2.008 -Spring 2004 1

2.008

Process Control

2.008 -Spring 2004 2

Outline

1. Optimization

2. Statistical Process Control

3. In-Process Control

2.008 -Spring 2004 3

What is quality?

2.008 -Spring 2004 4

Variation: Common and Special Causes

Pieces vary from each other:

But they form a pattern that, if stable, is called a distribution:

Size Size Size Size

Size Size Size

2.008 -Spring 2004 5

Common and Special Causes (cont’d)

Distributions can differ in…

…or any combination of these

Size Size Size

Location Spread Shape

2.008 -Spring 2004 6

Common and Special Causes (cont’d)

If only common causes of variation are present, the output of a process forms a distribution that is stabl

e over time and is predictable:

Size

Prediction

Time

2.008 -Spring 2004 7

Common and Special Causes (cont’d)

IF special causes of variation are present, the process output is not stable over time and is not predictable:

Size

Prediction

Time

2.008 -Spring 2004 8

Variation: Run Chart

Specified diameter = 0.490” +/-0.001”

Dia

met

er o

f p

art

Run numberProcess change

0.4915

0.4910

0.4905

0.4900

0.4895

0.4890

0.4885

0.48800 10 20 30 40 50

2.008 -Spring 2004 9

Process Control

In Control(Special causes eliminated)

Out of Control(Special causes present)

Time

Size

2.008 -Spring 2004 10

Statistical Process Control

1. Detect disturbances (special causes)

2. Take corrective actions

2.008 -Spring 2004 11

Central Limit Theorem

• A large number of independent events have a continuous probability density function that is normal in shape.

• Averaging more samples increases the precision of the estimate of the average.

2.008 -Spring 2004 12

Shewhart Control ChartA

vera

ge

(o

f 10

sam

ple

s) D

iam

eter

Run number

Upper control limit

Lower control limit

1.010

0.990

0.995

1.000

1.005

0 10 20 30 40 50 60 70 80 90 100

2.008 -Spring 2004 13

Sampling and Histogram Creation

Wheel of Fortune: Equal probability of outcome 1-

10, P=0.1 Taking 100 random samples, the

resulting histogram would look like this

Taking ∞ random samples, the resulting histogram would look li

ke this

1 2 3 4 5 6 7 8 9 10

Fre

quen

cy

20181614121086420

Outcome

1 2 3 4 5 6 7 8 9 10

Fre

quen

cyOutcome

2.008 -Spring 2004 14

Sampling and Histogram Creation (cont’d)

Wheel of Fortune: Equal probability of outcome 1-

10, P=0.1

Take 10 random samples, calculate their

average, and repeat 100 times, the resulting

histogram would resemble

Take 10 random samples, calculate their average, and repeat ∞ times, the resulting histogram would approach the continuous distributi

on shown

1 2 3 4 5 6 7 8 9 10

Fre

quen

cy

20

18

16

14

12

10

8

6

4

2

0

Outcome

1 2 3 4 5 6 7 8 9 10

2.008 -Spring 2004 15

Uniform DistributionsUnderlying Distribution

Pro

bab

ilit

y d

istr

ibu

tio

n 1.5

1

0.5

0

0 0.5 1

1.5

1

0.5

0

0 0.5 1

Pro

bab

ilit

y d

istr

ibu

tio

n

Distribution resulting from averaging of 2 random values

Distribution resulting from averaging of 3 random values

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Distribution resulting from averaging of 10 random values

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2.008 -Spring 2004 16

Normal Distribution

Distribution resulting from averaging of 3 random values

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Distribution resulting from averaging of 10 random values

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Underlying Distribution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Distribution resulting from averaging of 2 random values

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2.008 -Spring 2004 17

Precision

Not precise Precise

Not accurate

Accurate

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Shewhart Control Chart

• Upper Control Limit (UCL), Lower Control Limit (LCL)

• Subgroup size ( 5 < n < 20 )

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Shewhart Control Chart (cont’d)

Control chart based on 100 samples

Ave

rag

e (

of

10 s

amp

les)

