Session statistical process control (spc)

1

Total Quality Management

Statistical Process Control (SPC)

Need

X bar and R charts

P chart

C chart

Applications

Variation

Variation is natural - it is inherent in the world around us.

No two products or service experiences are exactly the same.

With a fine enough gauge, all things can be seen to differ.

One of the roles of management is work with all employees to reduce variation as much as possible.

The Presence of Variation

8’

4’ 4’ 4’ 4’ Measuring

Device

Tape Measure 4’ 4’ 4’ 4’

Engineer Scale 4.01’ 4.01’ 4.01’ 4.00’

Caliper 4.009’ 3.987’ 4.012’ 4.004’

Elec. Microscope 4.00913’ 3.98672’ 4.01204’ 4.00395’

Types of Variation

Common Cause Variation: The variation that naturally occurs

and is expected in the system

-- normal

-- random

-- inherent

-- stable

Special Cause Variation: Variation which is abnormal -

indicating something out of the ordinary has happened.

-- nonrandom

-- unstable

-- assignable cause variation

Type of Variation Travel Time to Work Example

Measurement of Interest: Time to get to work.

Common Cause Variation Sources:

-- traffic lights

-- traffic patterns

-- weather

-- departure time

Special Cause Variation Sources:

-- accidents

-- road construction detours

-- petrol refills

Total Product or Process Variation

Total variation = Common Cause + Special Cause

To reduce Total Variation

First reduce or eliminate special cause variation

Reduce common cause variation

Identify the source and remove the causes

2

Measures performance of a process

Uses mathematics (i.e., statistics)

Involves collecting, organizing, &

interpreting data

Objective: provide statistical when

assignable causes of variation are

present

Used to – Control the process as products are produced

– Inspect samples of finished products

Statistical Quality

Control

Statistical

Quality Control

Process

Control

Acceptance

Sampling

Variables

Charts

Attributes

Charts Variables Attributes

Types of

Statistical Quality Control

Has or Has not/Good

or Bad/Pass or

Fail/Accept or Reject

Characteristics for

which you focus on

defects

Categorical or

discrete random

variables

Attributes Variables

Quality

Characteristics

Measured values;

e.g., weight, length,

volume,voltage, current etc.

May be in whole or in

fractional numbers

Continuous random variables

Statistical technique used to ensure

process is making product to standard

All process are subject to variability

– Natural causes: Random variations

– Assignable causes: Correctable problems

Machine wear, unskilled workers, poor

material

Objective: Identify assignable causes

Uses process control charts

Statistical Process

Control (SPC)

Comparing Distributions Production Output Example

Plant A Plant B

99 100 100 100 101

Units Produced

90 90 100 110 110

No Differences!???

1005

500

n

XX 100

5

500

n

XX

Production Output Distributions What is the Difference?

90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

Plant A

90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

Plant B

Fre

qu

ency

F

req

uen

cy

3

Measure of Variation (Sigma) S = Standard Deviation

Plant A

99 99-100 = -1 12 = 1 100 100-100 = 0 02 = 0 100 100-100 = 0 02 = 0 100 100-100 = 0 02 = 0 101 101-100 = 1 12 = 1

X )( XX

0

1

)( 2

n

XXS

2)( XX

2

707.4

2S

X )( XX 2)( XX

90 90 -100= -10 -102 =100 90 90 -100= -10 -102 =100 100 100 -100 = 0 02 = 0 110 110 -100 = 10 102 =100 110 110 -100 = 10 102 =100

0 400

104

400S

Plant B

The Concept of Stability

99.7%

+ 3S - 3S - 2S - 1S +1S +2S

95%

68% X X X X X X

XX

Plant A

100X

707.1001 SX

414.1012 SX

121.1023 SX

707.4

2S

293.991 SX

586.982 SX

879.973 SX

X

Under Normal Conditions:

