chapter 14 - wessa.net · a control chart for monitoring the mean of a process: the x-chart two...

Chapter 14

Methods for Quality Improvement

Quality, Processes, and Systems

Quality of a good or service – the extent to which it satisfies user needs and preferences

8 Dimensions of Quality•Performance

•Features

•Reliability

•Conformance

•Durability

•Serviceability

•Aesthetics

•Other perceptions that influence judgment of quality


Process – series of actions or operations

that transforms input into outputs over time


System – collection of interacting processes

with an ongoing purpose


Two important points about systems

1. No two items produced by a process are

the same

2. Variability is an inherent characteristic of

the output of all processes


6 major sources of Process Variation

1. People

2. Machines

3. Materials

4. Methods

5. Measurement

6. Environment

Statistical Control

Control Charts – graphical

devices used for

•monitoring process variation

•Identifying when to take action to

improve the process

•Assisting in diagnosing the

causes of process variation

run chart, or time series

plot

Statistical Control

Run Chart enhanced

by

•Adding centerline

•Connecting plot

points in temporal

order

Enhancements aid

the eye in picking out

any patterns

Statistical Control

Output variable of interest can be described by a

probability distribution at any point in time.

Particular value of output variable at time t can be

thought of as being generated by these probability

distributions

The distribution may change over time, either the

mean, the variance or both.

Distribution of the process – distribution of the

output variable

Statistical Control

A process whose output distribution does not

change over time is said to be in statistical

control, or in control. Processes with changing

distributions are out of statistical control, or out

of control, or lacking stability.

Statistical Control

Patterns of Process Variation Patterns of Process Variation

– changing distributions

Statistical Control

The output of processes that are in

statistical control still have variability

associated with them, but there is no pattern

to this variability. It is random.

Statistical Control

Statistical Process Control – keeping a

process in statistical control or bringing a

process into statistical control through

monitoring and eliminating variation

Common Causes of variation – methods,

materials, machines, personnel and

environment that constitute a process and

the inputs required by the process

Statistical Control

Special Causes of Variation (Assignable Causes) – events or actions that are not part of the process design.

Processes in control still exhibit variation, from the common causes.

Processes out of control exhibit variation from both common causes and special causes of variation

Most processes are not naturally in a state of statistical control

Statistical Control

The Logic of Control Charts

Control charts are used to help differentiate between variation due to common and special causes

When a value falls outside the control limits, it is either a rare event or the process is out of control

Mean when process

is in control


Hypothesis testing with

control charts:H0: Process is under

control

Ha: Process is out of

control

Another view:H0: = centerline

Ha: centerline

Ha here indicates that the

mean has shifted


Control limits vs. Specification limits

Specification limits – set by customers,

management, product designers. Determined as

“acceptable values” for an output.

Control limits are dependent

on the process,

specification limits are not.

A Control Chart for Monitoring the

Mean of a Process: The x-Chart

- control chart

that plots sample means

•Often used in concert

with R-chart, which

monitors process

variation

•More sensitive to

changes in process

mean than a chart of

individual

measurements

x c h a r t



To construct, you need 20 samples of a sample

size of at least 2.

where A2 is found in a Table of Control Chart Constants,

and R is the mean range of the samples

1 2 3 . . .:

kx x x xC e n te r l in e x

k

2:L o w er co n tro l l im it x A R

2:U p p er co n tro l l im it x A R



Two important decisions in Constructing an x-chart

1.Determine sample size n

2.Determine the frequency with which samples are to be

drawn

Rational Subgroups – subgroups chosen with sample

size n and frequency to make it likely that process changes

will happen between rather than within samples

Rational Subgrouping strategy maximizes the chance for

measurements to be similar within each sample, and for

samples to differ from each other.



Summary of x-chart Construction

1. Collect at least 20 samples with sample size n ≥ 2,

utilizing rational subgrouping strategy

2. Calculate mean and range for each sample

3. Calculate mean of sample means x and mean of

sample ranges R

4. Plot centerline and control limits

5. Plot the k sample means in the order that the samples

were produced by the process



Constructing Zone Boundaries These zone

boundaries are used

in conjunction with

Pattern-Analysis rules

to help determine

when a process is out

of control

Using 3-sigma control limits

Upper A-B Boundary: 2

2

3x A R

Lower A-B Boundary: 2

2

3x A R

Upper B-C Boundary: 2

1

3x A R

Lower B-C Boundary: 2

1

3x A R

Using estimate standard deviation of x , 2R d

n


2R d

x

n


2R d

x

n


R dx

n


R dx

n



Any of the 6 rules being broken suggests an out of control process


Variation of a Process: The R-Chart

R-chart used to detect changes in process

variation

R-chart plots and monitors the variation of

sample ranges



To construct, you need 20 samples of a sample

size of at least 2.

where D3 and D4 are found in a Table of Control Chart

Constants. When n ≤ 6, there is only an upper control limit

1 2 3. . .

