part 4: statistical process control control charts

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2 Part 4: Statistical Process Control Control Charts Henriqueta Nóvoa | José Sarsfield Cabral Faculdade de Engenharia da Universidade do Porto Mestrado em Engenharia Mecânica Mestrado em Engenharia Industrial e Gestão Mestrado em Engenharia Electrotécnica e de Computadores

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Page 1: Part 4: Statistical Process Control Control Charts

2

Part 4: Statistical Process Control

Control Charts

Henriqueta Nóvoa | José Sarsfield Cabral

Faculdade de Engenharia da Universidade do Porto

Mestrado em Engenharia Mecânica

Mestrado em Engenharia Industrial e Gestão

Mestrado em Engenharia Electrotécnica e de Computadores

Page 2: Part 4: Statistical Process Control Control Charts

Deming

4

“Why are we here?”

“We are here to make

another world.”

Another quote :

“If I had to reduce my message for management

to just a few words, I’d say it all had to do

with reducing variation” (Neave 1990).

Page 3: Part 4: Statistical Process Control Control Charts

Timeline

1938 W. E. Deming invites Shewhart to present seminars on control charts at

the U.S. Department of Agriculture Graduate School.

1946 The American Society for Quality Control (ASQC) is formed as the

merger of various quality societies.

The International Standards Organization (ISO) is founded.

Deming is invited to Japan by the Economic and Scientific Services Section of the

U.S. War Department to help occupation forces in rebuilding Japanese industry.

The Japanese Union of Scientists and Engineers (JUSE) is formed.

1946–1949 Deming is invited to give statistical quality control seminars to Japanese

industry.

1951 A. V. Feigenbaum publishes the first edition of his book, Total Quality Control.

JUSE establishes the Deming Prize for significant achievement in quality

control and quality methodology.

W. Edwards Deming 1900- 1993

Page 4: Part 4: Statistical Process Control Control Charts

Major contributions:“have one aim: to make it possible for people to work with joy”

1. Create a constancy of purpose focused on the improvement of products

and services. Constantly try to improve product design and

performance. Investment in research, development, and innovation will have long-term payback to the organization.

2. Adopt a new philosophy of rejecting poor workmanship, defective products,

or bad service.

3. Do not rely on mass inspection to “control” quality.

4. Do not award business to suppliers on the basis of price alone, but also

consider quality.

W. Edwards Deming 1900- 1993

Page 5: Part 4: Statistical Process Control Control Charts

Major contributions:

5. Focus on continuous improvement. Constantly try to improve the

production and service system. Involve the workforce in these activities and

make use of statistical methods.

6. Practice modern training methods and invest in training for all

employees.

7. Improve leadership, and practice modern supervision methods.

8. Drive out fear. Many workers are afraid to ask questions, report problems,

or point out conditions that are barriers to quality and effective production.

9. Break down the barriers between functional areas of the business.

10. Eliminate targets, slogans, and numerical goals for the workforce.

W. Edwards Deming 1900- 1993

Page 6: Part 4: Statistical Process Control Control Charts

Major contributions:

11. Eliminate numerical quotas and work standards. These standards have

historically been set without regard to quality.

12. Remove the barriers that discourage employees from doing their jobs.

Management must listen to employee suggestions, comments, and

complaints.

13. Institute an ongoing program of training and education for all

employees. Education in simple, powerful statistical techniques should be

mandatory for all employees.

14. Create a structure in top management that will vigorously advocate the first

13 points.

W. Edwards Deming 1900- 1993

Page 7: Part 4: Statistical Process Control Control Charts

16

Statistical Process Control

Statistical process control is a collection of

tools that when used together can result in

process stability and variability reduction

Page 8: Part 4: Statistical Process Control Control Charts

18

Statistical Process Control

The seven major tools are

1) Histogram or Stem and Leaf plot

2) Check Sheet

3) Pareto Chart

4) Cause and Effect Diagram

5) Defect Concentration Diagram

6) Scatter Diagram

7) Control Chart

Douglas C. Mongomery “Introduction to Statistical Quality Control”

Page 9: Part 4: Statistical Process Control Control Charts

19

The rest of the “Magnificent Seven”

The control chart is most effective when

integrated into a comprehensive SPC program.

The seven major SPC problem-solving tools should

be used routinely to identify improvement

opportunities.

The seven major SPC problem-solving tools should

be used to assist in reducing variability and

eliminating waste.

Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley

Page 10: Part 4: Statistical Process Control Control Charts

20

The first control chart

In May 1924, Shewhart (1891-1967) proposes a method

for evaluating the stability overtime of the proportion

of defective appliances (p).

Upper Control Limit

Lower Control Limit

P

“this point indicates trouble”

Page 11: Part 4: Statistical Process Control Control Charts

The first control chart

21

…That all changed on May 16, 1924. Dr.

Shewhart's boss, George D. Edwards, recalled:

"Dr. Shewhart prepared a little memorandum only

about a page in length. About a third of that

page was given over to a simple diagram which

we would all recognize today as a

schematic control chart.”

That diagram, and the short text which preceded

and followed it, set forth all of the essential

principles and considerations which are involved

in what we know today as process quality

control.

Shewhart's work pointed out the importance of

reducing variation in a manufacturing process

and the understanding that continual process-

adjustment in reaction to non-conformance

actually increased variation and degraded

quality. Font: Wikipedia

Western Electric and Bell

Telephone Laboratories

1918-24

1891-1967

Page 12: Part 4: Statistical Process Control Control Charts

Tampering

https://www.youtube.com/watch?v=2VogtYRc9dA

Lessons from the Funnel Experiment

• Despite good intentions to fix a system and move it closer to the target of

the process, manipulations only made outcomes worse overtime;

• All systems have a certain level of inherent variability;

• Attempting to adjust a stable process will make things worse.

Font: Marco Santos

Page 13: Part 4: Statistical Process Control Control Charts

?

”A phenomenon will be said to be controlled when, through the use of past experience, we can

predict, at least within limits, how the phenomenon may be expected to vary in the future”

(W.A.Shewhart, 1931)

Management is prediction!

Font: Marco Santos

Page 14: Part 4: Statistical Process Control Control Charts

27

Chance causes vs. assignable causes of variation

The value of any quality charateristic varies from part to

part, from day to day…

The dispersion of the results is due to a variety of reasons acting randomly, that could be either known or unknown, whose

effect on the individual variable is small

Common causes are inherent to the processes (typically, their

elimination is expensive or difficult).

Together, their action determines that the quality characteristic

will behave in accordance with a stable probability distribution.

chance causes (or common)

Page 15: Part 4: Statistical Process Control Control Charts

28

Chance causes vs. assignable causes of variation

Definition of a Normal random variable:

“ A continuous random variable X follows (approximately) a Normal

distribution with parameters m and s if the sum of a large number of

effects caused by independent causes, in which the effect of each cause

is very small when compared to the sum of the effects of all the other

causes.”

Chance (or common) causes of variation

Page 16: Part 4: Statistical Process Control Control Charts

29

The variation may also reflect a situation of instability

This will happen if, in addition to common causes of variation,

other type of causes occur, possibly avoidable, which effects

acting together can cause a change in the parameters/shape of the

underlying distribution.

assignable (special) causes of variation

Chance causes vs. assignable causes of variation

Page 17: Part 4: Statistical Process Control Control Charts

30

Chance causes vs. assignable causes of variation

Page 18: Part 4: Statistical Process Control Control Charts

31

Shewhart Control Charts

Page 19: Part 4: Statistical Process Control Control Charts

Chance causes vs. assignable causes of variation

Appropriately reacting to the source of variation in a process provides the correct

economic balance between overreacting and under-reacting to variation from a process.

chance causes assignable causes

chance

causes

assignable

causes

Type I error

Change

(increases variability)

Focus on how to

fundamentally

change the process

Type II error

Sub-react

(lost prevention)

Focus in the research

of assignable causes

The

variability

is really

caused by…

How do you deal with variability...

Page 20: Part 4: Statistical Process Control Control Charts

33

Control Charts Objectives

A process that is operating with only common (chance)

causes of variation present is said to be in statistical

control.

A process that is operating in the presence of

assignable causes of variation is said to be out of

control.

O objectivo fundamental das Cartas de Controlo é o

de identificar Causas Assinaláveis de Variação,

balizando os momentos em que tais fenómenos

ocorrem.

The fundamental objective of Control Charts is to

identify assignable causes of variation, marking

the moments when they occur.

Douglas C. Mongomery “Introduction to Statistical Quality Control”

Page 21: Part 4: Statistical Process Control Control Charts

34

Control Charts Objectives

The control chart is the tool used to

distinguish between common and

assignable causes of variation;

The control limits represent the

expected variation due to common

causes.

Control Limits are often called “the voice of the process”

and used to identify assignable causes of variation

Page 22: Part 4: Statistical Process Control Control Charts

Common vs. Assignable causes of variation

Eliminate ACV Reduce CCV

A process with

assignable causes

A stable and

more capable

processA stable process

Out of control

Unpredictable

In statistical control

Predictable - stable

TUM, Slides on Quality Management

Page 23: Part 4: Statistical Process Control Control Charts

39

A brief introduction to process capability

Assume that a certain

quality characteristic follows

a normal distribution with a

mean m and a standard

deviation, s.The natural tolerance limits

of the process are:

UNTL = m + 3s

LNTL = m - 3s

O

Page 24: Part 4: Statistical Process Control Control Charts

41

A brief introduction to process capability

Whatever the distribution of the quality charateristic is, it can be

proved that

For k = 3, it turns out that the probability of having an

observation within the interval average ± 3s is at least 88.9%.

O

1k with

2

11

kkXkP msms

What if the distribution of the quality characteristic is non-normal?

Page 25: Part 4: Statistical Process Control Control Charts

A brief introduction to process capability

42

Specification Limits:

Often called “voice of the customer” and used to determine if the product

meets a customer requirement.

Page 26: Part 4: Statistical Process Control Control Charts

43

A brief introduction to process capability

A simple and quantitative way to express process capability is

through the Process Capability Ratio Cp:

USL: Upper Specification Limit

LSL: Lower Specification Limit

s: Standard deviation (s)

Meaning, the ability to produce products or provide services that

meet specifications defined by the customers' needs.

