Statistical Process Control Chapter 4. Chapter Outline Foundations of quality control Product launch...

Statistical Process Control

Chapter 4

Chapter Outline

Foundations of quality control Product launch and quality control activities Quality measures and control charts Transformation processes and variation Statistical process control (SPC) Variation and conformance quality SPC in services

Chapter Outline (2)

SPC overview Objectives of SPC Control chart format Hypothesis testing Terminology (what is n?)

Chapter Outline (3)

Control charts for variables x-bar charts R charts Control chart patterns

Control charts for attributes p charts

Chapter Outline (4)

Process capability re-visited Control limits vs. specification limits Process capability ratio, Cp

Cp does not work when the mean and the target are not equal

Process capability index, Cpk

Customer Requirements

Product Specifications

Statistical Process Control:

Measure & monitor quality

MeetsSpecifications

?

Process Specifications

Yes

Conformance Quality

Fix process or

inputs

No

Product launch

activities: Revise

periodically

Ongoing Activities

Quality Measuresand Control Charts

Discrete measures Good/bad, yes/no (p charts) Count of defects (c charts)

Variables – continuous numerical measures

Length, diameter, weight, height, time, speed, temperature, pressure

Controlled with

charts and Rx

Variation in aTransformation Process

Transformation Process

Inputs• Facilities• Equipment• Materials• Energy

OutputsGoods &Services

•Variation in inputs create variation in outputs• Variations in the transformation process create variation in outputs

Types of VariationCommon Cause Variation

Common cause (random) variation: systematic variation in a process. Results from usual variations in inputs, output rates, and procedures

Usually results from a poorly designed product or process, poor vendor selection, or other management issues

If the amount of common cause variation is not acceptable, it is management's responsibility to take corrective action.

Types of VariationSpecial Cause Variation

Special cause (non-random or assignable cause) variation: a short-term source of variation in a process. Results from changes or abnormal variations in inputs, outputs, or procedures.

Usually results from errors by workers, first-line supervisors, or vendors

The cause can and should be identified. Corrective action should be taken.

Statistical Process Control (SPC)

A process is in control if it has no assignable cause variation. The process is consistent or predictable.

SPC distinguishes between common cause and assignable cause variation

Measure characteristics of goods or services that are important to customers

Make a control chart for each characteristic The chart is used to determine whether the

process is in control

Specification Limits

The target is the ideal value Example: if the amount of beverage in a bottle

should be 16 ounces, the target is 16 ounces Specification limits are the acceptable range of values

for a variable Example: the amount of beverage in a bottle must be at

least 15.8 ounces and no more than 16.2 ounces. Range is 15.8 – 16.2 ounces. Lower specification limit = 15.8 ounces or LSPEC = 15.8

ounces Upper specification limit = 16.2 ounces or USPEC = 16.2

ounces

Specifications and Conformance Quality

A product which meets its specification has conformance quality.

Capable process: a process which consistently produces products that have conformance quality. Must be in control and meet specifications

Capable Transformation Process

Capable Transformation

Process

Inputs• Facilities• Equipment• Materials• Energy

OutputsGoods &Servicesthat meet

specifications

If the process is capable and the product specification is based on current customer requirements, outputs will meet customer requirements.

Copyright 2006 John Wiley & Sons, Inc. 4-15

Nature of defect is different in services

Service defect is a failure to meet customer requirements

Monitor times, customer satisfaction, quality of work, product availability

Applying SPC to Services


Applying SPC to Services (2)

Hospitals timeliness and quickness of care, staff responses to

requests, accuracy of lab tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance and 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 and luggage handling,

waiting time at ticket counters and check-in, agent and flight attendant courtesy, accurate flight information, passenger cabin cleanliness and maintenance


Applying SPC to Services (3)

Fast-Food Restaurants waiting time for service, customer complaints,

cleanliness, food quality, order accuracy, employee courtesy

Catalogue-Order Companies order accuracy, operator knowledge and

courtesy, packaging, delivery time, phone order waiting time

Insurance Companies billing accuracy, timeliness of claims

processing, agent availability and response time

Objectives of Statistical Process Control (SPC)

Determine Whether the process is in control Whether the process is capable Whether the process is likely to remain

in control and capable

Control Chart Format

Upper Control Limit (UCL)

Process Mean

Lower Control Limit (LCL) Sample

Mea

sure

Hypothesis Test

H0: The process mean (or range) has not changed. (null hypothesis)

H1: The process mean (or range) has changed. (alternative hypothesis).

If the process has only random variations and remains within the control limits, we accept H0. The process is in control.

Terminology

We take periodic random samples n = sample size = number of

observations in each sample

X and R Charts for Variables

X = Sample mean Measure of central tendency Central Limit Theorem: X is normally

distributed. R = Sample range

Measure of variation R has a gamma distribution (not

normal)

Data for Examples 4.3 and 4.4

Slip-ring diameter (cm)Sample 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 0.08… … … … … … … …10 5.01 4.98 5.08 5.07 4.99 5.03 0.1050.09 1.15

Note: n = number in each sample = 5

Calculate X and R for Each Sample

Sample 1:X = 5.02 + 5.01 + 4.94 + 4.99 + 4.96 5

= 4.98 R = range = maximum - minimum = 5.02 - 4.94 = 0.08 Repeat for all samples

Calculate X and R

X = 4.98 + 5.00 + 4.97 + … + 5.03 = 5.01

10

R = 0.08 + 0.12 + 0.08 + … + 0.10 = 0.115

10

The Normal Distribution

=0 1 2 3

95%

99.74%

-1-2-3

Control Limits for X

99.7% confidence interval for X: (X - 3, X + 3). This may be approximated as

(X - A2R, X + A2R). A2 is a factor which depends on n and is

obtained from a table.

