Download - 3.Control de Procesos ING - UC3M€¦ · Tema 3: Control EstadTema 3: Control Estadístico de Procesos (SPC)stico de Procesos (SPC)--IN ENGLISHIN ENGLISH ... (99.7%) willbe in theinterval:

Grado en Ingeniería

Asignatura: Estadística

Tema 3: Control Estadístico de Procesos (SPC)- IN ENGLISH

Grado en IngenierGrado en Ingenieríía a

Asignatura: EstadAsignatura: Estadíísticastica

Tema 3: Control EstadTema 3: Control Estadíístico de Procesos (SPC)stico de Procesos (SPC)-- IN ENGLISHIN ENGLISH

Grado en Ingeniería. Estadística. Tema 3

Grado en Ingeniería. Estadística. Tema 3Número de transparencia: 2

Statistical Process Control (SPC)StatisticalStatistical ProcessProcess Control (SPC)Control (SPC)

IntroductionThe variability of a production process

Assignable causesNon-assignable causes

Control by variablesThe capability of a process. Capability indexControl by variables graphsInitial study of a process. Determining the capacityMonitorization process

Control by attributes


SPC. IntroductionSPC. SPC. IntroductionIntroduction

ProductionprocessRaw material

Finished product


Does it have sufficient quality? Is it adequate?Does it have sufficientquality in order to be

competitive in the market?


ProductionprocessRaw material

Finished product


OBJECTIVE

To test that the process is operating under optimum conditions

To obtain information that allows the process to be improved

HOW?

Measurement during the production process

Specified measurement intervals

Graphical representation of the evolution of a quality characteristic




Suppose X is the variable of interest (for example: length).Periodic samples of X are takenIn the control graphs the evolution of a statistic of X is plotted. (For example, the sample mean)

1x 2x 3x

Upper control limit

Lower control limit

If any point is outside the control limits, or if non-random behaviour is observed, this isevidence that the process is not operating under optimum conditions.


Statistical Process control. TYPES:

By variables (ranges): we analyze a measureable characteristic

of the production (for example, the diameter of an axis)

By attributes (defective / non-comforming fraction): we analyze

a qualitative characteristic of the production (for example: accept /

defect)






Control by variablesThe capacity of a process. Capability indexControl by variables graphsInitial study of a process. Determining the capacityMonitorization process



SPC. Variability of a processSPC. SPC. VariabilityVariability ofof a a processprocess

PROCESS CHARACTERISTICS:

VARIABILITY: two outputs never the same.

REPEATABILITY: the capability of production with a “stable” level ofquality

processinputs outputs



VARIABILITY: Even though an exhaustive relationship of the causes of the

variablity cannot be given, it is interesting to classify as either causes assignable

to specific factors (and therefore controllable) and unassignable causes.

ASSIGNABLE CAUSES �

Produced by important factors that can be controlled in order to maintain

them at acceptable levels. The appear sporadically.

They produce unstable, significant and UNPREDICTABLE variability

NON-ASSIGNABLES CAUSES �

Produced by secondary factors whose effect is not considered significant.

They are a permanent and inherent part of the production process.

They produce stable, non-significant, accepted and PREDICTABLE

variablity


PROCESS IN CONTROL:PROCESS IN CONTROL:PROCESS IN CONTROL:PROCESS IN CONTROL:We say a process is IN CONTROL when only

non-assignable causes are present

NON- ASIGNABLES CAUSES � stable, non-significant, PREDICTABLE variability


The control graphs allow thepresence of assignable

causes in the process to be detected


SPC. Capacity of a processSPC. SPC. CapacityCapacity ofof a a processprocess

PREMISE:

The characteristic of production that we control (for example: thediameter of an axis, the packaged weight …) follows a normal distribution WHEN the process is under control.

X � N (µ, σ)

Almost all of the production (99.7%) will be in the interval:

(µ − 3σ ; µ + 3σ)

We call CAPACITY over the previous interval

CAPACITY = 6σ


SPC. Capacity of a processSPC. SPC. CapacityCapacity ofof a a processprocess

PROBLEMNot all of the products that we produce can be valid.

We only accept those that satisfy given specifications.

ToleranceToleranceToleranceTolerance intervalintervalintervalinterval:::: set of values of X that are considered to be

admissable or acceptable.

PRODUCT TOLERANCE INTERVAL (LT2 – LT1)

LT1 LT2

This is defined by the client, or on occasions by the designer of the product or the personresponsable for the process (it often represents an objective value, θ, and an acceptable error, L):

θ ± L


The relation between the CAPACITY of a process and theTOLERANCE INTERVAL of a product gives us information aboutthe defects produced in our process.