If Cp > 1, then a low number of nonconforming items will be produced;

If Cp = 1, assuming a normal distribution of QC, then we are producing

0,27% nonconforming;

If Cp < 1, then a large number of nonconforming items will be produced.

In practice, we look for a Cp > 1.3

s6

LSL-USL

pC

Page 27: Part 4: Statistical Process Control Control Charts

A brief introduction to process capability

45

Page 28: Part 4: Statistical Process Control Control Charts

46

Shewhart Control Charts

A machine produces cylindrical shafts at a rate of 4000 parts/hour.

Over a period of eight hours of uninterrupted work, a piece is

removed every four minutes (equivalent to 15 shafts per hour)

constituting a sample of 120 shafts.

The nominal (target) shaft diameter is 10.00 mm, and shafts whose

diameters do not deviate from the target value by more than

0.02mm (s = 0.01) are accepted by the client.

USL= 10.02

LSL = 9.98

N = 120

S = 0.01

Chance causes vs. assignable causes of variation

Exemplo 1:

Page 29: Part 4: Statistical Process Control Control Charts

47

Shewhart Control Charts

66.001.06

04.0

ˆ6

s

LSLUSLC

p

9.98

9.99

10.00

10.01

10.02 Limite Sup. Tolerância

1ª hora 2ª hora 3ª hora 4ª hora 5ª hora 6ª hora 7ª hora 8ª hora

(ii)

Limite Inf. Tolerância

10.00x

01.0s

10.029.98

0%

5%

10%

15%

20%

25%

30%

Fre

quên

cia

rela

tiva

LIT LST

66.0

01.06

04.0

6

s

Tpc

(i)

Chance causes vs. assignable causes of variation

s =0.01

USLLSL

Upper specification limit

Lower specification limit

Page 30: Part 4: Statistical Process Control Control Charts

48

Shewhart Control Charts

Conclusions:

The process is stable and (apparently) is operating with only

chance (or common) causes of variation;

The shaft variability is caused by a high number of potential

causes, such as: fluctuations in the temperature of the plant,

alterations in raw material, variations in the power supply of the

machine, vibrations in the support of the machine, etc.

Chance causes vs. assignable causes of variation

Page 31: Part 4: Statistical Process Control Control Charts

49

Shewhart Control Charts

Example 2:

assignable cause

10.00x

01.0s

10.029.98

0%

5%

10%

15%

20%

25%

30%

Fre

quên

cia

rela

tiva

LIT LST

66.0

01.06

04.0

6

s

Tpc

(i)

9.98

9.99

10.00

10.01

10.02 Limite Superior da Tolerância

1ª hora 2ª hora 3ª hora 4ª hora 5ª hora 6ª hora 7ª hora 8ª hora

(ii)

Limite Inferior da Tolerância

66.0ˆ6

s

LSLUSLC

p

Chance causes vs. assignable causes of variation

USLLSL

Upper specification limit

Lower specification limit

Page 32: Part 4: Statistical Process Control Control Charts

50

Shewhart Control Charts

Conclusions:

The process is not stabilised, suggesting the influence of

assignable causes of variation;

An analysis a posteriori identified the source of the problem: the

operator “adjusted”/fine-tuned the machine hourly, believing

that this procedure was effective to control the quality of the

final product.

Chance causes vs. assignable causes of variation

Page 33: Part 4: Statistical Process Control Control Charts

51

Shewhart Control Charts

Chance causes vs. assignable causes of variation

s = 0,01

10.029.98

x = 10,00

0.67

0.016

0.04

σ6 ˆ

USL-LSLC

p

LIT LST

9.98

9.99

10.00

10.01

10.02

1ªhora 2ªhora 3ªhora 4ªhora 5ªhora 6ªhora 7ªhora 8ªhora

LST

LIT

“In control”

9.98

9.99

10.00

10.01

10.02

1ª hora 2ª hora 3ª hora 4ª hora 5ª hora 6ª hora 7ª hora 8ª hora

LST

LIT

“Out-of-control”

USL

LSL

USL

LSL

USLLSL

Page 34: Part 4: Statistical Process Control Control Charts

52

Shewhart Control Charts

Variação associada a

Causa Especiais

Variação associada a

Causa Comuns

Fora do normal Normal

Perturbada Natural

Instável Estável

Não homogénea Homogénea

Mista Uma única distribuição

Errática Sem mudança

Aos “saltos” Constante

Imprevisível Previsível

Inconsistente Consistente

Pouco comum Habitual

Diferente Semelhante

Importante Pouco importante

Significativa Não significativa

Chance causes vs. assignable causes of variation

Page 35: Part 4: Statistical Process Control Control Charts

Shewhart Control Charts

In a process in statistical control with unknown parameters of the

distribution of the quality charateristic, the probability of having values

within this interval

deviation) standard valueexpected ()( sm k

is constant

Control Charts Basic Principles

Page 36: Part 4: Statistical Process Control Control Charts

Shewhart Control Charts

In a normal distribution:

Control Charts Basic Principles

Page 37: Part 4: Statistical Process Control Control Charts

Shewhart Control Charts

Control Charts Basic Principles

sample of size n

H0: m = m0

H1: m ≠ m0

xm = m0

x N(m0,sx)

xm = m0

ET = x N(m0,sx)

Don´t Reject H0 Reject H0Reject H0

a/2a/2

m0 + ksxm0 - ksx

Page 38: Part 4: Statistical Process Control Control Charts

56

Shewhart Control Charts

Control Charts Basic Principles

x3

x4

x1

x2

m0 + ksx

m0 - ksx

m0

H0 Rejection region

H0 Rejection region

1ª sample 2ª sample 3ª sample 4ª sample

H0: m = m0

H1: m ≠ m0

xm = m0

Don´t Reject H0 Reject H0Reject H0

a/2a/2

Shewhart Control Charts

Control Limits

Page 39: Part 4: Statistical Process Control Control Charts

57

Shewhart Control Charts

Control limits

Usual significance levels (typically a = 5%) would lead, in

most industrial environments, to an excessive number of

false alarms.

Therefore, Shewhart has proposed k = 3, which leads

(assuming normal distributions) to an a = 0,27%.

Page 40: Part 4: Statistical Process Control Control Charts

58

Shewhart Control Charts

Generic assumptions

Lower control limit

Upper control limit

+ 3s

- 3s

m0 → Central line

1 2 3 4 5 6 7 8 9 10 11 12 13

h

l

Usual scale lh 6LCL-UCL

Page 41: Part 4: Statistical Process Control Control Charts

59

Shewhart Control Charts

Statistical Basis of the control chart

Relationship between hypothesis testing and control charts:

Example:

o We have a process that we assume the true process mean

is m = 1.5 and the process standard deviation is s = 0.15.

o Samples of size 5 are taken giving a standard deviation of the

sample average, , as x

0671.05

15.0

nx

ss

Page 42: Part 4: Statistical Process Control Control Charts

60

Shewhart Control Charts

Statistical Basis of the control chart

Example:

Control limits are set at 3 standard deviations from the

mean;

The 3-Sigma Control Limits are:

UCL = 1.5 + 3(0.0671) = 1.7013

CL = 1.5

LCL = 1.5 - 3(0.0671) = 1.2987

Page 43: Part 4: Statistical Process Control Control Charts

61

Shewhart Control Charts

Statistical Basis of the control chart

Choosing the control limits is equivalent to setting up the

critical region for testing hypothesis:

H0: m = 1.5

H1: m 1.5

Douglas C. Mongomery “Introduction to Statistical Quality Control”

Page 44: Part 4: Statistical Process Control Control Charts

62

Shewhart Control Charts

Patterns for identifying assignable causes

If the control charts consist of a time series of

sample statistics, will the analysis of sequences of

observations enhance their diagnostic ability?

Page 45: Part 4: Statistical Process Control Control Charts

64

Shewhart Control Charts

(ii) Abnormal sequence of points

Patterns for identifying assignable causes

Tempo

A sequence of 8 points

bellow the central line

One point beyond control limits

Tempo

Linha +2s

Linha -2s

Linha -1s

Linha +1s

Four out of five points

bellow the line -1sigma

Two out of three points

above the line +2sigma

Page 46: Part 4: Statistical Process Control Control Charts

65

Shewhart Control Charts

(iii) - a) Trends – seven or more points going up or down

Tempo

Linha +2sigma

Linha -2sigma

(iii) - b) Big oscillations – between three consecutive points, one is

between the UCL and the line +2sigma and the other is between

the LCL and the line -2sigma

Tempo

Tendência ascendente

não-linear

Tendência descendente (sete

pontos sucessivos a descer)

Patterns for identifying assignable causes

Page 47: Part 4: Statistical Process Control Control Charts

66

Shewhart Control Charts

(iii) - c) Proximity to the center line – almost all points within the limits 1.5sigma

(indicating a probable mixture of populations with different expected values);

(iii) - b) Cyclic patterns: seasonality

Tempo

Linha +1.5sigma

Linha -1.5sigma

Tempo

Periodicidade

Patterns for identifying assignable causes

Line

Line

Page 48: Part 4: Statistical Process Control Control Charts

67

Shewhart Control Charts

Patterns for identifying assignable causes

Page 49: Part 4: Statistical Process Control Control Charts

68

Shewhart Control Charts

Patterns for identifying assignable causes

Page 50: Part 4: Statistical Process Control Control Charts

69

Shewhart Control Charts

Patterns for identifying assignable causes

Page 51: Part 4: Statistical Process Control Control Charts

70

Shewhart Control Charts

Patterns for identifying assignable causes

Although these pattern tests allow the Shewhart control charts

to be more sensitive, if some tests are used together, then

the probability of a type I error (a may become too large,

thus accentuating the risk of false alarm.

These effect will be even bigger when more rules

are considered simultaneously (it is common to find

recommendations of 6 or 8!)

E

Page 52: Part 4: Statistical Process Control Control Charts

71

Shewhart Control Charts

Minitab: Patterns for identifying assignable causes

Page 53: Part 4: Statistical Process Control Control Charts

72

Shewhart Control Charts

(i) process out-of-control and producing defective units;

(ii) process out-of-control and not producing defects units;

(iii) Process in control and producing defective units;

(iv) process in control and not producing defects units.