3 Control Chart Factors

Sample size x-chart R-chartn A2 D3 D4

2 1.88 0 3.273 1.02 0 2.574 0.73 0 2.285 0.58 0 2.116 0.48 0 2.007 0.42 0.08 1.928 0.37 0.14 1.86

Control Limits for X and R

For X: LCL = X - A2R = 5.01 - 0.58 (0.115) =

4.94

UCL = X + A2R = 5.01 + 0.58 (0.115) = 5.08

For R: LCL = D3R = 0 (0.115) = 0

UCL = D4R = 2.11 (0.115) = 0.243

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=

UCL = 0.243

LCL = 0

Ra

ng

e

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 –

R Chart

Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. 4-4-3232

Control Chart Pattern – Change in MeanControl Chart Pattern – Change in Mean

UCLUCL

LCLLCL

Sample observationsSample observationsconsistently above theconsistently above thecenter linecenter line

LCLLCL

UCLUCL

Sample observationsSample observationsconsistently below theconsistently below thecenter linecenter line


Control Chart Patterns: TrendControl Chart Patterns: Trend

LCLLCL

UCLUCL

Sample observationsSample observationsconsistently increasingconsistently increasing

UCLUCL

LCLLCL

Sample observationsSample observationsconsistently decreasingconsistently decreasing


Control Charts for Control Charts for AttributesAttributes

p-charts uses portion defective in a sample

c-charts uses number of defects in an item


p-Chartp-Chart

UCL = p + zp

LCL = p - zp

z = number of standard deviations from process averagep = sample proportion defective; an estimate of process averagep= standard deviation of sample proportion

pp = = pp(1 - (1 - pp))

nn


p-Chart Examplep-Chart Example

20 samples of 100 pairs of jeans20 samples of 100 pairs of jeans

NUMBER OFNUMBER OF PROPORTIONPROPORTIONSAMPLESAMPLE DEFECTIVESDEFECTIVES DEFECTIVEDEFECTIVE

11 66 .06.06

22 00 .00.00

33 44 .04.04

:: :: ::

:: :: ::

2020 1818 .18.18

200200


p-Chart Example (cont.)p-Chart Example (cont.)

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 =


0.020.02

0.040.04

0.060.06

0.080.08

0.100.10

0.120.12

0.140.14

0.160.16

0.180.18

0.200.20

Pro

po

rtio

n d

efec

tive

Pro

po

rtio

n d

efec

tive

Sample numberSample number22 44 66 88 1010 1212 1414 1616 1818 2020

UCL = 0.190

LCL = 0.010

p = 0.10p-Chart p-Chart Example Example

Process Capability Revisited

A process must be in control before you can decide whether or not it is capable.

Control charts measure the range of natural variability in a process (what the process is actually producing)

Specification limits are set to meet customer requirements.

Process cannot meet specifications if one or both control limits is outside specification limits

Process Meets Customer Requirements

UCL

LCL

X

Lower specification limit

Upper specification limit

Process Does Not Meet Customer Requirements

UCL

LCL

X

Lower specification limit

Upper specification limit

Process Capability Ratio Cp

For a product characteristic, letLSL = lower specification limitUSL = upper specification limit= mean, = standard deviationIf (1) The process is in control and (2) = target (or mean = target)we can use the process capability ratio, Cp to

determine whether the process is capable

Computing Cp

Given: LSL = 8.5, USL = 9.5, target = 9, = 9, = 0.12Note that = targetCompute:

If Cp > 1, the process is capable.

If Cp < 1, the process is not capable

Conclusion: Cp = 1.39 > 1 process is capable.

39.172.0

1

)12.0(6

5.85.9

6

LSLUSL

C p

Check on the Accuracy of Cp

LCL = - 3 = 9 – 3(0.12) = 8.64UCL = + 3 = 9 + 3(0.12) = 9.36The specification limits are 8.5 – 9.5The control limits are within the

specification limits. The process is capable.

Computing Cp – Another Example

Given: LSL = 8.5, USL = 9.5, target = 9, = 8.8, = 0.12Note that does not equal the target.Compute:

Conclusion: Cp = 1.39 > 1 process is capable.

Wrong! LCL = - 3 = 8.8 – 3(0.12) = 8.44 < LSL

Cp can give the wrong answer if does not equal the target. Use Cpk

39.172.0

1

)12.0(6

5.85.9

6

LSLUSL

C p

Computing Cpk

Given: LSL = 8.5, USL = 9.5, target = 9, = 8.8, = 0.12Compute:

Cpk < 1 the process is not capable

Cpk always tells you whether the process is capable.

Note: If is not given, use x instead of

83.094.1,83.0min

)12.0(3

8.85.9,

3(0.12)

8.5-8.8min

3,

3minimum

USLLSLC pk

Statistical Process Control Chapter 4. Chapter Outline Foundations of quality control Product launch...

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