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Chapter 15Statistical ProcessProcess Control

MGS3100Julie Liggett De Jong

Statistical Process Control is used to prevent quality problems

Statistical Process Control ….

How it works.

Take periodic samples from process

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Take periodic samples from process

Plot sample points on control chart

Determine if process is within limits

Take periodic samples from process

Plot sample points on control chart

Determine if process is within acceptable limits

Variation

1 Common Causes1. Common CausesVariation inherent in a processEliminated through system improvements

Variation

2 Special Causes2. Special CausesVariation due to identifiable factorsModified through operator or management action

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Attribute measures

ProductProduct characteristic evaluated with a discrete choice:Good / bad Yes / NoPass / Fail

Attribute measures

ProductProduct characteristics evaluated with a discrete choice:Good / bad Yes / NoPass / Fail

Attribute measures

ProductProduct characteristics evaluated with a discrete choice:

Good / bad Yes / NoPass / Fail

Variable measures

Measurable product characteristic:

Length size weightLength, size, weight, height, time, velocity

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Variable measures

Measurable product characteristics:

Length size weightLength, size, weight, height, time, velocity

Variable measures

Measurable product characteristics

Length size weightLength, size, weight, height, time, velocity

SPC Applied to Services

Hospitals

Timeliness

Responsiveness

Accuracy of lab tests

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Grocery Stores

Check-out time

Stocking

Cleanliness

AirlinesLuggage handling Waiting times Courtesy

Fast Food Restaurants

Waiting timesWaiting times

Food quality

Cleanliness

Internet Orders

Order accuracyOrder accuracy Packaging Delivery time Email confirmationconfirmationPackage tracking

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Insurance

Billing accuracy

Timeliness of claims processing

Agent availabilityg y

Response time

Control ChartsControl Charts

Graphs that establish process control limits

Attribute measures:P-ChartsC-Charts

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Variable measures:Mean (x-bar) control charts Range (R) control charts

A Process is in control if:

A Process is in control if:

No sample points are outside control limits

A Process is in control if:

Most points are near process average

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A Process is in control if:

About equal number of points are above & below centerline

A Process is in control if:

Points appear to be randomly distributed

Process Control Chart

Uppercontrol

li it

Out of control

limit

Processaverage

Lowercontrol

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

controllimit

Figure 15.1

To develop Control Charts:

Use in-control data

If non-random causes are present, find them and discard data related to them

Correct control chart limits

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Control ChartsControl Charts Measures Description p Chart Attributes Calculates percent defectives

in samplep

r Chart (range chart)

Variables Reflects the amount of dispersion in a sample

x bar Chart (mean chart)

Variables Indicates how sample results relate to the process average

Cp Process Measures the capability of a p (Process Capability

Ratio)

Capability p y

process to meet design specifications

Cpk (Process Capability

Index)

Process Capability

Indicates if the process mean has shifted away from design target

Control ChartsControl Charts Measures Description p Chart Attributes Calculates percent defectives

in samplep

r Chart (range chart)

Variables Reflects the amount of dispersion in a sample

x bar Chart (mean chart)

Variables Indicates how sample results relate to the process average

Cp Process Measures the capability of a p (Process Capability

Ratio)

Capability p y

process to meet design specifications

Cpk (Process Capability

Index)

Process Capability

Indicates if the process mean has shifted away from design target

p-Chart

UCL = p + zσp

LCL = p - zσp

where

p = the sample proportion defective; an estimate of the process average

p

p-Chart

UCL = p + zσp

LCL = p - zσp

where

p = the sample proportion defective; an estimate of the process average

p

p =total defectives

total sample observations

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p-Chart

UCL = p + zσp

LCL = p - zσp

where

p = the sample proportion defective; an estimate of the process average

z = the number of standard deviations from the process average

p

average

The Normal Distribution

μ=0 1σ 2σ 3σ-1σ-2σ-3σ

95%99.74%

Control Chart Z Values

Smaller Z values make more narrow control limits and more sensitive chartsZ = 3.00 is standardCompromise between sensitivity and errors

μ=0 1σ 2σ 3σ-1σ-2σ-3σ

95%99.74%

p-Chart

UCL = p + zσp

LCL = p - zσp

where

p = the sample proportion defective; an estimate of the process average

z = the number of standard deviations from the process average

p

averageσp = the standard deviation of the sample proportion

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p-Chart

UCL = p + zσp

LCL = p - zσp

σp = p(1 - p)

np

p =total defectives

total sample observationsp = total sample observations

p-Chart Example ~Western Jeans Company p337

20 samples of 100 pairs of jeans (n = 100)