Limite Superior de Controlo

Limite Inferior de Controlo

Limite Superior da Tolerância

Limite Inferior da Tolerância

1x

Defeituosa

Amostra 1

Amostra 2

2x

Specification Limits are set up taking into account customers’s requests and the

characteristics of the productive process.

Control Limits are calculated taking into account the variability of the process.

Difference between specification limits and control limits

Upper Specification Limit

Lower Specification Limit

Upper Control Limit

Lower Control Limit

sample 1

sample 2

defect

Page 54: Part 4: Statistical Process Control Control Charts

73

Relationship between Natural Tolerance Limits, Control

Limits ans Specification Limits

Page 55: Part 4: Statistical Process Control Control Charts

74

Shewhart Control Charts

The Warning Limits are specified at two-sigma;

If one or more points fall between the warning

limits and the control limits, or very close to

the warning limit, we should be suspicious that

the process may not be operating properly;

Advantage: the use of warning limits can

increase the sensitivity of the control chart;

Disadvantage: the use of warning limits can

also result in an increased risk of false alarms.

Warning Limits

Page 56: Part 4: Statistical Process Control Control Charts

78

Shewhart Control Charts

Guidelines:

In designing a control chart, both the sample size to be

selected and the frequency of selection must be specified;

Larger samples make it easier to detect small shifts in the

process;

Current practice tends to favor smaller, more frequent

samples.

Sample Size and Sampling Frequency

Page 57: Part 4: Statistical Process Control Control Charts

79

Page 58: Part 4: Statistical Process Control Control Charts

80

Shewhart Control Charts

Guidelines:

The collection of initial data must be preceded by a careful

study of the process in order to identify potential assignable

causes;

Pilot samples (K samples of size N) for the initial set-up of the

control limits must be collected during time periods when it is

assured that assignable potential causes are absent.

Guidelines for the design of the control chart

It is essential to update the control limits regularly.

Page 59: Part 4: Statistical Process Control Control Charts

81

Shewhart Control Charts

Example:Frozen orange juice concentrate is packed in 6-oz cardboard cans. These cans are

formed on a machine by spinning them from cardboard stock and attaching a metal

bottom panel. By inspection of a can, we may determine whether, when filled, it

could possibly leak either on the side seam or around the bottom joint. Such a

nonconforming can has an improper seal on either the side seam or the bottom

panel.

Set up a control chart to improve the fraction of nonconforming cans produced by

this machine.

Guidelines for the design of the control chart

Page 60: Part 4: Statistical Process Control Control Charts

82

Shewhart Control Charts

The power of the test is reduced, thus reducing the sensitivity of

the control chart.

Sensitivity of the control chart

By doing k = 3, it decreases a but increases up b !

What are the practical consequences of such a rule?

Page 61: Part 4: Statistical Process Control Control Charts

83

Shewhart Control Charts

Sensitivity of the control chart

By doing k = 3, it

decreases a but

increases up b !

b 1b

a

Page 62: Part 4: Statistical Process Control Control Charts

84

Shewhart Control Charts

How to evaluate the ability of Shewhart control charts

to detect out-of-control situations?

The capacity of a control chart for detecting an out-of-control

situation can be measured by the power of the test (1-b).

Sensitivity of the control chart

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85

Shewhart Control Charts

Sensitivity of the control chart – Operating Characteristic Curves (OCC)

H0: m= 1200

H1: m≠ 1200

s =300

N=100

a= 0,05

Page 64: Part 4: Statistical Process Control Control Charts

86

Shewhart Control Charts

Example: Control Charts for Averages

.2

34

333 0000

sm

sm

smsm

NX

Assuming X Normal (m0, s), and N = 4

LC = 0mm X

s 5.0

H0: m = m0

H1: m m0

Sensitivity of the control chart – Operating Characteristic Curves (OCC)

If the expected value suffers a deviation of between the time elapsed

between the first and second sample, what is the probability of having an out-of-control

signal in the sample following the shift?

Page 65: Part 4: Statistical Process Control Control Charts

87

Shewhart Control Charts

2)()3()( 000

x

xx

x

x

x

x LSCxz

s

smsm

s

sm

s

m

0228.02zP )-(1 Pd b

Sensitivity of the control chart – Operating Characteristic Curves (OCC)

Pd = (1- b)

m1 = m0 +

LIC

LSC

a/2

m0

H0: m = m0

H1: m m0

Page 66: Part 4: Statistical Process Control Control Charts

88

Page 67: Part 4: Statistical Process Control Control Charts

89

Shewhart Control Charts

In Statitical Process Control it is common to talk about the Operating

Characteristic Curves (OCC), a curve that reflects the ability (or sensitivity)

of the chart to detect deviations from H0.

Sensitivity of the control chart – Operating Characteristic Curves (OCC)

H0: m = m0

(H1: m = m0 ± sx)

0.2

0.4

0.6

0.8

1.0

b

%

a = 0.27%

0.0-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0

(50.0%)

(84.1%)

(97.7%)

OCC for a control chart for averages with N = 4

The OCC represents the

probability that a point is

within the control limits, as a

function of each possible

value of the parameter of the

underlying distribution.

Page 68: Part 4: Statistical Process Control Control Charts

90

Shewhart Control Charts

Sensitivity of the control chart – Operating Characteristic Curves (OCC)

H0: m= 1200

H1: m≠ 1200

s =300

N=100

a= 0,0027

Page 69: Part 4: Statistical Process Control Control Charts

91

Shewhart Control Charts

ARL- Average Run Length

For a certain deviation of the value of the parameter in relation to H0,

the average run length is the expected value of the number of samples that we

need to gather until having an out-of-control signal.

Pd = probability of detecting the deviation on the 1º sample

(1- Pd). Pd = probability of detecting the deviation on the 2º sample …

= probability of detecting the deviation on the K sample

d

d

1k

d1k

d

1k

dP

1PP1kPP1E

dP

1ARL

Sensitivity of the control chart

1 − 𝑃𝑑𝑘−1. 𝑃𝑑

Page 70: Part 4: Statistical Process Control Control Charts

92

Shewhart Control Charts

What is the ARL corresponding to a deviation of ?Xs

Pd = 0.0228

86.430228.0

1

P

1ARL

d

meaning that, if the expected value of the x variable suffers a shift

of 0.5s, you need 44 samples, in average, in order to detect this

shift (N=4).

Sensitivity of the control chart

Page 71: Part 4: Statistical Process Control Control Charts

93

Shewhart Control Charts

ARL for a control chart for averages with N = 4

= 0

ARL = 6.3s 0.1

37.3700027.0

1110

ad

HP

ARL

Sensitivity of the control chart

AR

L

0

5

10

15

20

25

30

35

40

45

50

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0

H0: m = m0

(H1: m = m0 ± sx)

(6.3)

(43.9)

Page 72: Part 4: Statistical Process Control Control Charts

94

Shewhart Control Charts

Sensitivity of the control chart

if the process is in statistical control, then:

control-of-out point 1 P

1ARL

a

1ARL0

Page 73: Part 4: Statistical Process Control Control Charts

95

Shewhart Control Charts

Measures of sensitivity of the control chart

if the process is out-of-control, then:

b

1

1ARL1

Page 74: Part 4: Statistical Process Control Control Charts

96

Control Charts for Attributes

Measures of sensitivity of the control chart

If ARL is considered unsatisfactory, then action must be taken to

reduce the out-of-control ARL1.

Possible alternatives:

1. Increasing the sample size would result in a smaller value of b and a

shorter out-of-control ARL1;

2. Reduce the interval between samples (e.g. if we are currently

sampling every hour, it will take about seven hours, on the average, to

detect the shift. If we take the sample every half hour, it will require

only three and a half hours, on the average, to detect the shift);

3. Use a control chart that is more responsive to small shifts, such as the

cumulative sum charts (CUSUM).

Page 75: Part 4: Statistical Process Control Control Charts

97

Types of Control Charts

Mean and variation (moving range)

( 𝒙; Rmoving)

Number nonconforming (Np)(Número de defeituosas)

Number of nonconformities (c)(Número de defeitos)

Number of nonconformities per unit (u)(average number of non-conformities per unit |

número de defeitos por unidade)

Fraction nonconforming (p)(Proporção de defeituosas)

Mean and variation (std deviation)

( 𝒙; s)N < 8

Mean and variation (range)

( 𝒙; R)

Control

Charts

Page 76: Part 4: Statistical Process Control Control Charts

Types of Control Charts

98

Page 77: Part 4: Statistical Process Control Control Charts

99

Types of Control Charts

Np and p control charts

Control charts for counting one random variable that only has two

possible outcomes:

defective (non conforming) or nondefective (conforming)

Sample size (N) constant Np Chart (number of defective items )

Sample size (N) variable p Chart (fraction nonconforming)

Y (number of defectives in a subgroup ) Binomial (N, p)

H0: p = p0

H1: p p0

Control Charts for Attributes

Page 78: Part 4: Statistical Process Control Control Charts

100

Types of Control Charts

c and u control charts

Control charts for variables dealing with the number of defects or

nonconformities for a given “opportunity area” (piece, component,

time or space unit)

“opportunity area” constant c control chart (nº of defects)

“opportunity area” variable u control chart (nº of defects/unity)

Y (number of occurrences per opportunity area) Poisson (l)

H0: l = l0

H1: l l0

Control Charts for Attributes

Page 79: Part 4: Statistical Process Control Control Charts

101

Types of Control Charts

Control charts and

Sample size N >8 control chart best estimator of s is

the sample standard deviation

Sample size N 8 control chart best estimator of s is

the sample range

H0: s = s0

H1: s s0

)( s;x )( R;x

)( s;x

H0: m = m0

H1: m m0

Assumptions of the and control charts: )( s;x

x ),( 0 xN sm

),( 0 smN

Control Charts for Variables

x

)( R;x

)( R;x

Page 80: Part 4: Statistical Process Control Control Charts

102

Types of Control Charts

Control chart for individual measurements

Moving range: a statistic consisting in the absolute difference between

successive values of the variable under analysis

H0: s = s0

H1: s s0

H0: m = m0

H1: m m0.