Mean,Std. dev.1,5

Normal Distribution

x

density

-24 -14 -4 6 16 26

0

0,02

0,04

0,06

0,08

µ − 3σ µ + 3σ

SPC. Capacity - Capability indexSPC. SPC. CapacityCapacity -- CapabilityCapability indexindex


Mean,Std. dev.1,5

Normal Distribution

x

density

-24 -14 -4 6 16 26

0

0,02

0,04

0,06

0,08

µ − 3σ µ + 3σ

LT1 LT2




Mean,Std. dev.1,5

Normal Distribution

x

density

-24 -14 -4 6 16 26

0

0,02

0,04

0,06

0,08

µ − 3σ µ + 3σ

LT1 LT2 DEFECTS!!




Mean,Std. dev.1,5

Normal Distribution

x

density

-24 -14 -4 6 16 26

0

0,02

0,04

0,06

0,08

µ − 3σ µ + 3σ

LT2LT1




Mean,Std. dev.1,5

Normal Distribution

x

density

-24 -14 -4 6 16 26

0

0,02

0,04

0,06

0,08

µ − 3σ µ + 3σ

LT2LT1 NO DEFECTS!!




Mean,Std. dev.1,5

Normal Distribution

x

density

-24 -14 -4 6 16 26

0

0,02

0,04

0,06

0,08

µ − 3σ µ + 3σLT1 LT2




Mean,Std. dev.1,5

Normal Distribution

x

density

-24 -14 -4 6 16 26

0

0,02

0,04

0,06

0,08

µ − 3σ µ + 3σLT1 LT2

FEWDEFECTS




The CAPABILITY INDEX is the relation between the CAPACITYof the process and the TOLERANCE INTERVAL of the product:

CI =LT2 – LT16 σ

CI < 1 Defects!! Inspection of all the units

Mean,Std. dev.1,5

Normal Distribution

x

density

-24 -14 -4 6 16 26

0

0,02

0,04

0,06

0,08

CI > 1 Few defects. Less rigorous inspection

Mean,Std. dev.1,5

Normal Distribution

xdensity

-24 -14 -4 6 16 26

0

0,02

0,04

0,06

0,08




Example:We manufacture metallic pieces. The weight of the pieces is a qualityvariable.

The person responsible for the process establishes a tolerance interval of(0.76 ± 0.02) kg.We suppose that when the process is under control the weight of the pieceshas a normal distribution with mean 0.76 and standard deviation 0.028

We require:1. The probability that a piece will be defective

2. The capability index

3. If the machine’s operation changed, and this increased the process mean by 0.02kg, what would the proportion of defective products be?

4. If the machine’s operation changed, and this increased the standarddeviation by 0.01kg (without a change in the mean), what would theproportion of defective products be?


Control charts for rangesControl Control chartscharts forfor rangesranges

The presence of assignablecauses produces a change eitherin the mean or the standarddeviation, or in both. These implyan increase in the proportion ofdefective articles.

The control charts for rangesallow the process to be monitoredin order to detect the presence ofassignable causes

Mean,Std. dev.

17,2

20,2

Normal Distribution

x

density

0 5 10 15 20 25 30

0

0,04

0,08

0,12

0,16

0,2

Mean,Std. dev.

17,2

17,5

Normal Distribution

x

density

-8 2 12 22 32 42

0

0,04

0,08

0,12

0,16

0,2

Mean,Std. dev.

17,2

20,5

Normal Distribution

x

density

-5 5 15 25 35 45

0

0,04

0,08

0,12

0,16

0,2


Control charts for rangesControl Control chartscharts forfor rangesranges

In order to monitor the process, we take samples periodically andplot the evolution of the samplemean and the sample range (orthe standard deviation)

If the mean or the range (or thestandard deviation) exceeds theestablished control limits on thechart, or a non-random behaviouris seen, ….

then we have statistical evidencethat the mean or the variance ofthe process has changed.

That is:the process is not in control

X-bar Chart for Col_1

Subgroup

X-bar

CTR = 74,00

UCL = 74,01

LCL = 73,99

0 5 10 15 20 25

73,98

73,99

74

74,01

74,02

Range Chart for Col_2

0 5 10 15 20 25

Subgroup

0

0,01

0,02

0,03

0,04

0,05

Range

CTR = 0,02

UCL = 0,05

LCL = 0,00


Control charts for ranges. - Justification.Control Control chartscharts forfor rangesranges. . -- JustificationJustification..