NUMBER OF PROPORTION

p-Chart Example ~Western Jeans Company

NUMBER OF PROPORTIONSAMPLE DEFECTIVES DEFECTIVE

1 6 .062 0 .003 4 .04: : :: : :

20 18 .18200

Ex 1, P337

0 16

0.18

0.20

UCL = 0.190

p-Chart Example ~Western Jeans Company

0.08

0.10

0.12

0.14

0.16

Prop

ortio

n de

fect

ive

p = 0.10

0.02

0.04

0.06

P

Sample number2 4 6 8 10 12 14 16 18 20

LCL = 0.010

Ex 1, P337

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Control ChartsControl Charts Measures Description p Chart Attributes Calculates percent defectives in

samplesample

r Chart (range chart)

Variables Reflects the amount of dispersion in a sample

x bar Chart (mean chart)

Variables Indicates how sample results relate to the process average

Cp Process Measures the capability of a p(Process Capability

Ratio)

Capability y

process to meet design specifications

Cpk (Process Capability

Index)

Process Capability

Indicates if the process mean has shifted away from design target

Range ( R ) Chart

UCL = D R LCL = D RUCL = D4R LCL = D3R

R = ∑Rk

where:where:

R = range of each samplek = number of samples

n A2 D3 D4

SAMPLE SIZE FACTOR FOR x-CHART FACTORS FOR R-CHART

2 1.88 0.00 3.273 1.02 0.00 2.574 0 73 0 00 2 28

Factors for R-Chart: D3 & D4

4 0.73 0.00 2.285 0.58 0.00 2.116 0.48 0.00 2.007 0.42 0.08 1.928 0.37 0.14 1.869 0.44 0.18 1.82

10 0.11 0.22 1.7811 0.99 0.26 1.7412 0.77 0.28 1.7213 0.55 0.31 1.6914 0.44 0.33 1.6715 0.22 0.35 1.6516 0.11 0.36 1.6417 0.00 0.38 1.6218 0.99 0.39 1.6119 0.99 0.40 1.6120 0.88 0.41 1.59

Table 1, P343

R-Chart Example ~ Goliath Tool Company (p345)

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R-Chart Example ~ Goliath Tool Company

OBSERVATIONS (SLIP-RING DIAMETER, CM)SAMPLE k 1 2 3 4 5 x RSAMPLE 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 0.14

R = max – min = 5.02 – 4.94 = 0.08

8 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 0.10total 50.09 1.15

Ex 3, P344

n A2 D3 D4

SAMPLE SIZE FACTOR FOR x-CHART FACTORS FOR R-CHART

2 1.88 0.00 3.273 1.02 0.00 2.574 0 73 0 00 2 28

Factors for R-Chart: D3 & D4

4 0.73 0.00 2.285 0.58 0.00 2.116 0.48 0.00 2.007 0.42 0.08 1.928 0.37 0.14 1.869 0.44 0.18 1.82

10 0.11 0.22 1.7811 0.99 0.26 1.7412 0.77 0.28 1.7213 0.55 0.31 1.6914 0.44 0.33 1.6715 0.22 0.35 1.6516 0.11 0.36 1.6417 0.00 0.38 1.6218 0.99 0.39 1.6119 0.99 0.40 1.6120 0.88 0.41 1.59

Table 1, P343

R-Chart Example ~Goliath Tool Company

UCL = 0.243

ange R = 0.115

0.28 –

0.24 –

0.20 –

0.16 –

Example 15.3

LCL = 0

Ra

Sample number

|1

|2

|3

|4

|5

|6

|7

|8

|9

|10

0.12 –

0.08 –

0.04 –

0 –

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x-Chart Calculations

UCL = x + A2R LCL = x - A2R= =

x = x1 + x2 + ... xk

k=

where

x = the average of the sample meansR bar = the average range values

=

x-Chart Example ~ Goliath Tool Company

OBSERVATIONS (SLIP-RING DIAMETER, CM)SAMPLE k 1 2 3 4 5 x RSAMPLE 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 0.14