)(moving

R;x

k

iii xx

kAM

21

1

1

Individual values control chart (x)

Moving range control chart (MR)

Control Charts for Variables

Page 81: Part 4: Statistical Process Control Control Charts

106

Types of Control Charts

Mean and variation (moving range)

( 𝒙; Rmoving)

Number nonconforming (Np)(Número de defeituosas)

Number of nonconformities (c)(Número de defeitos)

Number of nonconformities per unit (u)(average number of non-conformities per unit |

número de defeitos por unidade)

Fraction nonconforming (p)(Proporção de defeituosas)

Mean and variation (std deviation)

( 𝒙; s)N < 8

Mean and variation (range)

( 𝒙; R)

Control

Charts

Page 82: Part 4: Statistical Process Control Control Charts

107

Control Charts for Attributes

Control Chart for number nonconforming (Np) and

fraction nonconforming (p)

If Y represent the number of time a certain event occurs over N identical

experiments, and if

(i) correspond to each experience only two possible outcomes

(“sucess” and “failure”),

(ii) if the probability of each outcome is constant for all the N experiences and if,

(iii) the results of each experiment are independent

then Y (number of “sucesses”) B(N, p), with parameters

pN m )1( ppN s

Page 83: Part 4: Statistical Process Control Control Charts

108

Control Charts for Attributes

Number nonconforming control chart – Np Chart

Standard Given

For standard given

)1(3 ppNpN

pN

)1(3 ppNpN

UCL =

CL =

LCL =

Page 84: Part 4: Statistical Process Control Control Charts

109

Control Charts for Attributes

Fraction Nonconforming Control Chart– p Chart

Standard Given

For standard given

Assuming that the proportion of nonconforming (defectives) in a

subgroup = Y/N has the parameters:

UCL =

CL =

LCL =

ppNN

1

mN

ppppN

N

)1()1(

12

2 s

then,Nppp )1(3

p

Nppp )1(3

Page 85: Part 4: Statistical Process Control Control Charts

110

Control Charts for Attributes

In a boiler assembly line, 200 boilers

are produced per day. After assembly,

the boilers undergo through a large set

of tests and verifications. When a

faulty boiler is detected, it is diverted

from the assembly line, entering a

particular station for repairs.

The responsible of the process wants

to evaluate if some recent changes in

the process had any impact on the

fraction nonconfoming (or the number

of defects) of the process.

The number of nonconforming boilers

was steady for a long time, being of 5

boilers a day.

Np and p chart (standard given – N constant sample size) : Example

Daily boiler rejections in the final tests (N = 200)

Day Number of

Defects

Percentage of

defects

Day Number of

Defects

Percentage of

defects

1 7 0.035 16 4 0.020

2 7 0.035 17 6 0.030

3 6 0.030 18 7 0.010

4 2 0.010 19 5 0.025

5 11 0.055 20 1 0.005

6 2 0.010 21 2 0.010

7 4 0.020 22 5 0.025

8 3 0.015 23 10 0.050

9 4 0.020 24 5 0.025

10 1 0.005 25 2 0.010

11 3 0.015 26 4 0.020

12 7 0.035 27 2 0.010

13 7 0.035 28 3 0.015

14 4 0.020 29 8 0.040

15 5 0.025 30 4 0.020

Daily boiler rejections 30 days AFTER process changes

Page 86: Part 4: Statistical Process Control Control Charts

111

Control Charts for Attributes

Y (number of rejected boilers) B(N, p) assuming that Np= 5

Np and p chart (standard given – N constant sample size) : Example

208.2975.0025.0200)1( ppNs

624.11208.235)1(3 ppNpN

5 pN

0624.1208.235)1(3 ppNpN

UCL =

CL =

LCL =

Dia

0

2

4

6

8

10

12

5 10 15 20 25 30

LC = 5

LSC = 11.624

LSC = 0

UCL

LCL

CL

Day

There is no evidence that the changes performed in the process had any impact in the boiler’s defect rate

Page 87: Part 4: Statistical Process Control Control Charts

112

Control Charts for Attributes

Y (number of rejected boilers) B(N, p) assuming that Np= 5

UCL = 11,624

CL = 5

LCL = 0

Np and p chart (standard given – N constant sample size) : Example

There is no evidence that the changes performed in the process had any impact in the boiler’s defect rate

28252219161310741

12

10

8

6

4

2

0

Sample

Sam

ple

Co

un

t

__NP=5

UCL=11,62

LCL=0

Np Chart Number Nonconforming Boilers

An estimated historical parameter is used in the calculations.

Page 88: Part 4: Statistical Process Control Control Charts

113

Control Charts for Attributes

p control chart - fraction nonconforming (proporção de defeituosas)

Np and p chart (standard given – N constant sample size) : Example

LSC =

LC =

LIC =

0581.0200975.0025.03025.0)1(3 Nppp

025.0p

00081.0200975.0025.0305.0)1(3 Nppp

Dia

0.00

0.01

0.02

0.03

0.04

0.05

0.06

5 10 15 20 25 30

LC = 0.025

LSC = 0.0581

LIC = 0

UCL

CL

LCL

Day

There is no evidence that the changes performed in the process had any impact in the boiler’s defect rate

Page 89: Part 4: Statistical Process Control Control Charts

114

Control Charts for Attributes

p control chart - fraction nonconforming (proporção de defeituosas)

Np and p chart (standard given – N constant sample size) : Example

UCL = 0.058

CL = 0.025

LCL = 0

There is no evidence that the changes perormed in the process had any impact in the boiler’s defect rate

28252219161310741

0,06

0,05

0,04

0,03

0,02

0,01

0,00

Sample

Pro

po

rtio

n

_P=0,025

UCL=0,05812

LCL=0

p Chart for Fraction Nonconforming Boilers

An estimated historical parameter is used in the calculations.

Page 90: Part 4: Statistical Process Control Control Charts

115

Control Charts for Attributes

Number nonconforming control chart – Np Chart

No Standard Given

No Standard Given

items ofnumber total

items defective ofnumber totalˆ

1

1

K

k

k

K

k

k

N

y

p

)ˆ1(ˆ3ˆ ppNpN

pN ˆ

)ˆ1(ˆ3ˆ ppNpN

UCL =

CL =

LCL =

Page 91: Part 4: Statistical Process Control Control Charts

116

Control Charts for Attributes

Fraction Nonconforming Control Chart – standard given vs. no standard given

28252219161310741

0,06

0,05

0,04

0,03

0,02

0,01

0,00

Sample

Pro

port

ion

_P=0,025

UCL=0,05812

LCL=0

28252219161310741

0,06

0,05

0,04

0,03

0,02

0,01

0,00

Sample

Pro

port

ion

_P=0,0235

UCL=0,05563

LCL=0

p Chart for Fraction Nonconforming Boilers

An estimated historical parameter is used in the calculations.

p Chart for Fraction Nonconforming Boilers

Page 92: Part 4: Statistical Process Control Control Charts

117

Control Charts for Attributes

Np and p chart: Control Limits

Guidelines for the definition of control limits:

Number of samples

N ( sample size) should be such that, preferably and, at

least ;

In situations where p is very low (or very high), the previous rule

might lead to impossible sample sizes

(ex. if p = 1 ppm the size of the sample would be N= 1 million).

But, although you might follow all the rules, you can never be sure that the

process was in control while the samples were gathered.

3025 k

1ˆ pN

5ˆ pN

Page 93: Part 4: Statistical Process Control Control Charts

118

Control Charts for Attributes

Np and p chart: Control Limits

Basic rules for the elimination of “outliers”:

If control limits for current or future production are to be meaningful, they

must be based on data from a process that is in control. Therefore, when

the hypothesis of past control is rejected, it is necessary to revise the trial

control limits, examining each of the out-of-control points, looking for an

assignable cause;

Any points that exceed the trial control limits should be investigated. If

assignable causes for these points are discovered, they should be discarded

and new trial control limits determined;

Whenever possible, the number of discarded samples should not exceed

more than 10% of the total number of samples.

The computed control limits based on the preliminar samples are known as

Trial Control Limits – Phase I of control chart usage.

Page 94: Part 4: Statistical Process Control Control Charts

119

Types of Control Charts

Mean and variation (moving range)

( 𝒙; Rmoving)

Number nonconforming (Np)(Número de defeituosas)

Number of nonconformities (c)(Número de defeitos)

Number of nonconformities per unit (u)(average number of non-conformities per unit |

número de defeitos por unidade)

Fraction nonconforming (p)(Proporção de defeituosas)

Mean and variation (std deviation)

( 𝒙; s)N < 8

Mean and variation (range)

( 𝒙; R)

Control

Charts

Page 95: Part 4: Statistical Process Control Control Charts

120

Control Charts for Attributes

Number of Nonconformities control chart – c chart

There are many instances where an item will contain nonconformities

but the item itself is not classified as nonconforming (or defective)

(nonconformities vs nonconforming ).

It is often important to construct control charts for the total number

of nonconformities or the average number of nonconformities for a

given “area of opportunity”. The inspection unit must be the same for

each unit.

The number of nonconformities in a given area of opportunity can be

modeled by the Poisson distribution.

Page 96: Part 4: Statistical Process Control Control Charts

121

Control Charts for Attributes

Control Charts for Nonconformities – c and u charts

Nonconformities vs Nonconforming

Page 97: Part 4: Statistical Process Control Control Charts

122

Control Charts for Attributes

C: number of occurences that a given phenomenon occurs in a continuous

interval – “area of opportunity” or inspection unit (volume, area, lenght or

space; time intervals or any other entity considered a continuum)

The Poisson distribution has the following characteristics:

It is a discrete distribution.

It describes rare events.

Each occurrence is independent of the other occurrences.

It describes discrete occurrences over a continuum or interval.

The occurrences in each interval can range from zero to infinity.

The expected number of occurrences must hold constant throughout the experiment.

C (number of occurences in an area of opportunity) Poisson (l)

l: average number of nonconformities by inspection unit

Control Charts for Nonconformities – c and u charts

Page 98: Part 4: Statistical Process Control Control Charts

123

Control Charts for Attributes

Properties of the Poisson distribution:

Any linear combination of independent poisson variables still follows a poisson

distribution (ex. total number of painting defects, i.e. non-conformities);

Certain types of binomial distribution problems can be approximated by using

the Poisson distribution. Binomial problems with large sample sizes and small

values of p, which then generate rare events, are potential candidates for use

of the Poisson distribution. As a rule of thumb:

Control Charts for Nonconformities – c and u charts

20N 7 pN

UCL = CC mm 3

CL = Cm

LCL = CC mm 3

cXVar

cXE

l

l

)(

)(

Page 99: Part 4: Statistical Process Control Control Charts

124

Control Charts for Attributes

Control Charts for Variables

Standard Given No Standard Given

Number of Nonconformities control chart | constant sample size –> c chart

c3cLCL

cCL

c3cUCL

c3cLCL

cCL

c3cUCL

Page 100: Part 4: Statistical Process Control Control Charts

125

Control Chart for Nonconformities: no standard given

The table bellow represents the number of nonconformities observed in 26

successive samples of 100 printed circuit boards. For reasons of convenience, the

inspection unit is defined as 100 boards.

Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley

Page 101: Part 4: Statistical Process Control Control Charts

126

Control Chart for Nonconformities: no standard given

The table bellow represents the number of nonconformities observed in 26

successive samples of 100 printed circuit boards. For reasons of convenience, the

inspection unit is defined as 100 boards.

C (number of nonconformities by inspection unit) Poisson (c)

c: average number of nonconformities by inspection unit

Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley

Page 102: Part 4: Statistical Process Control Control Charts

127

Control Chart for Nonconformities: no standard given

48.619.85385.193

85.19 lineCenter

33.2219.85319.85c3

85.1926

516

ccLCL

c

cUCL

c

Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley

Page 103: Part 4: Statistical Process Control Control Charts

128

Control Chart for Nonconformities: no standard given

36.619.67367.193

67.19 lineCenter

32.9719.67319.67c3

67.1924

472

ccLCL

c

cUCL

c

Twenty new samples, each consisting in one inspection unit (i.e. 100 boards) are

subsequently collected.

Revised Control Limits

Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley

Page 104: Part 4: Statistical Process Control Control Charts

129

Control Chart for Nonconformities: no standard given

By analysing the nonconformities by type, we can often gain insight into their cause.

Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley

Page 105: Part 4: Statistical Process Control Control Charts

Control Chart for Nonconformities: no standard given

Further Analysis of Nonconformities (defect data for 500 boards)

60% of the total number of defects is due to two defect types: solder insufficiency

and solder cold joints Problems with the wave soldering process!

As the process manufactures several different types of printed circuit boards, it may be

helpful to examine the occurrence of defect type by type of printed circuit board.

Page 106: Part 4: Statistical Process Control Control Charts

Control Chart for Nonconformities: no standard given

Further Analysis of Nonconformities (defect data for 500 boards)

As the process manufactures several different types of printed circuit boards, it may be

helpful to examine the occurrence of defect type by type of printed circuit board.

Page 107: Part 4: Statistical Process Control Control Charts

Control Chart for Nonconformities: no standard given

Further Analysis of Nonconformities (defect data for 500 boards)

As the process manufactures several different types of printed circuit boards, it may be

helpful to examine the occurrence of defect type by type of printed circuit board.

Page 108: Part 4: Statistical Process Control Control Charts

Control Chart for Nonconformities: no standard given

Further Analysis of Nonconformities

This tool is useful in focusing the attention of operators, manufacturing

engineers and managers on quality problems. Developing a good cause-and-effect

diagram usually advances the level of technological understanding of

the process.Douglas C. Montgomery “Introduction to Statistical Quality Control”, Wiley

Page 109: Part 4: Statistical Process Control Control Charts

134

Types of Control Charts

Mean and variation (moving range)

( 𝒙; Rmoving)

Number nonconforming (Np)(Número de defeituosas)

Number of nonconformities (c)(Número de defeitos)

Number of nonconformities per unit (u)(average number of non-conformities per unit |

número de defeitos por unidade)

Fraction nonconforming (p)(Proporção de defeituosas)

Mean and variation (std deviation)

( 𝒙; s)N < 8

Mean and variation (range)

( 𝒙; R)

Control

Charts

Page 110: Part 4: Statistical Process Control Control Charts

135

Control Charts for Attributes

UCL =

CL =

LCL =

uu sm 3

um

uu sm 3

Number of Nonconformities control chart | variable sample size –> u chart

Nk : number of “areas of opportunity” or inspection units in a given sample k (or

the size of the sample k in multiples of the same unit). For example, number

of square meters, of equal pieces, production hours;

yk : number of nonconformities (defects) present in the Nk “areas of opportunity”

of sample K;

uk : average number of nonconformities (defects) per unit, in the Nk “areas of

opportunity” of sample K ( );

mu : expected value of the average number of nonconformities (defects) per

inspection unit

su : standard deviation of the average number of nonconformities (defects) per

inspection unit

kkk Nyu

Page 111: Part 4: Statistical Process Control Control Charts

136

If we assume that follows a Poisson distribution, thenkkk uNy

ukyy N msm 2

kkk Nyu

2

2

2 1)( y

k

ukN

uVar ss k

uuk

k

uN

NN

mms

2

2 1

UCL =

CL =

LCL =

um

kuu Nmm 3

kuu Nmm 3

K

k

K

k

k

u

kN

y

1

1m̂

kuu Nmm ˆ3ˆ

um̂

kuu Nmm ˆ3ˆ

UCL =

CL =

LCL=

No standard given

k

uu

N

ms

Number of Nonconformities control chart | variable sample size –> u chart

Control Charts for Attributes

Page 112: Part 4: Statistical Process Control Control Charts

138

Numa fábrica têxtil que fabrica tecidos de várias texturas e padrões, foi instituído um

processo de amostragem que analisa a ocorrência de defeitos por 50 m2 de tecido. Os

dados de 10 rolos encontram-se na tabela seguinte. Com estes dados pretende-se construir

uma carta de controlo do número de não-conformidades por unidade de inspeção.

Number of Nonconformities control chart | variable sample size –> u chart

Control Charts for Attributes

K

k

K

k

k

u

kN

y

1

1m̂Número

do rolo Nº de m2

Nº Total de não-

conformidades

1 500 14

2 400 12

3 650 20

4 500 11

5 475 7

6 500 10

7 600 21

8 525 16

9 600 19

10 625 23

kuu Nmm ˆ3ˆ

um̂

kuu Nmm ˆ3ˆ

LSC =

LC =

LIC =

153

Nº de unidades

de inspecção

por rolo

10.0

8.0

13.0

10.0

9.5

10.0

12.0

10.5

12.0

12.5

107.5

42.15.107

153ˆ um

UCL LCL

2.555 0.291

2.689 0.158

2.416 0.431

2.555 0.291

2.584 0.262

2.555 0.291

2.456 0.390

2.528 0.319

2.456 0.390

2.436 0.411

k

kk

N

yu

Nº de não-conformidades

por unidade de inspecção

1.40

1.50

1.54

1.10

0.74

1.00

1.75

1.52

1.58

1.84

Page 113: Part 4: Statistical Process Control Control Charts

Number of Nonconformities control chart | variable sample size –> u chart

Control Charts for Attributes

191715131197531

1,4

1,2

1,0

0,8

0,6

0,4

0,2

0,0

Sample

Sam

ple

Co

un

t P

er

Un

it

_U=0,701

UCL=1,249

LCL=0,153

U Chart of Total Number of Imperfections

Tests performed with unequal sample sizes

Page 114: Part 4: Statistical Process Control Control Charts

140

Control Charts for Attributes

Measures of sensitivity of the control chart

The sensitivity of the control charts is associated with its ability to

detect shifts in the process parameters;

The power of the tests with the binomial and Poisson distribution is

low;

Control Charts are not the best tool to control processes with low

process capability.

Page 115: Part 4: Statistical Process Control Control Charts

141

Control Charts for Attributes

Measures of sensitivity of the control chart

The Operating-Characteristic Function

It is a graphical display of the probability of incorrectly accepting the

hypothesis of statistical control (i.e., a type II or b-error) against the process

fraction nonconforming.

The OCC provides a measure of the sensitivity of the control chart.

i.e. for N=50 and =0.20, UCL=0.3697 and LCL=0.0303

𝑝 =𝐷

𝑛D is a binomial random variable with parameters n and p

Note that when the LCL is negative, the second term on the right-hand side of

equation should be dropped.

p

pp

pp

LCLDPUCLDP

LCLpPUCLpP

ˆˆb

Page 116: Part 4: Statistical Process Control Control Charts

142

Control Charts for Attributes

Measures of sensitivity of the control chart

The Operating-Characteristic Function

pp

LCLDPUCLDP b

Introduction to Statistical Quality Control, Douglas C. Montgomery.

485.180.369750UCLN

515.10.030350LCLN

Page 117: Part 4: Statistical Process Control Control Charts

143

Control Charts for Attributes

Measures of sensitivity of the control chart

The Operating-Characteristic Curve

For N=50 and =0.20, UCL=0.3697 and LCL=0.0303p

Page 118: Part 4: Statistical Process Control Control Charts

144

Control Charts for Attributes

Measures of sensitivity of the control chart

Calculate the Average Run Lenght (ARL) for the fraction nonconforming

control chart (p chart)

if the process is in control, then:

controlof out plots point sample P

1ARL

a

1ARL0

Page 119: Part 4: Statistical Process Control Control Charts

145

Control Charts for Attributes

Measures of sensitivity of the control chart

if the process is in control, then:

a

1ARL0

UCLLCL

Page 120: Part 4: Statistical Process Control Control Charts

146

Control Charts for Attributes

Measures of sensitivity of the control chart

Calculate the Average Run Lenght (ARL) for the fraction

nonconforming control chart (p chart)

if it is out of control, then:

b

1

1ARL1

Page 121: Part 4: Statistical Process Control Control Charts

147

Control Charts for Attributes

Measures of sensitivity of the control chart

Example:

Suppose that the process shifts out of control to p = 0.3

(in control p= 𝒑 =0,2)

From the table b= 0.8594

b

1

1ARL

1

7

0.8594-1

1ARL1

Page 122: Part 4: Statistical Process Control Control Charts

148

Control Charts for Attributes

Measures of sensitivity of the control chart

Example:

Suppose that the process shifts out of control to p = 0.3

(in control p= 𝒑 =0,2)|From the table b= 0.8594

Page 123: Part 4: Statistical Process Control Control Charts

149

Control Charts for Attributes

Measures of sensitivity of the control chart

Suppose that the process shifts out of control to p = 0.3

b

1

1ARL

1 7

0.8594-1

1ARL1

Page 124: Part 4: Statistical Process Control Control Charts

150

Control Charts for Attributes

Measures of sensitivity of the control chart

If ARL is considered unsatisfactory, then action must be taken to

reduce the out-of-control ARL1.