Dia

met

er

Run number

Upper control limit

Lower control limit

1.010

0.990

0.995

1.000

1.005

0 10 20 30 40 50 60 70 80 90 100

Step disturbance

2.008 -Spring 2004 20

Shewhart Control Chart (cont’d)A

vera

ge

(o

f 10

sam

ple

s) D

iam

eter

Run number

Upper control limit

Lower control limit

1.010

0.990

0.995

1.000

1.005

0 10 20 30 40 50 60 70 80 90 100

Step disturbance

Control chart based on average of 10 samples, note the step change that occurs at run 51

2.008 -Spring 2004 21

Control Charts

1. Collection:． Gather data and plot on chart.

2. Control． Calculate control limits from process data, using simple formulae.

． Identify special causes of variation; take local actions to correct.

3. Capability． Quantify common cause variation; take action on the system.

These three phases are repeated for continuing process improvement.

Upper control limit Process average Lower control limit

0 10 20 30 40 50 60 70 80 90 100Run number

2.008 -Spring 2004 22

Setting the Limits

Idea: Points outside the limits will signal that something is wro

ng-an assignable cause. We want limits set so that assignabl

e causes are highlighted, but few random causes are highligh

ted accidentally.

Convention for Control Charts:• Upper control limit (UCL) = x + 3σsg

• Lower control limit (LCL) = x-3σsg

(Where σsg represents the standard deviation of a subgroup of samples)

2.008 -Spring 2004 23

Setting the Limits (cont’d)

Convention for Control Charts (cont’d):

As n increases, the UCL and LCL move closer to the center line, making the control chart more sensitive to shifts in the mean.

processsggroup sub

n/processsubgroup

nx /3UCL process

nx /3LCL process

2.008 -Spring 2004 24

Benefits of Control Charts

Properly used, control charts can:

• Be used by operators for ongoing control of a process

• Help the process perform consistently, predictably, for quality and cost

• Allow the process to achieve:

– Higher quality

– Lower unit cost

– Higher effective capacity

• Provide a common language for discussing process performance

• Distinguish special from common causes of variation; as a guide to local or management action

2.008 -Spring 2004 25

Process Capability

Lower specification limit

Upper specification limit

In Control and Capable(Variation from common cause reduced)

In Control but not Capable(Variation from common cause excessive)

Size

Time

2.008 -Spring 2004 26

Process Capability Index

1. Cp = Range/6

2. Cpk

2.008 -Spring 2004 27

Process Capability

• Take an example with:

Mean = .738

Standard deviation, σ = .0725

UCL = 0.900

LCL = 0.500

• Normalizing the specifications:

23.20725.0

738.0900.0UCLUCL

x

Z

28.30725.0

738.0500.0LCLLCL

x

Z

basis) absolutean (on 23.2MIN Z

738.0

x

0.500

LCL

0.900

UCL

28.3LCL

Z

23.2UCLZ

2.008 -Spring 2004 28

Process Capability (cont’d)

• Using the tables of areas under the Normal Curve, the proportions out of specification would be:

PUCL= 0.0129

PLCL = 0.0005

Ptotal = 0.0134

• The Capability Index would be:CPK = Zmin/3 = 2.23 / 3 = 0.74

2.008 -Spring 2004 29

Process Capability (cont’d)

• If this process could be adjusted toward the center of the specification, the proportion of parts falling beyond either or both specification limits might be reduced, even with no change in Standard deviation.

• For example, if we confirmed with control charts a new mean = 0.700, then:

700.0

x

0.500

LCL

0.900

UCL

76.2LCL

Z

76.2UCLZ

76.20725.0

700.0900.0UCL newUCL

x

Z

76.20725.0

700.0500.0LCL newLCL

x

Z

76.2MIN Z

2.008 -Spring 2004 30

Process Capability (cont’d)

• The proportions out of specification would be:PUCL= 0.0029

PLCL = 0.0029

Ptotal = 0.0058

• The Capability Index would be:CPK= Zmin/3 = 2.76 / 3 = 0.92

2.008 -Spring 2004 31

Improving Process Capability

• To improve the chronic performance of the process, concentrate on the common causes that affect all periods. These will usually require management action on the system to correct.

• Chart and analyze the revised process:– Confirm the effectiveness of the system by continued

monitoring of the Control Chart