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Process Capability and Statistical Process Control

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Chapter 3

Lecture Outline

• Basics of Statistical Process Control• Control Charts• Control Charts for Attributes• Control Charts for Variables• Control Chart Patterns• SPC with Excel and OM Tools• Process Capability

Copyright 2011 John Wiley & Sons, Inc. 3-2

Binomial Experiment

• A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:

• The experiment consists of n repeated trials.

• Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.

• The probability of success, denoted by P, is the same on every trial.

• The trials are independent

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The binomial distribution

• The binomial distribution gives the discrete probability distribution of obtaining exactly n successes out of N trials. The binomial distribution is therefore given by


A soccer (or hockey) player may attempt to score multiple goals from multiple shots. If he or she has a shooting success probability of 0.25 and takes 4 shots in a match, then the number of goals he or she scores can be modeled as B(4, 0.25); note that just as p represents the probability of any given shot becoming a goal, 1 − p represents—depending on how the model is set up and which sport is chosen—the shot missing the net or the goalkeeper making the save. The probability of the player scoring 0, 1, 2, 3, or 4 goals on 4 shots are:

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Statistical Process Control (SPC)

• Statistical Process Control• monitoring production process to

detect and prevent poor quality• is a tool for identifying problems in

order to make improvements

• Sample• subset of items produced to use

for inspection

• Control Charts• process is within statistical control

limits

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Process Variability

• Random• inherent in a process

• depends on equipment and machinery, engineering, operator, and system of measurement

• natural occurrences

• Non-Random• special causes• identifiable and

correctable• include equipment out of

adjustment, defective materials, changes in parts or materials, broken machinery or equipment, operator fatigue or poor work methods, or errors due to lack of training


SPC in Quality Management

• SPC uses• Is the process in control?

• Identify problems in order to make improvements

• Contribute to the TQM goal of continuous improvement

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Quality Measures:Attributes and Variables

• Attribute• A characteristic which is evaluated with a discrete

response• good/bad; yes/no; correct/incorrect• Color, cleanliness, surface texture, taste or smell

• Variable measure• A characteristic that is continuous and can be

measured• Weight, length, voltage, volume

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SPC Applied to Services

• Nature of defects is different in services

• Service defect is a failure to meet customer requirements

• Monitor time and customer satisfaction



• Hospitals• timeliness & quickness of care, staff responses to requests,

accuracy of lab tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance & checkouts

• Grocery stores• waiting time to check out, frequency of out-of-stock items, quality

of food items, cleanliness, customer complaints, checkout register errors

• Airlines• flight delays, lost luggage & luggage handling, waiting time at

ticket counters & check-in, agent & flight attendant courtesy, accurate flight information, cabin cleanliness & maintenance



• Fast-food restaurants• waiting time for service, customer complaints, cleanliness, food

quality, order accuracy, employee courtesy

• Catalogue-order companies• order accuracy, operator knowledge & courtesy, packaging,

delivery time, phone order waiting time

• Insurance companies• billing accuracy, timeliness of claims processing, agent

availability & response time


Where to Use Control Charts

• Process • Has a tendency to go out of control

• Is particularly harmful and costly if it goes out of control

• When to use control charts:• At beginning of process because of waste to begin

production process with bad supplies• Before a costly or irreversible point, after which product is

difficult to rework or correct• Before and after assembly or painting operations that

might cover defects• Before the outgoing final product or service is delivered

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Control Charts

• A graph that monitors process quality

• Control limits• upper and lower bands of a control chart

• Attributes chart• p-chart• c-chart

• Variables chart• mean (x bar – chart)• range (R-chart)

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Process Control Chart


1 2 3 4 5 6 7 8 9 10Sample number

Uppercontrol

limit

Processaverage

Lowercontrol

limit

Out of control

Normal Distribution

• In probability theory, the normal (or Gaussian) distribution, is a continuous probability distribution that is often used as a first approximation to describe real-valued random variables that tend to cluster around a single mean value. The graph of the associated probability density function is “bell”-shaped, and is known as the Gaussian function or bell curve:

where parameter μ is the mean value and σ 2 is the variance.

The distribution with μ = 0 and σ 2 = 1 is called the standard normal.

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Normal Distribution

• Probabilities for Z= 2.00 and Z = 3.00


=0 1 2 3-1-2-3

95%

99.74%

A Process Is in Control If …


1. … no sample points outside limits

2. … most points near process average

3. … about equal number of points above and below centerline

4. … points appear randomly distributed

Control Charts for Attributes

• p-chart• uses portion defective in a sample

• c-chart• uses number of defects (non-conformities) in a

sample


p-Chart


UCL = p + zp

LCL = p - zp

z =number of standard deviations from process average

p =sample proportion defective; estimates process mean

p = standard deviation of sample proportionp =

p(1 - p)

nn

Construction of p-Chart


20 samples of 100 pairs of jeans

NUMBER OF PROPORTIONSAMPLE # DEFECTIVES DEFECTIVE

1 6 .06

2 0 .00

3 4 .04

: : :