Possible alternatives:

1. Increasing the sample size would result in a smaller value of b and a

shorter out-of-control ARL1;

2. Reduce the interval between samples (e.g. if we are currently

sampling every hour, it will take about seven hours, on the average, to

detect the shift. If we take the sample every half hour, it will require

only three and a half hours, on the average, to detect the shift);

3. Use a control chart that is more responsive to small shifts, such as the

cumulative sum charts (CUSUM).

Page 125: Part 4: Statistical Process Control Control Charts

151

Control Charts for Attributes

Measures of sensitivity of the control chart

1. Increasing the sample size would result in a smaller value of b and a

shorter out-of-control ARL1;

Page 126: Part 4: Statistical Process Control Control Charts

152

Types of Control Charts

Mean and variation (moving range)

( 𝒙; Rmoving)

Number nonconforming (Np)(Número de defeituosas)

Number of nonconformities (c)(Número de defeitos)

Number of nonconformities per unit (u)(average number of non-conformities per unit |

número de defeitos por unidade)

Fraction nonconforming (p)(Proporção de defeituosas)

Mean and variation (std deviation)

( 𝒙; s)N < 8

Mean and variation (range)

( 𝒙; R)

Control

Charts

Page 127: Part 4: Statistical Process Control Control Charts

153

Control Charts for Variables

Control charts for variables are built based on quantitative data –

Quality characteristics are measured on a numerical scale.

Average control chart - evaluates process center

Range control chart - evaluates process variation

X

x

x

x LSC

LIC

LC

0

0

ss

mm

0

0

ss

mm

0

0

ss

mm

Page 128: Part 4: Statistical Process Control Control Charts

154

Shewhart Control Charts

Control Charts for Variables and R

Notation for variables control charts:

n - size of the sample (sometimes called a subgroup) chosen at a

given point in time

k - number of samples selected

m is the true process mean

s is the true process standard deviation

x

Page 129: Part 4: Statistical Process Control Control Charts

155

Shewhart Control Charts

Control Charts for Variables

Notation for variables control charts:

= average of the observations in the ith sample (where i = 1, 2, ..., k)

= grand average or “average of the averages”

(this value is used as the center line of the control chart)

Ri = range of the values in the ith sample

Ri = xmax - xmin

= average range for all k samplesR

Control Charts for Variables and R x

ix

x

Shewhart Control Charts

Page 130: Part 4: Statistical Process Control Control Charts

Standard Given No Standard Given

What do you know by now?

Control Charts for VariablesControl Charts for Variables and R x

Shewhart Control Charts

Page 131: Part 4: Statistical Process Control Control Charts

Control Charts for Variables

mm xNx ss X ( , )

UCL = Nsm 3

CL = m

LCL = Nsm 3

Control Charts for Variables : Standard Given and No Standard Givenx

Control Chart: standard given X

Page 132: Part 4: Statistical Process Control Control Charts

Standard Given No Standard Given

What do you know by now?

Control Charts for VariablesControl Charts for Variables and R x

Shewhart Control Charts

Page 133: Part 4: Statistical Process Control Control Charts

159

Control Charts for Variables

R Control Chart: standard given | Sample size N < 8

UCL = RR

sm 3

CL = R

m

LCL = RR

sm 3

For Normal variables with (m, s) and sample sizes N

UCL = ssss 23232 )3(3 Ddddd

CL = s2d

LCL = ssss 13232 )3(3 Ddddd .

sm 2

dR

ss 3

dR

Control Charts for VariablesControl Charts for Variables R: Standard Given

UCL = 𝑫𝟐.s

CL = 𝒅𝟐.s

𝐋𝐂𝐋 = 𝑫𝟏.s

Control Limits for the R chart: standard given

Page 134: Part 4: Statistical Process Control Control Charts

160

Control Charts for Variables

R Control Chart: no standard given | Sample size N 8

If s is unknown, then:

Control Charts for VariablesControl Charts for Variables R: No Standard Given

Control Limits for the R chart: no standard given

2

ˆd

Rs

K

kk

RK

R1

1where

LSC = RDd

RD

4

2

2

LC = Rd

Rd

2

2

LIC = RDd

RD

3

2

1.

N D4 D3

2 3,267 0

3 2,575 0

4 2,282 0

values that only depend on the

sample size N

<

Page 135: Part 4: Statistical Process Control Control Charts

Standard Given No Standard Given

What do you know by now?

Control Charts for VariablesControl Charts for Variables and R x

Shewhart Control Charts

Page 136: Part 4: Statistical Process Control Control Charts

162

Control Charts for Variables

If the parameters of X are unknown, then

mm xNx ss X ( , )

K

k

kxK

x1

1m̂

and if X follows a Normal distribution

2

ˆd

Rs

UCL = Nsm 3

CL = m

LCL = Nsm 3

UCL = RAxNd

Rx

2

2

3

CL = x

LCL = RAxNd

Rx

2

2

3

Control Charts for Variables : Standard Given and No Standard Givenx

Control Chart: standard given X

Control Chart: no standard given X

Page 137: Part 4: Statistical Process Control Control Charts

163

Shewhart Control Charts

Control Charts for Variables

Control Limits for the chartx

UCL = 𝒙 + 𝑨𝟐. 𝑅

CL = 𝑥

𝑳𝑪𝑳 = 𝒙 − 𝑨𝟐. 𝑅

UCL = 𝑫𝟒. 𝑅

CL = 𝑅

𝑳𝑪𝑳 = 𝑫𝟑. 𝑅

Control Limits for the R chart

Control Charts for Variables and R: No Standard Givenx

Small sample sizes N < 8

No standard given

Page 138: Part 4: Statistical Process Control Control Charts

164

Shewhart Control Charts

Control Charts for Variables

Control Limits for the chartx

UCL = 𝒙 + 𝑨𝟐. 𝑅

CL = 𝑥

𝑳𝑪𝑳 = 𝒙 − 𝑨𝟐. 𝑅

UCL = 𝑫𝟒. 𝑅

CL = 𝑅

𝑳𝑪𝑳 = 𝑫𝟑. 𝑅

Control Limits for the R chart

Control Charts for Variables and R: No Standard Givenx

Small sample sizes N < 8

No standard given

K

k

kxK

x1

1m̂

2

ˆd

Rs

If x follows a Normal distribution

Page 139: Part 4: Statistical Process Control Control Charts

165

Control Charts for Variables

Its is recomended that and estimates are calculated having a

minimum of 100 values (K: number of samples and typically n is

between 3 and 5);

The control limits of the control chart should only be considered definite

when all points on the plot in-control.

x R

20K

x

Control Limits for the chartx

UCL = 𝒙 + 𝑨𝟐. 𝑅

CL = 𝑥

𝑳𝑪𝑳 = 𝒙 − 𝑨𝟐. 𝑅

UCL = 𝑫𝟒. 𝑅

CL = 𝑅

𝑳𝑪𝑳 = 𝑫𝟑. 𝑅

Control Limits for the R chart

R

Page 140: Part 4: Statistical Process Control Control Charts

Shewhart Control Charts

167

Page 141: Part 4: Statistical Process Control Control Charts

168

Control Charts for Variables

Samples should be selected in such a way that maximizes the chances for

shifts in the process average to occur between samples, and thus to

show up as out-of-control points on the averages chart;

The R chart, on the other hand, measures the variability within a

sample. Therefore, samples should be selected so that variability within

samples measures only chance or random causes;

Another way of saying this:

X bar chart monitors the between sample variability

the variability of the process over time

R chart monitors the within sample variability

the “instantaneous” variability in a given moment

Page 142: Part 4: Statistical Process Control Control Charts

171

Control Charts for Variables

In a casting process, two samples of molten steel are taken from

the ladle per day (the first is taken early in the morning and the

second one in the afternoon).

For each sample of molten steel, two dog-bone specimens

(provetes) are casted. These specimens are used to perform tensile

tests and to obtain their corresponding mechanical properties,

mainly the yield stress (tensão de cedência).

There are reasons to suspect that, from the point of view of the

mechanical properties of steel, significant alterations have occurred

between morning and the afternoon.

control charts: exampleRx,

Page 143: Part 4: Statistical Process Control Control Charts

172

Control Charts for Variables

Numa determinado fundição, são retiradas da colher de vazamento

duas amostras de aço por dia (uma ao princípio da manhã e outra ao

fim da tarde). A partir de cada amostra são vazados dois provetes

que, posteriormente são ensaiados à tracção.

Há razões para suspeitar que, do ponto de vista das propriedades

mecânicas do aço, se verificam alterações significativas de manhã

para a tarde.

Na Tabela seguinte registam-se os valores da tensão de cedência

(em MPa) obtidos nos ensaios efectuados sobre os provetes

recolhidos nos últimos 25 dias úteis.

Cartas de Médias e Amplitudes (Cartas ) - ExemploAx,

Page 144: Part 4: Statistical Process Control Control Charts

173

Yield Stress (MPa) in two specimen (provetes) of molten steel

Morning Afternoon Morning Afternoon

Day Spec. 1 Spec. 2 Spec. 1 Spec. 2 Day Spec. 1 Spec. 2 Spec. 1 Spec. 2

19/04 197.6 193.7 199.7 203.5 06/05 197.6 198.7 204.8 203.8

20/04 195.7 199.8 199.5 198.5 07/05 198.1 198.9 203.0 196.0

21/04 192.5 190.3 209.8 210.3 10/05 197.6 200.1 196.4 197.5

22/04 198.5 199.1 201.1 196.5 11/05 206.8 205.7 195.5 192.7

23/04 201.7 199.2 201.8 201.4 12/05 199.0 205.4 194.5 196.6

26/04 204.7 199.4 196.5 197.8 13/05 196.9 198.7 199.2 194.6

27/04 197.8 195.4 199.6 196.1 14/05 197.9 198.6 200.9 201.8

28/04 200.4 200.6 197.5 198.1 17/05 200.5 201.3 209.4 208.6

29/04 204.3 202.2 197.7 198.4 18/05 199.5 202.5 193.6 197.7

30/04 200.6 206.9 205.5 197.8 19/05 202.3 202.0 199.5 196.6

03/05 198.8 202.7 194.9 200.3 20/05 200.4 196.3 199.4 199.6

04/05 195.1 199.2 202.7 200.9 21/05 212.8 211.1 191.7 193.5

05/05 203.1 200.9 193.8 192.8

Control Charts for Variables

control charts: exampleRx,

The following table record the values of yield stress (MPa) obtained in tests carried out

on samples collected in the last 25 days.