(5.02 + 5.01 + 4.95 + 4.99 + 4.96)/5 = 4.98

8 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 0.10total 50.09 1.15

Ex 4, P345

n A2 D3 D4

SAMPLE SIZE FACTOR FOR x-CHART FACTORS FOR R-CHART

2 1.88 0.00 3.273 1.02 0.00 2.574 0 73 0 00 2 28

Factors for R-Chart: D3 & D4

4 0.73 0.00 2.285 0.58 0.00 2.116 0.48 0.00 2.007 0.42 0.08 1.928 0.37 0.14 1.869 0.44 0.18 1.82

10 0.11 0.22 1.7811 0.99 0.26 1.7412 0.77 0.28 1.7213 0.55 0.31 1.6914 0.44 0.33 1.6715 0.22 0.35 1.6516 0.11 0.36 1.6417 0.00 0.38 1.6218 0.99 0.39 1.6119 0.99 0.40 1.6120 0.88 0.41 1.59

Table 1, P343

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UCL = 5.08

5.10 –

5.08 –

5 06

x-Chart Example ~ Goliath Tool Company

Mea

n

5.06 –

5.04 –

5.02 –

5.00 –

4.98 –

x = 5.01=

Example 15.4

LCL = 4.94

Sample number

|1

|2

|3

|4

|5

|6

|7

|8

|9

|10

4.96 –

4.94 –

4.92 –

Using x- and R-charts together

Each measures the process differently Both process average (x bar chart) and variability (R chart) must be in y ( )control

Sample Size Determination

Attribute control charts (p chart)• 50 to 100 parts in a sample

Sample Size Determination

Variable control charts (R- & x bar- charts)• 2 to 10 parts in a sample

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Process Capability

• Control limits (the “Voice of the Process” or the “Voice of the Data”): based on natural variations (common causes)

• Tolerance limits (the “Voice of the Customer”): customer requirements

Process Capability: A measure of how• Process Capability: A measure of how “capable” the process is to meet customer requirements; compares process limits to tolerance limits

Range of natural variability in process• Measured with control charts.

Process Capability

Process cannot meet specifications if natural variability exceeds tolerances3-sigma quality

• Specifications equal the process control limits.

6 i li6-sigma quality• Specifications twice as large as control

limits

Design Specifications

Process Capability

Design Specifications

(a) Natural variation exceeds design specifications; process is not capable of meeting specifications all the time.

Process

(b) Design specifications and natural variation the same; process is capable of meeting specifications most the time.

Process

Figure 15.5

Design Specifications

Process Capability

(c) Design specifications greater than natural variation; process is capable of always conforming to specifications.

ProcessDesign

Specifications

Figure 15.5

(d) Specifications greater than natural variation, but process off center; capable but some output will not meet upper specification.

Process

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Process Capability MeasuresProcess Capability Ratio (Cp )

t lCCpp ==

==

tolerance rangeprocess range

upper specification limit -lower specification limit

66σ

Design Specifications

a) Cp < 1.0

Process Capability MeasuresProcess Capability Ratio ( Cp )

Design S ifi i

Process

Design Specifications

Processb) Cp = 1.0

c) Cp > 1.0

Specifications

ProcessFigure 15.5

Computing Cp

Net weight specification = 9.0 oz ± 0.5 ozP 8 80

Munchies Snack Food Company

Process mean = 8.80 ozProcess standard deviation = 0.12 oz

Cp =

upper specification limit -lower specification limit

6σ

= = 1.399.5 - 8.56(0.12)

Ex 6, P 354

Process Capability MeasuresProcess Capability Index ( Cpk )

Cpk = minimum

x - lower specification limit3σ

=

upper specification limit - x3σ

=,

D iDesign Specifications

Process

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C > 1 00: Process is capable of meeting design

Process Capability MeasuresProcess Capability Index ( Cpk )

Cpk > 1.00: Process is capable of meeting design specifications

Cpk < 1.00: Process mean has moved closer to one of the upper or lower design specifications and will generate defects

Cpk = 1.00: The process mean is centered on the design target.

Computing Cpk

Net weight specification = 9.0 oz ± 0.5 ozProcess mean = 8.80 oz

Munchies Snack Food Company

Process standard deviation = 0.12 oz

Cpk = minimum

x - lower specification limit3σ

=

upper specification limit - x=,

= minimum , = 0.83

3σ

8.80 - 8.503(0.12)

9.50 - 8.803(0.12)

Ex 7, P354

The Process Capability Index

Cpk < 1 Not Capablepk p

Cpk > 1 Capable at 3σ

Cpk > 1.33 Capable at 4σ

Cpk > 1.67 Capable at 5σ

Cpk > 2 Capable at 6σ