68 percent of the time output will be between 99.293 and

100.707 units

95 percent of the time output will be between 98.586 and

101.414 units

99.7 percent of the time output will be between 97.879 units

and 102.121 units

Plant B

100X

1101 SX

1202 SX

1303 SX

104

400S

901 SX

802 SX

703 SX

X

Under Normal Conditions:

68 percent of the time output will be between 90 and 110 units

95 percent of the time output will be between 80 and 120 units

99.7 percent of the time output will be between 70 units and

130 units

Control Limits

Control Limits are the statistical boundaries of a process

which define the amount of variation that can be considered

as normal or inherent variation

3 sigma control limits are most common

+ 3S from the mean If the process is in control, a value outside the control limit will occur only 3 time in 1000 ( 1 - .997 = .003)

Process Control Limits

Lower Control Limit

Upper Control Limit

Special Cause Variation

Special Cause Variation

Average

Com

mon

Cau

se UCL= +3

LCL = - 3

X

X

4

Relationship Between

Population and Sampling

Distributions

Uniform

Normal

Beta

Distribution of sample means

x means sample of Mean

n

xx

Standard deviation of

the sample means

(mean)

x2 withinfall x all of 95.5%

x3 withinfall x all of 99.7%

x

3 x

2 x

1 σx x

1 x

2 x

3

Three population distributions

Sampling Distribution of

Means, and Process

Distribution

Sampling

distribution of the

means

Process

distribution of

the sample

)mean(

mx

X

As sample size

gets

large

enough,

sampling distribution

becomes almost

normal regardless of

population

distribution.

Central Limit Theorem

XX

Theoretical Basis

of Control Charts

X

Mean

Central Limit Theorem

xx

n

x

x

nX X

Standard deviation

X X

Theoretical Basis

of Control Charts

Process Control Limit Concepts

Control Limits Define the limits of stability

The ULC and LCL are calculated so that, if the process is stable, almost all of the process output will be located within the control limits.

3 sigma control limits

The most commonly used

UCL is 3 standard deviations above the average

LCL is 3 standard deviations below the average

If the process is stable, only about 3 out of 1000 process outputs will fall outside the control limits.

Process Control Limit Concepts (continued)

Measures inside control limits are assumed to come from a stable process - Measures outside the control limits are unexpected and considered the result of a special cause

The control limits are computed directly from the sample data selected from the process -- The limits and the average are not the choice of management or the operator - Formulas exist.

The control limits define the range of inherent variation for the process as it currently exists, not how we would like it to be

5

Show changes in data pattern

– e.g., trends

Make corrections before process is out of

control

Show causes of changes in data

– Assignable causes

Data outside control limits or trend in data

– Natural causes

Random variations around average

Control Chart Purposes

Control

Charts

R Chart

Variables

Charts

Attributes

Charts

X Chart

P Chart

C Chart

Continuous

Numerical Data

Categorical or Discrete

Numerical Data

Control Chart Types

Produce Good

Provide Service

Stop Process

Yes

No

Assign. Causes? Take Sample

Inspect Sample

Find Out Why Create

Control Chart

Start

Statistical Process

Control Steps Commonly Used Control Charts

Variables data

x-bar and R-charts

x-bar and s-charts

Charts for individuals (x-charts)

Attribute data

For “defectives” (p-chart, np-chart)

For “defects” (c-chart, u-chart)

Type of variables control chart

Shows sample means over time

Monitors process average

Example: Weigh samples of coffee &

compute means of samples; Plot

X Chart X Chart

Control Limits

Sample

Range at

Time i

# Samples

Sample

Mean at

Time i

From

Table

RAxx

LCL

RAxx

UCL

2

2

n

R

Ri

n

1i

n

xi

n

ix

6

Factors for Computing

Control Chart Limits

Sample

Size, n

Mean

Factor, A2

Upper

Range, D4

Lower

Range, D3

2 1.880 3.268 0

3 1.023 2.574 0

4 0.729 2.282 0

5 0.577 2.115 0

6 0.483 2.004 0

7 0.419 1.924 0.076

8 0.373 1.864 0.136

9 0.337 1.816 0.184

10 0.308 1.777 0.223 0.184

Type of variables control chart

– Interval or ratio scaled numerical data

Shows sample ranges over time

– Difference between smallest & largest

values in inspection sample

Monitors variability in process

Example: Weigh samples of coffee

& compute ranges of samples; Plot

R Chart

Sample Range at

Time i

# Samples

From Table

R Chart

Control Limits

n

R

R

R D LCL

R D UCL

i

n

1i

3R

4R

Out-of-control…when?