Graphical justification of the means

Thus if the process is in control, the sample mean is in the interval with a probability of 99.7% ,

Therefore, we establish control chart limits for the mean:

−=

=

+=

nLCL

CL

nUCL

σµ

µ

σµ

3

3

),( σµNX → controlunder ),(n

NXσµ → Controlunder

-5 -3 -1 1 3 5

0

0,1

0,2

0,3

0,4

n

σµ 3+n

σµ 3−

0.997

X

n

σµ 3±


SPC. Initial study: Determining the capabilitySPC. SPC. InitialInitial studystudy: : DeterminingDetermining thethe capabilitycapability

IMPORTANT:We will determine the capacity of a process when the PROCESS

is IN CONTROL (i.e, there is only variability due to non-assignablecauses),PROCEDURE:1. Take data (samples) from the process:

2. The samples are plotted on the control charts.3. We check that all of the samples come from a situation that isin control (and we reject those that are not). 4. We calculate the capability using the σ of the samples“in control”

CAPABILITY = 6 σ

kknkkk

n

n

sxxxx

sxxxx

sxxxx

or r

or r

or r

k21

22222212

11112111

,),...,,(

....

,),...,,(

,),...,,(

,,,

,,,

,,,

→

→

→



We check that all the samples come from a situation “in control” (wereject those that are not)

1st:

2nd:

nx

σ)3+

nx

σ)3−

2d

R=σ)

k

xx

i∑=k

rR

i∑=

x

ir i sample value,minimum- valuemaximum

RD4

R

RD3

Chart of means Chart of ranges



3rd:Means Ranges

nx

σ)3+

nx

σ)3−

x

RD4

R

RD3



3rd:Means Ranges

NOT IN CONTROL

nx

σ)3+

nx

σ)3−

x

RD4

R

RD3



3rd:Means Ranges

NOT IN CONTROL

We eliminate the samples that are not in control, and we calculate thecapacity as: CAPABILITY = 6

4th:

nx

σ)3+

nx

σ)3−

x

RD4

R

RD3

σ)


We eliminate the samples that are not in control, and we calculate thecapacity as: CAPABILITY = 6


3ºMeans Standard deviations

NOT IN CONTROL

ALTERNATIVE

nx

σ)3+

nx

σ)3−

x

2c

s=σ)

σ)

SB4

S

SB3

4th:


SPC. Process monitoringSPC. SPC. ProcessProcess monitoringmonitoring

Once the capability of the process has been determined we can define the control charts that enable us to monitor the process.

MONITORING

Take samples at regular intervals.

Look for abnormal behaviour.

X-bar Chart for PESO LINEA

Subgroup

X-bar

CTR = 99,98

UCL = 100,46

LCL = 99,50

0 10 20 30 40

99,4

99,6

99,8

100

100,2

100,4

100,6

Range Chart for PESO LINEA

0 10 20 30 40

Subgroup

0

0,3

0,6

0,9

1,2

1,5

1,8

Range

CTR = 0,84

UCL = 1,77

LCL = 0,00

Not only a point outside the control lines gives us statistical evidence ofNOT IN CONTROL,-also non-random like behaviour ofthe points



MONITORING:

We divide the control chart into three zones of equal size

A(3σ); B(2σ); C(σ)

σ

σ

2σ

2σ

3σ

3σ A

A

B

B

C

C



We define the following types of behaviour abnormal:

Points outside zone A.

σ

σ

2σ

2σ

3σ

3σ A

A

B

B

C

C




Three consecutive points either inside or outside zone A.

σ

σ

2σ

2σ

3σ

3σ A

A

B

B

C

C




Four out of five consecutive points either in or outsidezone B.

σ

σ

2σ

2σ

3σ

3σ A

A

B

B

C

C




“Run”: Eight consecutive points on the same side of thecentral line

σ

σ

2σ

2σ

3σ

3σ A

A

B

B

C

C




Tendency: continuous movement towards the upper orlower zone.

σ

σ

2σ

2σ

3σ

3σ A

A

B

B

C

C




Change of level

σ

σ

2σ

2σ

3σ

3σ A

A

B

B

C

C




Systematic variability.

σ

σ

2σ

2σ

3σ

3σ A

A

B

B

C

C




Cycles.

σ

σ

2σ

2σ

3σ

3σ A

A

B

B

C

C




Over-stability.

σ

σ

2σ

2σ

3σ

3σ A

A

B

B

C

C


Example: Control chartExampleExample: Control : Control chartchart

We want to control a packaging process of a cosmetic product fromthe packaged weight.

1. Initial study of the process: the determination of the process capability. 20 samples are taken of size 5. Shortly, we will present the resultsobtained using Statgraphics

2. Process monitoring: It is decided to monitor the process in order todetect the presence of assignable causes. Samples of size 5 are takenevery hour and the mean and the range are plotted on a control chart. Shortly, we will present the results obtained using Statgraphics.