Page 145: Part 4: Statistical Process Control Control Charts

174

Control Charts for Variables

Taking into considerations that each group of observations comprehends two

specimens (provetes) of the same ladle (colher de vazamento) (N=2)

51.250

1 50

1

k

RR

Control Limits for the R chart

Control Limits for the chart

UCL = 42.20451.2880.170.1992

RAx

CL = 70.199x

LCL = 98.19451.2880.170.1992

RAx

UCL = 20.851.2267.34

RD

CL = 51.2R

LCL = 0

x

control charts: exampleRx,

Page 146: Part 4: Statistical Process Control Control Charts

Shewhart Control Charts

175

Page 147: Part 4: Statistical Process Control Control Charts

176

Control Charts for Variables

Process is

out-of-

control

control charts: exampleRx,

464136312621161161

210

205

200

195

Sample

Sa

mp

le M

ea

n

__

X=199,69

UCL=204,40

LCL=194,98

464136312621161161

7,5

5,0

2,5

0,0

Sample

Sa

mp

le R

an

ge

_R=2,504

UCL=8,181

LCL=0

1

1

1

1

1

1

1

1

subgroup size 2

Xbar-R Chart of Ordenado

Page 148: Part 4: Statistical Process Control Control Charts

177

Control Charts for Variables

The standard deviation of the yield stress of the specimen can be calculated

according to:

22.2128.1

51.2ˆ

2

d

Rs << 31.4

1100

100

1

2

i

iT xxss

control charts: exampleRx,

Page 149: Part 4: Statistical Process Control Control Charts

Control Charts for Variables

If it was not considered the information that there are suspicions considering

potential alterations of the yield stress between morning and afternoon,

then, with N = 4:

control charts: exampleRx,

Yield Stress (MPa) in two specimen (provetes) of molten steel

Morning Afternoon Morning Afternoon

Day Prov. 1 Prov. 2 Prov. 1 Prov. 2 Day Prov. 1 Prov. 2 Prov. 1 Prov. 2

19/04 197.6 193.7 199.7 203.5 06/05 197.6 198.7 204.8 203.8

20/04 195.7 199.8 199.5 198.5 07/05 198.1 198.9 203.0 196.0

21/04 192.5 190.3 209.8 210.3 10/05 197.6 200.1 196.4 197.5

22/04 198.5 199.1 201.1 196.5 11/05 206.8 205.7 195.5 192.7

23/04 201.7 199.2 201.8 201.4 12/05 199.0 205.4 194.5 196.6

26/04 204.7 199.4 196.5 197.8 13/05 196.9 198.7 199.2 194.6

27/04 197.8 195.4 199.6 196.1 14/05 197.9 198.6 200.9 201.8

28/04 200.4 200.6 197.5 198.1 17/05 200.5 201.3 209.4 208.6

29/04 204.3 202.2 197.7 198.4 18/05 199.5 202.5 193.6 197.7

30/04 200.6 206.9 205.5 197.8 19/05 202.3 202.0 199.5 196.6

03/05 198.8 202.7 194.9 200.3 20/05 200.4 196.3 199.4 199.6

04/05 195.1 199.2 202.7 200.9 21/05 212.8 211.1 191.7 193.5

05/05 203.1 200.9 193.8 192.8

Page 150: Part 4: Statistical Process Control Control Charts

179

Control Charts for Variables

If it was not considered the information that there are suspicions considering

potential alterations of the yield stress between morning and afternoon,

then, with N = 4:

92.725

1 25

1

k

RR >> 51.2R (N = 2)

UCL = 47.20592.7729.070.1992

RAx

CL = 70.199x

LCL = 93.19392.7729.070.1992

RAx

UCL = 07.1892.7282.24

RD

CL = 92.7R

LCL = 0

control charts: exampleRx,

Page 151: Part 4: Statistical Process Control Control Charts

252321191715131197531

205

200

195

Sample

Sa

mp

le M

ea

n

__X=199,69

UCL=205,45

LCL=193,93

252321191715131197531

20

10

0

Sample

Sa

mp

le R

an

ge

_R=7,90

UCL=18,03

LCL=0

1

1

subgroup size 4

Xbar-R Chart of Ordenado

method for estimating within subgroup variation - R bar

180

Control Charts for Variables

Previous limit: 8.2

control charts: exampleRx,

Process is out-of-

control (assignable

causes are present

between morning

and night)

Page 152: Part 4: Statistical Process Control Control Charts

181

Control Charts for Variables

Selection of the sample size

The ability of the and R charts to detect shifts in process quality

is described by their operating characteristic(OC) curves

OCC of the control chart (ii) – CCO da Carta AX

control charts: exampleRx,

OCC of the R control chart

Page 153: Part 4: Statistical Process Control Control Charts

182

Types of Control Charts

Mean and variation (moving range)

( 𝒙; Rmoving)

Number nonconforming (Np)(Número de defeituosas)

Number of nonconformities (c)(Número de defeitos)

Number of nonconformities per unit (u)(average number of non-conformities per unit |

número de defeitos por unidade)

Fraction nonconforming (p)(Proporção de defeituosas)

Mean and variation (std deviation)

( 𝒙; s)N < 8

Mean and variation (range)

( 𝒙; R)

Control

Charts

Page 154: Part 4: Statistical Process Control Control Charts

OCAP- Out of Control Action Plan

When a control chart is

introduced, an initial OCAP

should accompany

it. Control charts without an

OCAP are not likely to be

useful as a process

improvement tool.

Page 155: Part 4: Statistical Process Control Control Charts

184

Shewhart Control Charts

Control Charts for Variables

Notation for variables control charts:

si = standard deviation of the values in the ith sample

= average standard deviation for all k sampless

Control Charts for Variables and R x

k

i

is sk

s1

1m̂

Page 156: Part 4: Statistical Process Control Control Charts

185

Shewhart Control Charts

Control Charts for Variables

Control Limits for the chartx

UCL = 𝒙 + 𝑨𝟑. 𝑠

CL = 𝑥

𝑳𝑪𝑳 = 𝒙 − 𝑨𝟑. 𝑠

UCL = 𝑩𝟒. 𝑠

CL = 𝑠

𝑳𝑪𝑳 = 𝑩𝟑. 𝑠

Control Limits for the s chart

Control Charts for Variables and sx

Small sample sizes N ≥8

No standard given

K

k

kxK

x1

1m̂

4

ˆc

ss

If x follows a Normal distribution

Page 157: Part 4: Statistical Process Control Control Charts

186

Shewhart Control Charts

Control Charts for Variables

Control Limits for the chartx

UCL = 𝒙 + 𝑨𝟑. 𝑠

CL = 𝑥

𝑳𝑪𝑳 = 𝒙 − 𝑨𝟑. 𝑠

UCL = 𝑩𝟒. 𝑠

CL = 𝑠

𝑳𝑪𝑳 = 𝑩𝟑. 𝑠

Control Limits for the s chart

Control Charts for Variables and s x

Small sample sizes N > 8

No standard given

Page 158: Part 4: Statistical Process Control Control Charts

187

Types of Control Charts

Mean and variation (moving range)

( 𝒙; Rmoving)

Number nonconforming (Np)(Número de defeituosas)

Number of nonconformities (c)(Número de defeitos)

Number of nonconformities per unit (u)(average number of non-conformities per unit |

número de defeitos por unidade)

Fraction nonconforming (p)(Proporção de defeituosas)

Mean and variation (std deviation)

( 𝒙; s)N < 8

Mean and variation (range)

( 𝒙; R)

Control

Charts

Page 159: Part 4: Statistical Process Control Control Charts

Shewhart Control Charts

188

There are many situations in which the sample size used for process monitoring is n = 1; that

is, the sample consists of an individual unit. Examples:

1. Automated inspection and measurement technology is used, and every unit manufactured

is analyzed so there is no basis for rational subgrouping;

2. Data comes available relatively slowly, and it is inconvenient to allow sample sizes of

n > 1 to accumulate before analysis. The long interval between observations will cause

problems with rational subgrouping. This occurs frequently in both manufacturing and

nonmanufacturing situations;

3. Repeat measurements on the process differ only because of laboratory or analysis error,

as in many chemical processes;

4. In process plants, such as papermaking, measurements on some parameter such as coating

thickness across the roll will differ very little and produce a standard deviation that

is much too small if the objective is to control coating thickness along the roll.

Control Charts for Individual Measurements

Page 160: Part 4: Statistical Process Control Control Charts

190

Shewhart Control Charts

Control Charts for VariablesControl Charts for Individual Measurements

No Standard Given

Control Limits for Individual Units (x chart)

UCL = 𝒙 + 3.𝑴𝑹

𝑑2CL = 𝒙

𝑳𝑪𝑳 = 𝒙 − 3.𝑴𝑹

𝑑2

But, how can we estimate process variability? A good estimator is based on the moving range (MR):

iiixxMR

1i = 2, 3, ..., k

k

iii

xxk

MR2

1

1

1

UCL = 𝑫𝟒.𝑀𝑅

CL = 𝑀𝑅

𝑳𝑪𝑳 = 𝑫𝟑.𝑀𝑅

Control Limits for the Moving Range chart

A good estimator of s is then

with N=2, d2=1.128

2

ˆd

MRs

Page 161: Part 4: Statistical Process Control Control Charts

Shewhart Control Charts

191

Control Charts for Individual Measurements

The mortgage loan processing unit of a bank monitors the

costs of processing loan applications. The quantity tracked is

the average weekly processing costs, obtained by dividing

total weekly costs by the number of loans processed during

the week. The processing costs for the most recent 20 weeks

are shown in the table.

Set up individual and moving range control charts for these

data.