Process Control Chart

0

20

40

60

80

100

120

140

160

180

200

200 201 202 203 204 205 206 207 208 209

Sample Number

Me

asu

re

Process is Out of Control

UCL

LCL

Average

Shift in Process Average


0

20

40

60

80

100

120

140

160

180

200

200 201 202 203 204 205 206 207 208 209 210

Sample Number

Me

as

ure


Trend: 8 or more points moving in the same direction - up or down

UCL

LCL

Average

Process Average Trend Up

7


0

20

40

60

80

100

120

140

160

180

200

200 201 202 203 204 205 206 207 208 209 210

Sample Number

Me

as

ure


Nonrandom Patterns Present in the Data

Average

UCL

LCL


60

70

80

90

100

110

120

130

140

150

200 201 202 203 204 205 206 207 208 209 210

Sample Number

Me

as

ure


Nonrandom Patterns Present in the Data

UCL

LCL

Average

Signals of Control Problems

A point outside the control limits

7 or more points in a row above or below the average (center-line) Shift

8 or more points in a row moving in the same direction, up or down. Trend

Nonrandom patterns in the data

Use Common sense and Good Judgment

Using X and R Process Control Charts

Situation: Boise Cascade is interesting in monitoring the

length of logs that arrive at a mill yard. In the long run, they

want the average to be 18 feet and the variation should

continue to decline

The process output measure is length of the logs.

An X and R chart will be developed to monitor the

log lengths.

Developing and R Charts

Define Process Measurement of Interest

Determine Subgroup (sample) size (3-6)

Determine data gathering methods

where, how, who

Determine number of subgroups (20-30)

Collect Data

Compute X and R for each subgroup

Plot X and R on separate charts

Compute Control Limits

Draw Control Limits and Centerline on Charts

X

Log length Example: Data 30 days (subgroups) -- subgroup size = 4

Day Log Length (feet)

1 2 3 4

1 16 18 21 23

2 26 20 19 19

3 20 22 18 18

4 24 16 22 20

5 17 19 24 17

6 17 17 15 18

7 22 12 20 22

8 24 19 19 17

9 18 18 20 14

10 17 23 19 15

11 20 20 17 21

12 21 17 21 23

13 22 17 22 17

14 16 19 18 19

15 17 18 15 23

8

Log Length Data (continued)


1 2 3 4

16 19 17 21 17

17 19 19 13 16

18 21 14 17 16

19 18 17 25 18

20 20 18 20 19

21 23 21 23 21

22 20 20 20 14

23 18 18 26 15

24 20 22 23 21

25 23 22 21 24

26 22 14 21 19

27 18 20 18 22

28 19 20 16 14

29 21 19 16 20

30 22 22 19 21

Compute X for Each Subgroup

n

X =

X

Where:

X = the values in

the subgroups

n = subgroup size

First Subgroup:

19.5 = 4

23 + 21 + 18 + 16 = 1X

Compute R for Each Subgroup

R = Subgroup High - Subgroup Low

First Subgroup:

R1 = 23 - 16

= 7

Log Length Example: Data

30 days (subgroups) -- subgroup size = 4


1 2 3 4 Average = X Range = R

1 16 18 21 23 19.5 7

2 26 20 19 19 21 7

3 20 22 18 18 19.5 4

4 24 16 22 20 20.5 8

5 17 19 24 17 19.25 7

6 17 17 15 18 16.75 3

7 22 12 20 22 19 10

8 24 19 19 17 19.75 7

9 18 18 20 14 17.5 6

10 17 23 19 15 18.5 8

11 20 20 17 21 19.5 4

12 21 17 21 23 20.5 6

13 22 17 22 17 19.5 5

14 16 19 18 19 18 3

15 17 18 15 23 18.25 8

Log Length Data (continued)