: : :

20 18 .18

200



UCL = p + z = 0.10 + 3p(1 - p)

n

0.10(1 - 0.10)

100

UCL = 0.190

LCL = 0.010

LCL = p - z = 0.10 - 3p(1 - p)

n

0.10(1 - 0.10)

100

= 200 / 20(100) = 0.10total defectives

total sample observationsp =



0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Pro

por

tion

defe

ctiv

e

Sample number2 4 6 8 10 12 14 16 18 20

UCL = 0.190

LCL = 0.010

p = 0.10

p-Chart in Excel


Click on “Insert” then “Charts”

to construct control chart

I4 + 3*SQRT(I4*(1-I4)/100)

I4 - 3*SQRT(I4*(1-I4)/100)

Column values copiedfrom I5 and I6

c-Chart


UCL = c + zc

LCL = c - zc

where

c = number of defects per sample

c = c

c-Chart


Number of defects in 15 sample rooms

1 122 83 16

: :: :15 15 190

SAMPLE

c = = 12.6719015

UCL = c + zc

= 12.67 + 3 12.67= 23.35

LCL = c - zc

= 12.67 - 3 12.67= 1.99

NUMBER OF

DEFECTS

c-Chart


3

6

9

12

15

18

21

24

Num

ber

of

defe

cts

Sample number

2 4 6 8 10 12 14 16

UCL = 23.35

LCL = 1.99

c = 12.67

Control Charts for Variables


Range chart ( R-Chart ) Plot sample range (variability)

Mean chart ( x -Chart ) Plot sample averages

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x-bar Chart: Known

UCL = x + z x LCL = x - z x

-

-

==

Where

= process standard deviation

x = standard deviation of sample means =/k = number of samples (subgroups)n = sample size (number of observations)

x1 + x2 + ... + xk

kX = =

- - -

n

Copyright 2011 John Wiley & Sons, Inc.

Observations(Slip-Ring Diameter, cm) n

Sample k 1 2 3 4 5 -

x-bar Chart Example: Known


x

We know σ = .08

x-bar Chart Example: Known


= 5.01 - 3(.08 / )10

= 4.93

_____10

50.09 = 5.01X =

=

LCL = x - z x = -UCL = x + z x

= -= 5.01 + 3(.08 / )10

= 5.09

x-bar Chart Example: Unknown


_UCL = x + A2R LCL = x - A2R

= =_

where

x = average of the sample means

R = average range value

=_

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Control Chart Factors

n A2 D3 D42 1.880 0.000 3.2673 1.023 0.000 2.5754 0.729 0.000 2.2825 0.577 0.000 2.1146 0.483 0.000 2.0047 0.419 0.076 1.9248 0.373 0.136 1.8649 0.337 0.184 1.81610 0.308 0.223 1.77711 0.285 0.256 1.74412 0.266 0.283 1.71713 0.249 0.307 1.69314 0.235 0.328 1.67215 0.223 0.347 1.65316 0.212 0.363 1.63717 0.203 0.378 1.62218 0.194 0.391 1.60919 0.187 0.404 1.59620 0.180 0.415 1.58521 0.173 0.425 1.57522 0.167 0.435 1.56523 0.162 0.443 1.55724 0.157 0.452 1.54825 0.153 0.459 1.541

Factors for R-chartSample

Size Factor for X-chart



OBSERVATIONS (SLIP- RING DIAMETER, CM)

SAMPLE k 1 2 3 4 5 x R

1 5.02 5.01 4.94 4.99 4.96 4.98 0.082 5.01 5.03 5.07 4.95 4.96 5.00 0.123 4.99 5.00 4.93 4.92 4.99 4.97 0.084 5.03 4.91 5.01 4.98 4.89 4.96 0.145 4.95 4.92 5.03 5.05 5.01 4.99 0.136 4.97 5.06 5.06 4.96 5.03 5.01 0.107 5.05 5.01 5.10 4.96 4.99 5.02 0.148 5.09 5.10 5.00 4.99 5.08 5.05 0.119 5.14 5.10 4.99 5.08 5.09 5.08 0.15

10 5.01 4.98 5.08 5.07 4.99 5.03 0.10

50.09 1.15Totals



∑ Rk

1.1510R = = = 0.115

_ ____ ____

_UCL = x + A2R

=

= 50.0910

_____x = = = 5.01 cmx

k___

= 5.01 + (0.58)(0.115) = 5.08

_LCL = x - A2R

== 5.01 - (0.58)(0.115) = 4.94

x- bar Chart

Example


UCL = 5.08

LCL = 4.94

Mea

n

Sample number

|1

|2

|3

|4

|5

|6

|7

|8

|9

|10

5.10 –

5.08 –

5.06 –

5.04 –

5.02 –

5.00 –

4.98 –

4.96 –

4.94 –

4.92 –

x = 5.01=

R- Chart


UCL = D4R LCL = D3R

R = Rk

WhereR = range of each samplek = number of samples (sub groups)