Example: Control chart – Initial studyExampleExample: Control : Control chartchart –– InitialInitial studystudyInitial study of the process: the determination of the process capability. 20 samples are taken of size 5.

Statgraphics. Menu: Special / Quality control / Variable Control Chart / X-bar and R …(Size = 5)Right button: Select “Initial study”.


Example: Control chart – Initial studyExampleExample: Control : Control chartchart –– InitialInitial studystudyInitial study of the process: the determination of the process capability. 20 samples are taken ofsize 5.

1. We plot the 20 samples on the control chart.

2. We eliminate all the samples that are outside the control lines

3. Using the remaining samples we estimate the parameters of the normal distribution.

X-bar Chart for PESO

0 4 8 12 16 20

Subgroup

99

99,3

99,6

99,9

100,2

100,5

100,8

X-bar

CTR = 100,01

UCL = 100,51

LCL = 99,50

Range Chart for PESO

0 4 8 12 16 20

Subgroup

0

0,4

0,8

1,2

1,6

2

2,4

Range

CTR = 0,88

UCL = 1,85

LCL = 0,00

X-bar Chart for PESO

Subgroup

X-bar

CTR = 99,99

UCL = 100,43

LCL = 99,55

0 4 8 12 16 20

99

99,3

99,6

99,9

100,2

100,5

100,8

Range Chart for PESO

Subgroup

Range

CTR = 0,76

UCL = 1,61

LCL = 0,00

0 4 8 12 16 20

0

0,4

0,8

1,2

1,6

2

2,4

Estimates---------Process mean = 99,9908Process sigma = 0,326455


Example: Control chart - Process monitoringExampleExample: Control : Control chartchart -- ProcessProcess monitoringmonitoringInitial study of the process: the determination of the process capability. 20 samples are taken ofsize 5. Statgraphics.

Menu: Special / Quality control / Variable Control Chart / X-bar and R …Right button: Select “Control to Standard”.“Specific Parameters” – we enter those estimated in the initial study

---------Process mean = 99,9908Process sigma = 0,326455


Example: Control chart - Process monitoringExampleExample: Control : Control chartchart -- ProcessProcess monitoringmonitoring

Process monitoring: It is decided to monitor the process in order to detect the presence ofassignable causes. Samples of size 5 are taken every hour and the mean and the range are plotted on a control chart. (Load new data)

For sample 39, even though this is not outside the 3-sigma limits, it is already showing an alarm(due to its range).

X-bar Chart for PESO LINEA

Subgroup

X-bar

CTR = 99,99

UCL = 100,43

LCL = 99,55

0 10 20 30 40

99,5

99,7

99,9

100,1

100,3

100,5

Range Chart for PESO LINEA

Subgroup

Range

CTR = 0,76

UCL = 1,61

LCL = 0,00

0 10 20 30 40

0

0,3

0,6

0,9

1,2

1,5

1,8


Control charts for attributesControl Control chartscharts forfor attributesattributes

Acceptable/Defective

Record of simple data (less cost!)

Easier to count defects than calculate the mean andstandard deviation

Samples sizes are normally greater

Only necessary to plot one control chart

np Chart for TAPONADO

0 4 8 12 16 20

Subgroup

0

2

4

6

8

np

CTR = 1,75

UCL = 5,54

LCL = 0,00


Control charts for attributesControl Control chartscharts forfor attributesattributes

Chart for the NUMBER of defects in the sample : NP-chart

Chart for the PROPORTION of defects in the sample : P-chart

−+=

=

−+=

n

pppLCL

pCL

n

pppUCL

)(

)(

13

13

−−=

=−+=

)(

)(

pnpnpLCL

npCL

pnpnpUCL

13

13

np Chart for Col_5

0 5 10 15 20 25 30

Subgroup

0

4

8

12

16

20

24

np

CTR = 11,57

UCL = 20,51

LCL = 2,62

p Chart for Col_6

0 5 10 15 20 25 30

Subgroup

0

0,1

0,2

0,3

0,4

0,5

p

CTR = 0,23

UCL = 0,41

LCL = 0,05


SPC for attributes. Determining the process capabilitySPC SPC forfor attributesattributes. . DeterminingDetermining thethe processprocess capabilitycapability

If we call the proportion of defects of a process, p, thenthe CAPABILITY OF THE PROCESS is defined as

CAPABILITY = 1 – p

To determine the capacity means to determine theproportion of defects, p, in controlled conditions

Premises:

1. The proportion of defective components is stable in thelong-term

2. The fact that each component is defective is independentof whether the previous or next component is defective


SPC for attributes. Determining the process capabilitySPC SPC forfor attributesattributes. . DeterminingDetermining thethe processprocess capabilitycapability

Process:

1. We take samples and count the number of defects thatoccur in each

2. We estimate the proportion of defects

3. We check that all of the samples come from a controlledsituation (and we reject/do not use those that do not). We plot the number of defects in a control chart.