Day Log length (feet)

1 2 3 4 Average = X Range = R

16 19 17 21 17 18.5 4

17 19 19 13 16 16.75 6

18 21 14 17 16 17 7

19 18 17 25 18 19.5 8

20 20 18 20 19 19.25 2

21 23 21 23 21 22 2

22 20 20 20 14 18.5 6

23 18 18 26 15 19.25 11

24 20 22 23 21 21.5 3

25 23 22 21 24 22.5 3

26 22 14 21 19 19 8

27 18 20 18 22 19.5 4

28 19 20 16 14 17.25 6

29 21 19 16 20 19 5

30 22 22 19 21 21 3

Plot of Subgroup Ave ra ge s

0

5

10

15

20

25

30

35

40

45

50

1 3 5 7 9

11

13

15

17

19

21

23

25

27

29

Subgroup

Su

bg

rou

p A

ve

rag

e

Plot the X Values

9

Plot of R Values (Ranges)

Plot of R Values

0

2

4

6

8

10

12

14

16

18

1 3 5 7 9

11

13

15

17

19

21

23

25

27

29

Subgroup

Ran

ge

(R)

Compute Centerlines for Each Chart

X Chart:

19.25 = 30

577.5 =

k

X =

iX

R = R

k =

171

30 = 5.7

i

R Chart:

Plot of Subgroup Ave ra ge s

0

5

10

15

20

25

30

35

40

45

50

1 3 5 7 9

11

13

15

17

19

21

23

25

27

29

Subgroup

Su

bg

rou

p A

ve

rag

e

Plot the the Centerline on X Chart

X = 19.25

Plot of R Values

0

2

4

6

8

10

12

14

16

18

1 3 5 7 9

11

13

15

17

19

21

23

25

27

29

Subgroup

Ran

ge

(R)

Plot of Centerline on R Chart

R= 5.7

Compute the Control Limits on the X Chart

Table

n A2 D3 D4

1 2.66

2 1.88 0.0 3.27

3 1.02 0.0 2.57

4 0.73 0.0 2.28

5 0.58 0.0 2.11

6 0.48 0.0 2.00

Compute X Control Limits

n A2 D3 D4

1 2.66

2 1.88 0.0 3.27

3 1.02 0.0 2.57

4 0.73 0.0 2.28

5 0.58 0.0 2.11

6 0.48 0.0 2.00

UCL R = X + = 19.25 + .73(5.7) = 23.412A

LCL = X - = 19.25 - .73(5.7) = 15.092A R

Now Plot the Control Limits on the X Chart

10

Plot of Subgroup Averages

0

5

10

15

20

25

30

1 3 5 7 9

11

13

15

17

19

21

23

25

27

29

Subgroup

Su

bg

ro

up

Av

era

ge

Plot Control Limits on X Chart

X = 19.25

23.41 UCL

LCL 15.09

Compute Control Limits for R Chart

n A2 D3 D4

1 2.66

2 1.88 0.0 3.27

3 1.02 0.0 2.57

4 0.73 0.0 2.28

5 0.58 0.0 2.11

6 0.48 0.0 2.00

UCL = = 2.28(5.7) = 13.004D R

LCL = = 0.00(5.7) = 0.003D R

Plot the Control Limits on R Chart

Plot of R Values

0

2

4

6

8

10

12

14

16

18

1 3 5 7 9

11

13

15

17

19

21

23

25

27

29

Subgroup

Ra

ng

e (

R)