R-Chart Example


OBSERVATIONS (SLIP- RING DIAMETER, CM)

SAMPLE k 1 2 3 4 5 x R

1 5.02 5.01 4.94 4.99 4.96 4.98 0.082 5.01 5.03 5.07 4.95 4.96 5.00 0.123 4.99 5.00 4.93 4.92 4.99 4.97 0.084 5.03 4.91 5.01 4.98 4.89 4.96 0.145 4.95 4.92 5.03 5.05 5.01 4.99 0.136 4.97 5.06 5.06 4.96 5.03 5.01 0.107 5.05 5.01 5.10 4.96 4.99 5.02 0.148 5.09 5.10 5.00 4.99 5.08 5.05 0.119 5.14 5.10 4.99 5.08 5.09 5.08 0.15

10 5.01 4.98 5.08 5.07 4.99 5.03 0.10

50.09 1.15Totals

R-Chart Example


Retrieve chart factors D3 and D4

UCL = D4R = 2.11(0.115) = 0.243

LCL = D3R = 0(0.115) = 0_

_

R-Chart Example


UCL = 0.243

LCL = 0

Ra

nge

Sample number

R = 0.115

|1

|2

|3

|4

|5

|6

|7

|8

|9

|10

0.28 –

0.24 –

0.20 –

0.16 –

0.12 –

0.08 –

0.04 –

0 –

X-bar and R charts – Excel & OM Tools


Using x- bar and R-Charts Together

• Process average and process variability must be in control

• Samples can have very narrow ranges, but sample averages might be beyond control limits

• Or, sample averages may be in control, but ranges might be out of control

• An R-chart might show a distinct downward trend, suggesting some nonrandom cause is reducing variation


Control Chart Patterns

• Run• sequence of sample values that display same

characteristic

• Pattern test• determines if observations within limits of a control

chart display a nonrandom pattern



• To identify a pattern look for:• 8 consecutive points on one side of the center line• 8 consecutive points up or down• 14 points alternating up or down• 2 out of 3 consecutive points in zone A (on one side

of center line)• 4 out of 5 consecutive points in zone A or B (on one

side of center line)



Copyright 2011 John Wiley & Sons, Inc. 3-3-4545

UCL

LCL

Sample observationsconsistently above thecenter line

LCL

UCL

Sample observationsconsistently below thecenter line



LCL

UCL

Sample observationsconsistently increasing

UCL

LCL

Sample observationsconsistently decreasing

Zones for Pattern Tests

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UCL

LCL

Zone A

Zone B

Zone C

Zone C

Zone B

Zone A

Process average

3 sigma = x + A2R=

3 sigma = x - A2R=

2 sigma = x + (A2R)= 23

2 sigma = x - (A2R)= 23

1 sigma = x + (A2R)= 13

1 sigma = x - (A2R)= 13

x=

Sample number

|1

|2

|3

|4

|5

|6

|7

|8

|9

|10

|11

|12

|13

Copyright 2011 John Wiley & Sons, Inc.

Performing a Pattern Test


1 4.98 B — B

2 5.00 B U C

3 4.95 B D A

4 4.96 B D A

5 4.99 B U C

6 5.01 — U C

7 5.02 A U C

8 5.05 A U B

9 5.08 A U A

10 5.03 A D B

SAMPLE x ABOVE/BELOW UP/DOWN ZONE

Sample Size Determination

• Attribute charts require larger sample sizes• 50 to 100 parts in a sample

• Variable charts require smaller samples• 2 to 10 parts in a sample


Process Capability

• Compare natural variability to design variability • Natural variability

• What we measure with control charts• Process mean = 8.80 oz, Std dev. = 0.12 oz

• Tolerances• Design specifications reflecting product

requirements• Net weight = 9.0 oz 0.5 oz • Tolerances are 0.5 oz


Process Capability


Process Capability Ratio


Cp =tolerance range

process range

upper spec limit - lower spec limit

6=

Computing Cp


Process Capability Index


Cpk = minimum

x - lower specification limit

3

=

upper specification limit - x

3

=

,

Computing Cpk


Net weight specification = 9.0 oz 0.5 ozProcess mean = 8.80 ozProcess standard deviation = 0.12 oz

Cpk = minimum

= minimum , = 0.83

x - lower specification limit

3

=

upper specification limit - x

3

=

,

8.80 - 8.50

3(0.12)

9.50 - 8.80

3(0.12)

Process Capability With Excel


=(D6-D7)/(6*D8)

See formula bar

Process Capability With OM Tools



Copyright 2011 John Wiley & Sons, Inc.All rights reserved. Reproduction or translation of this work beyond that permitted in section 117 of the 1976 United States Copyright Act without express permission of the copyright owner is unlawful. Request for further information should be addressed to the Permission Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information herein.

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