control limits

process knowledge

presentthe process