R Chart with Control Limits

13.0

0.0

5.7

UCL

LCL

Utilizing the Control Charts

Continue to Collect Subgroup data

Plot Values to X and R charts

Examine the R Chart First - Then the X Chart

Look for Signals

A point outside the control limits

7 points in a row above or below the centerline

8 points in a row moving in the same direction

any nonrandom patterns

Take action when signal indicates

Update Control limits when appropriate

59

Special Variables Control Charts

x-bar and s charts

x-chart for individuals

Special control charts for variable data X bar and s-Chart

1

)( 2

n

XXS

Sample S.D.

at Time i

# Samples

From Table

n

S S

SB LCL

SB UCL

i

n

1i

3S

4S

sAxx

LCL

sAxx

UCL

3

3

For the associated x-chart, the control limits are derived from the overall standard deviation are:

11

X chart for individuals

2/3

2/3

dRxxUCL

dRxxUCL

RD LCL

RD UCL

3R

4R

Samples of size 1, however, do not furnish enough information for process variability measurement. Process variability can be determined by using a moving average of ranges, or a moving range, of n successive observations. For example, a moving range of n=2 is computed by finding the absolute difference between two successive observations. The number of observations used in the moving range determines the constant d2; hence, for n=2, from appendix b, d2=1.128.

Set of observations measuring the percentage of cobalt in a chemical process

Fraction nonconforming (p-chart) Fixed sample size

Variable sample size

np-chart for number nonconforming

Charts for defects c-chart

u-chart

Charts for Attributes

12

P Charts

Used When the Variable of Interest is an Attribute and

We are Interested in Monitoring the Proportion of Items

in Sample that have this Attribute -

Can accommodate unequal sample sizes.

Sample sizes are usually 50 or greater.

Need 20-30 samples to construct the P-chart. Examples: Proportion of Invoices with errors Proportion of Incorrectly Sorted Logs Proportion of Items Requiring Rework

P Chart Example

Plywood Veneer is graded when it comes out of the

dryer. Sheets that graded incorrectly cause

problems later in the process. Management is

interested in monitoring the rate of incorrectly

graded veneer.

The variable of interest is the proportion of

incorrectly graded veneer.

Each shift, n=100 sheets are selected and evaluated

for grade. The number of mis-grades are

recorded.

P Charts

Step 1:

Collect appropriate data.

Attribute data of the “yes/no” type

A Sheet is inspected. Is it incorrectly graded - Yes or No?

Record the number of “Yes” occurrences

P-Chart Data

P Charts

Step 2:

Calculate the fraction defective for each

subgroup.

The fraction defective is known as the p value:

subgroup theof size

subgroup in the ancesnonconform ofnumber = p

Key Point: The fraction defective is always expressed as a decimal value.

Using the percentage value (i.e. 4.7% rather than .047) will

cause later computations to be inaccurate.

Fraction Nonconformance - p- Values

13

P Charts

Step 3:

Plot the data on a graph.

Plot each p value

Plot of the p-Values

p Charts

Step 4:

Compute the center line for the p chart

and plot on the chart

The center line of the p chart is p

subgroups allin examined items ofnumber total

subgroups allin ancesnonconform ofnumber total = p

215.2,000

429 = p

P-Values and Centerline

CL = .215

p Charts Step 5

If the sample sizes are equal, compute the 3 sigma control limits using the following formulas - plot on control chart:

092.100

)215.1(215.3215.LCL

)-(13 - = LCL

Limit ControlLower

338.100

)215.1(215.3215.UCL

)-(13 + = UCL

Limit Control Upper

n

ppp

n

ppp

P Control Chart

UCL = .338

CL=.215

LCL = .092

14

P Charts

Analyzing p Charts

p charts are analyzed using the standard tests

for special cause variation: A Point located outside the control limits

7 or more points above or below the centerline

8 or more points moving in the same direction

Other evidence of nonrandom patterns

P Charts

Step 5: Alternative - When sample sizes are not equal Compute the 3-sigma upper and lower control limits for the p chart. If the size of the subgroup size varies, the control limit

calculations can be accomplished by two methods:

Compute multiple control limits based on the largest and smallest subgroup sizes

The two sets of control limits are plotted on the p chart. By calculating control limits based on the largest and smallest subgroups, both the narrowest limits (largest subgroup size) and the widest limits (smallest subgroup size) are plotted.