:k

R R R RC e n te r l in e R

k

3:L o w e r c o n tr o l l im i t R D

4:U p p e r c o n tr o l l im i t R D



Summary of R-Chart Construction

1. Collect at least 20 samples with sample size n ≥ 2,

utilizing rational subgrouping strategy

2. Calculate the range for each sample

3. Calculate mean of sample ranges R

4. Plot centerline and control limits. When n ≤ 6, there is

only an upper control limit

5. Plot the k sample ranges in the order that the samples

were produced by the process

A Control Chart for Monitoring the Variation

of a Process: The R-Chart

Constructing Zone Boundaries

These zone boundaries are used in conjunction with

Pattern-Analysis rules 1-4 to help determine when a

process is out of control


2

2R

R dd


2

2R

R dd


2

RR d

d


2

RR d

d

Note: when n ≤ 6, the R-chart has no lower control limit, but boundaries

can still be plotted if non-negative

A Control Chart for Monitoring the Proportion of

Defectives Generated by a Process: The p-Chart

p-chart used to detect changes in process

proportion when output variable is

categorical

As long as process proportion remains

constant, process is in statistical control



Sample-Size determination

Choose n such that

where

n = Sample Size

p0 = an estimate of the process proportion p

0

0

9 1 pn

p



Calculations for p-chart Construction

N u m b e r o f d e fe c t iv e i te m s in sa m p lep

N u m b e r o f i te m s in sa m p le

:T o ta l n u m b e r o f d e fe c t iv e i te m s in a l l k s a m p le s

C e n te r l in e pT o ta l n u m b e r o f u n i ts in a l l k s a m p le s

1

: 3

p p

U p p e r c o n tr o l l im i t pn

1

: 3

p p

L o w e r c o n tr o l l im i t pn



Summary of p-Chart Construction

1. Collect at least 20 samples utilizing rational

subgrouping strategy and appropriate sample size

2. Calculate proportion of defective units for each sample

3. Plot centerline and control limits.

4. Plot the k sample proportions on the control chart in

the order the samples were produced by the process



Constructing Zone Boundaries

These zone boundaries are used in conjunction with

Pattern-Analysis rules 1-4 to help determine when a

process is out of control


2

p p

pn


2

p p

pn

Upper B-C Boundary: 1p p

pn

Lower B-C Boundary: 1p p

pn

Note: when LCL is negative it should not be plotted. Lower zone

boundaries can be plotted if non-negative

Diagnosing the Causes of Variation

If monitoring phase identifies that problems

exist, diagnosis is needed to determine what

the problems are.


Cause-and-Effect diagrams used to assist in

process diagnosis

Basic Cause-and-Effect diagram:


Cause-and-Effect diagram applied to specific problem:

Capability Analysis

Used when a process

is in statistical control,

but level of variation is

unacceptably high.

Capability Analysis

•A Capability Analysis diagram is used to assess process capability.

•This diagram builds on a frequency distribution of a large sample of individual measurements from the process by adding specification limits and target value

Capability Analysis

From this, 2 approaches

1. Report percentage of outcomes that fall

outside of specification limits

2. Construct a capability index Cp where

6p

S p e c i f i c a t io n s p r e a d U S L L S LC

P r o c e s s S p r e a d

Capability Analysis

Interpretation of CpIf Cp=1, (specification spread = process spread) process is capable

If Cp>1, (specification spread > process spread) process is capable

If Cp<1, (specification spread < process spread) process is not capable

If the process follows a normal distribution

Cp=1.00 means about 2.7 units per 1000 will be unacceptable

Cp=1.33 means about 63 units per million will be unacceptable

Cp=1.67 means about .6 units per million will be unacceptable

Cp=2.00 means about 2 units per billion will be unacceptable

chapter 14 - wessa.net · a control chart for monitoring the mean of a process: the x-chart two...

Documents