Example - Determining the process capabilityExampleExample -- DeterminingDetermining thethe processprocess capabilitycapability

We want to control the “capping” process of bottles using the number of defects detected. It isrequired to calculate the process capability of the capping with samples of size 20 using thevariable “taponado”

Statgraphics. Menu: Special/Quality control / Attributes Control Chart / np-chartSubgroup sizes = 20Right button : Select “Initial study”.



1.1.1.1. WeWeWeWe estimateestimateestimateestimate pppp

08.0=p)

np - Initial Study for TAPONADO

Number of subgroups = 20Subgroup size = 20,00 subgroups excluded

np Chart--------UCL: +3,0 sigma = 5,54103Centerline = 1,75LCL: -3,0 sigma = 0,0

1 beyond limits

Estimates---------Mean np = 1,75Sigma = 1,26368

( np / p = 0.0875 )



2. 2. 2. 2. WeWeWeWe calculatecalculatecalculatecalculate thethethethe control control control control chartchartchartchart andandandand wewewewe plotplotplotplot thethethethe samplessamplessamplessamples....WeWeWeWe eliminateeliminateeliminateeliminate thethethethe samplessamplessamplessamples thatthatthatthat are are are are outsideoutsideoutsideoutside thethethethe control control control control limitslimitslimitslimits

=−−=

===−+=

0)1(3

75.1

54.5)1(3

ppnpnLCS

pnLC

ppnpnLCS

)))

)

)))


Subgroup

np

CTR = 1,75

UCL = 5,54

LCL = 0,00

0 4 8 12 16 20

0

2

4

6

8

We must eliminate sample 11 because it is outside thecontrol limits



3. 3. 3. 3. WeWeWeWe eliminateeliminateeliminateeliminate samplesamplesamplesample 11 11 11 11 ((((StatgraphicsStatgraphicsStatgraphicsStatgraphics: : : : ExcludeExcludeExcludeExclude / / / / automaticautomaticautomaticautomatic))))

TheTheTheThe restrestrestrest are are are are insideinsideinsideinside thethethethe limitslimitslimitslimits


Subgroup

np

CTR = 1,42

UCL = 4,87

LCL = 0,00

0 4 8 12 16 20

0

2

4

6

8

07110.=p) Processcapability = 0.93

Estimates---------Mean np = 1,42105Sigma = 1,14895


SPC for attributes. Process monitoringSPC SPC forfor attributesattributes. . ProcessProcess monitoringmonitoring

Samples are taken periodically

The number of defective components are plotted on a control chart

Any point outside the comtrol limits, or any non-randombehaviour is an indication of statistical evidence that theprocess is not in control. (The same as we did forcontrol of variables)


Example. Process monitoringExampleExample. . ProcessProcess monitoringmonitoring

It is required to control a bottle capping process the number of defects detected. For this, wetake a sample of size 20 every hour. Now we show how to obtain the control chart for the numberof defects

Which samples do not demonstrate statistical control?Statgraphics.

Menu: Special/Quality control / Attributes Control Chart / np-chartData: TAPONADO LINEASubgroup size = 20Right button: Select “control to Standard”. “Specify parameter” , np=20*0.07=1.4


Example. Process monitoringExampleExample. . ProcessProcess monitoringmonitoring

np Chart for TAPONADO LINEA

Subgroup

np

CTR = 1,42

UCL = 4,87

LCL = 0,00

0 4 8 12 16 20

0

2

4

6

8

10

12np Chart--------UCL: +3,0 sigma = 4,82316Centerline = 1,4LCL: -3,0 sigma = 0,0

It is required to control a bottle capping process the number of defects detected. For this, wetake a sample of size 20 every hour. Now we show how to obtain the control chart for the numberof defects

Which samples do not demonstrate statistical control?Statgraphics.

Menu: Special/Quality control / Attributes Control Chart / np-chartData: TAPONADO LINEASubgroup size = 20Right button: Select “control to Standard”. “Specify parameter” , np=20*0.07=1.4

Download - 3.Control de Procesos ING - UC3M€¦ · Tema 3: Control EstadTema 3: Control Estadístico de Procesos (SPC)stico de Procesos (SPC)--IN ENGLISHIN ENGLISH ... (99.7%) willbe in theinterval:

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