Compute separate control limits for each fraction nonconformance.

p Charts

Using Multiple Control Limits: In analyzing a control chart with multiple limits, it must be clear

that:

Any value plotting outside the widest control limits is considered out of control

Any value plotting inside the narrowest control limits is considered in control

Only those values, if any, which plot between the two upper or two lower control limits raise questions needing further evaluation (calculate their individual limits)

np-Charts for number Nonconforming

The np-chart is a useful alternative to the p-chart because it is often easier to understand for production personnel-the number of nonconforming items is more meaningful than a fraction.

To use the np-chart, the size of each sample must be constant.

15

k

y ....... y y pn n21

)-(13 - n = LCL

Limit ControlLower

)-(13 + n = UCL

Limit Control Upper

pn

pn

ppnp

ppnp

npnp

ppn

/)( ere wh

)-(1 = s

deviation standard theof Estimate

pn

87

Chart for defects

A defect is a single nonconforming characteristics of an item, while a defective refers to an item that has one or more defects.

In some situation, quality assurance personnel mat be interested not only in whether an item is defective but also in how many defects it has. For example, in complex assemblies such as electronics, the number of defects is just as important as whether the product is defective.

The c-chart is used to control the total number of defects per unit when subgroup size is constant. If subgroup sizes are variable, a u-chart is used to control the average number of defects per unit.

c Charts

A c chart is a process control tool for charting and monitoring the number of attributes per unit. Each unit must be like all other units with respect to size, volume, height, or other measurement.

c Charts

Necessary Characteristics Subgroups must be the same size (in practical use, if

they vary less than + 15% from the average it is acceptable to use the average subgroup size to compute the chart)

Subgroup size must be large enough to provide an average of at least 5 nonconformities per subgroup

The attribute of interest is the number of nonconformities per unit

Each unit may have one or more nonconformities

The actual number of nonconformities is small compared with the number of opportunities for nonconformities

16

c Charts

Step 1:

Collect appropriate data.

Attribute data of the “counting” type

Issue is Re-patch requirements.

Subgroup size is 3 sheets of plywood

Variable of interest is the combined number of re-patch spots in the three sheets

C-Chart Example

Boise Cascade Plywood Plant has to re-patch sheets

when knot patches become loose or are missed during

the initial patch line operation. The department is

monitoring the number of re-patches.

3 Sheets are grouped to make sure that the average

number > 5

Re-Patch Data c Charts

Step 2:

Graph the data.

The number of nonconformities is on the vertical axis

The sample number is on the horizontal axis

Plot The Nonconformatives c Charts

Step 3:

Compute the average number and

standard deviation of defects per unit. Average:

Standard deviation:

c = total nonconformities in all samples

number of samples

s c =

17

Compute the Mean and Standard Deviation

Total = 277

33.308.11

08.1125

277

S

C

c Charts

Step 4:

Compute the 3-sigma upper and lower

control limits. Upper Control Limit

Lower Control Limit

08.21)33.3(308.113 + = UCL cc

08.1)33.3(308.113 - = LCL cc

c Charts

Step 5:

Plot the center line, , and the upper and

lower control limits. c

c Control Chart

UCL=21.08

CL=11.08

LCL=1.08

c Charts

Analyzing c Charts The c chart utilizes the standard tests for signaling

when a process is out of control:

Points located outside the control limits

7 or more points above or below the centerline

8 or more points moving in the same direction

Other evidence of nonrandom patterns

c Charts

Common Mistakes Plotting specification limits instead of control limits

Not taking action to determine the special cause when one of the rules for process control has been violated

Not plotting the data immediately after it is collected

Collecting data on defects per unit when the units are not of the same size, height, etc.

Using a desired value to develop control limits rather than actual data from the process

c

18

u-chart

Used when the subgroup size is not constant or the nature of the production process does not yield discrete, measurable units.

For example, suppose that in an auto assembly plant, several different models are produced that vary in surface area. The number of defects will not then be valid comparison among different models.

Other applications, such as the production of textiles, photographic film, or paper, have no convenient set of items to measure. In such cases, a standard unit of measurement is used, such as defects per square foot or defects per square inch. The control chart for these situations is u-chart.

k21

n21

n..........nn

c ....... c c u

iu nus / =

inuu /3 + = UCLu

inu /3 - u = LCLu

105 106

107

Control Chart Selection

Quality Characteristic

variable attribute

n>1?

n>=10 or

computer?

x and MR no

yes

x and s

x and R no

yes

defective defect

constant

sample

size?

p-chart with

variable sample

size

no

p or

np

yes constant

sampling

unit?

c u

yes no

108

Control Chart Design Issues

Basis for sampling

Sample size

Frequency of sampling

Location of control limits

19

109 110

SPC Implementation Requirements

Top management commitment

Project champion

Initial workable project

Employee education and training

Accurate measurement system

111

Process Capability

The range over which the natural variation of a process occurs as determined by the system of common causes

Measured by the proportion of output that can be produced within design specifications

112

Types of Capability Studies

• Peak performance study - how a process performs under ideal conditions

• Process characterization study - how a process performs under actual operating conditions

• Component variability study - relative contribution of different sources of variation (e.g., process factors, measurement system)

113

Process Capability Study

1. Choose a representative machine or process

2. Define the process conditions

3. Select a representative operator

4. Provide the right materials

5. Specify the gauging or measurement method

6. Record the measurements

7. Construct a histogram and compute descriptive statistics: mean and standard deviation

8. Compare results with specified tolerances 114

Process Capability

specification specification

specification specification

natural variation natural variation

(a) (b)

natural variation natural variation

(c) (d)

20

115

Process Capability Index

Cp = UTL - LTL

6

Cpl, Cpu }

UTL -

3

Cpl = - LTL

3

Cpk = min{

Cpu =

Process Capability

Nominal

value

800 1000 1200 Hours

Upper

specification

Lower

specification

Process distribution

(a) Process is capable

Process Capability

Nominal

value

Hours

Upper

specification

Lower

specification

Process distribution

(b) Process is not capable

800 1000 1200

Process Capability

Lower

specification

Mean

Upper

specification

Six sigma

Four sigma

Two sigma

Nominal value

Upper specification = 1200 hours

Lower specification = 800 hours

Average life = 900 hours = 48 hours

Process Capability

Light-bulb Production

Cp = Upper specification - Lower specification

6

Process Capability Ratio

Process Capability

Light-bulb Production Upper specification = 1200 hours



Cp = 1200 - 800

6(48)

Process Capability Ratio

= 1.39

CP = 1.33 4 Sigma

CP =2.0 6 Sigma

21

Process Capability Analysis

The process is centered at the target of 200 and the Cp = 2.00. All

is well.


The process has shifted to an average of 205, but Cp is still at 2.00.

Target


Real problems exist -- the process is centered at 230. Now, even

though Cp = 2.00, much of the output is defective.

Target

Process Capability




Cp = 1.39

Cpk = Minimum of

Upper specification - x

3

x - Lower specification

3

Process

Capability

Index

,

Process Capability




Cp = 1.39

Cpk = Minimum of

1200 - 900

3(48)

900 - 800

3(48)

Process

Capability

Index

= .69

=2.08

Process Capability




Cp = 1.39 Cpk = 0.69

Process

Capability

Index

Process

Capability

Ratio

22

Process Capability




Cp = 1.39

Cpk = Minimum of

1200 - 900

3(48)

900 - 800

3(48)

Process

Capability

Index

,

Technology

Session statistical process control (spc)