UCL = 𝒙 + 3.𝑴𝑹

𝑑2CL = 𝒙

𝑳𝑪𝑳 = 𝒙 − 3.𝑴𝑹

𝑑2

UCL = 𝒙 + 3.𝑴𝑹

𝑑2= 300.5 + 3

7.79

1.1.28= 321.22

CL = 𝑥 = 300.5

𝑳𝑪𝑳 = 𝒙 − 3.𝑴𝑹

𝑑2= 300.5 − 3

7.79

1.128= 279.78

Page 162: Part 4: Statistical Process Control Control Charts

Shewhart Control Charts

192

Control Charts for Individual Measurements

Page 163: Part 4: Statistical Process Control Control Charts

Shewhart Control Charts

193

Control Charts for Individual Measurements

Disadvantages of the (x, MR) control charts

Less sensitive than the xbar-R charts;

One of the assumptions is that the observations follow a Normal distribution;

The successive values of the ranges could be highly correlated;

Advantages of the (x, MR) control charts

Simple interpretation;

Immediate update;

Direct comparison with the specifications limits.

Page 164: Part 4: Statistical Process Control Control Charts

204

Shewhart Control Charts

Control charts are used in two diferent ways:

1) As a monitoring and surveillance online process tool,

functioning as an early warning system for the occurrence of

uncontrollable situations;

2) As a retrospective process analysis tool, with the aim of

identifying the presence (or absence) of special causes of

variation and provide their elimination.

Page 165: Part 4: Statistical Process Control Control Charts

205

Process Analysis with control charts

The analysis of processes through control charts has two main

purposes:

(1) To estimate the real and the potential capability of

processes;

(2) To reduce the process’s variability.

Page 166: Part 4: Statistical Process Control Control Charts

206

Process Analysis with control charts

(1) Process Capability Estimation

The head of the packing division was having

problems with the detergent powder

packing machine, and thought that the

machine didn´t have sufficient accuracy to

meet customers' requirements.

The specifications for the detergent weight

are 4 grams around the nominal value of 504

grams.

504g ± 4𝑔

There is a suspicion that, besides the equipment, the supplier and the origin of the raw

materials is a factor that might influence the weight of the packages.

Page 167: Part 4: Statistical Process Control Control Charts

207

2Process Analysis with control charts

(1) Process Capability Estimation

In order to evaluate the problem, the net

weights (in grams) observed in samples of

size 4 taken randomly from 25 different lots

were gathered (each lot has 5000 packets).

The origin of the raw materials might

change from lot to lot.

1 lot = 5000 packetsRaw materials of the same

supplier

25 lots

Page 168: Part 4: Statistical Process Control Control Charts

208

Tabela– Pesos (em gramas) de detergente em pó

Lote no. x1 x2 x3 x4x A

1 504.9 506.0 505.2 504.7 505.2 1.32 501.2 499.5 501.1 500.8 500.7 1.73 506.6 503.4 505.6 503.7 504.8 3.24 503.2 502.5 503.2 503.5 503.1 1.05 498.5 500.2 499.1 500.7 499.6 2.2

6 504.0 503.1 502.9 504.4 503.6 1.57 503.7 502.6 502.9 502.9 503.0 1.18 502.7 503.7 503.7 504.5 503.7 1.89 505.0 504.2 506.7 504.9 505.2 2.510 504.9 504.4 505.2 504.0 504.6 1.211 506.3 505.9 504.7 507.0 506.0 2.312 504.2 503.5 504.2 502.5 503.6 1.713 500.3 501.7 500.3 502.6 501.2 2.3

14 501.6 504.0 501.3 502.7 502.4 2.715 502.5 503.5 502.8 503.3 503.0 1.016 504.0 503.3 504.4 506.5 504.6 3.217 502.5 498.9 501.8 501.3 501.1 3.618 502.3 500.2 503.7 503.6 502.5 3.519 503.8 505.2 504.6 504.7 504.6 1.420 506.4 505.8 508.0 505.2 506.4 2.821 505.2 506.6 504.8 504.1 505.2 2.5

22 505.3 504.6 505.9 506.4 505.6 1.823 506.9 503.7 504.0 505.9 505.1 3.224 504.9 506.1 505.7 504.4 505.3 1.725 509.1 506.8 507.8 507.0 507.7 2.3

503.9x 14.2A

Process Analysis with control charts

(1) Process Capability Estimation

Weight (grams) of the 25 samples

R

𝑅

Page 169: Part 4: Statistical Process Control Control Charts

209

Equipment Raw materials

In order to ensure that we meet specifications, we must understand:

Process Analysis with control charts

(1) Process Capability Estimation

Page 170: Part 4: Statistical Process Control Control Charts

210

Peso

499 500 501 502 503 504 505 506 507 508 509

Lim

ite

Infe

rior

da

Tol

erân

cia

Lim

ite

Sup

erio

r da

Tol

erân

cia

0

5

10

15

20

Fre

quên

cia

x = 503.9

s = 2.09

The weight distribution is centred, unimodal and not too asymmetrical

Specifications: 504g ± 4𝑔

Process Analysis with control charts

(1) Process Capability Estimation

LSL USL

Page 171: Part 4: Statistical Process Control Control Charts

211

Process Analysis with control charts

(1) Process Capability Estimation

An advantage of using the histogram to estimate process capability is

that it gives an immediate, visual impression of process performance.

But, they do not necessarily display the potential capability of the

process because they do not address the issue of statistical control.

8

Page 172: Part 4: Statistical Process Control Control Charts

212

Knowing that , the process capability index is (capacidade do processo): 09.2 ss

64.009.26

8

6

s

LSLUSLC

p

Conclusion: The head of the packing division is right, the equipment is not capable

of meeting specifications.

When estimating global variability through the standard deviation s, what factors of

variation are included in this value?

09.2s

Equipment ? Raw materials ?

Question

Process Analysis with control charts

(1) Process Capability Estimation

Page 173: Part 4: Statistical Process Control Control Charts

213

2Process Analysis with control charts

(1) Process Capability Estimation

S = 2.09

Includes the variation:

-> within subgroups;

-> between subgroups, due to changes in

raw materials.

The origin of the raw materials might

change from lot to lot.

So, we need to compare this estimate of s with another estimate that only takes into account

the within subgroup variation.

Page 174: Part 4: Statistical Process Control Control Charts

214

Tabela– Pesos (em gramas) de detergente em pó

Lot no. x1 x2 x3 x4 x R

1 504.9 506.0 505.2 504.7 505.2 1.3

2 501.2 499.5 501.1 500.8 500.7 1.7

3 506.6 503.4 505.6 503.7 504.8 3.2

4 503.2 502.5 503.2 503.5 503.1 1.0

5 498.5 500.2 499.1 500.7 499.6 2.2

6 504.0 503.1 502.9 504.4 503.6 1.5

7 503.7 502.6 502.9 502.9 503.0 1.1

8 502.7 503.7 503.7 504.5 503.7 1.8

9 505.0 504.2 506.7 504.9 505.2 2.5

10 504.9 504.4 505.2 504.0 504.6 1.2

11 506.3 505.9 504.7 507.0 506.0 2.3

12 504.2 503.5 504.2 502.5 503.6 1.7

13 500.3 501.7 500.3 502.6 501.2 2.3

14 501.6 504.0 501.3 502.7 502.4 2.7

15 502.5 503.5 502.8 503.3 503.0 1.0

16 504.0 503.3 504.4 506.5 504.6 3.2

17 502.5 498.9 501.8 501.3 501.1 3.6

18 502.3 500.2 503.7 503.6 502.5 3.5

19 503.8 505.2 504.6 504.7 504.6 1.4

20 506.4 505.8 508.0 505.2 506.4 2.8

21 505.2 506.6 504.8 504.1 505.2 2.5

22 505.3 504.6 505.9 506.4 505.6 1.8

23 506.9 503.7 504.0 505.9 505.1 3.2

24 504.9 506.1 505.7 504.4 505.3 1.7

25 509.1 506.8 507.8 507.0 507.7 2.3

503.9x 14.2R

Raw material constant =

same supplier

14.2R

The average range only

takes into account

within subgroup

variation

Process Analysis with control charts

(1) Process Capability Estimation

Looking at the available information, this means:

Page 175: Part 4: Statistical Process Control Control Charts

215

Process Analysis with control charts

(1) Process Capability Estimation

252321191715131197531

508

506

504

502

500

Sample

Sa

mp

le M

ea

n

__X=503,903

UCL=505,462

LCL=502,344

252321191715131197531

4,8

3,6

2,4

1,2

0,0

Sample

Sa

mp

le R

an

ge

_R=2,14

UCL=4,882

LCL=0

1

1

1

11

1

1

1

Detergent powder packing machine

Raw material might change between subgroups

But, is the process stable?

Process variability

is in-control

Page 176: Part 4: Statistical Process Control Control Charts

216

Re-estimating the standard deviation with the Average Range (amplitude média):

04.1059.2

14.2ˆ

2

d

Rd

s where sd only accounts for within subgroup variation

28.104.16

8

ˆ6

d

p

TC

s

The equipment is capable for producing packages according

to specifications.

Nevertheless, instability due the changes in raw materials,

or due to any other assignable cause that might occur

between subgroups, should be eliminated.

Cp represents the process capability that can be achieved if all assignable

causes are removed between subgroups, i.e. the potential process

capability

Process Analysis with control charts

(1) Process Capability Estimation

Page 177: Part 4: Statistical Process Control Control Charts

21728.1

04.16

8

ˆ6

d

p

TC

s

Process Analysis with control charts

(1) Process Capability Estimation

64.009.26

8

6

s

TPp

Page 178: Part 4: Statistical Process Control Control Charts

218

28.104.16

8

ˆ6

d

p

TC

s

Process Analysis with control charts

(1) Process Capability Estimation

64.009.26

8

6

s

TPp

Cp: is the best indicator of potential process capability because it assumes a

stable and on-target process.

Pp: is an indicator of actual process performance if the process is on-target.

It does take into consideration the long-term variability.

Page 179: Part 4: Statistical Process Control Control Charts

220

Procedure for Process Improvement with control charts:

(i) Identification of possible and plausible assignable causes of variation

(i.e. factors) and collect samples in rational subgroups;

(ii) Confirmation or rejection of the hypothesis considered in (i);

(iii) Create an hierarchy of assignable causes of variation/ factors according

to the magnitude of its effects;

(iv) Analyze and eliminate the causes that are at the root of such factors of

variation;

(v) Confirm the results.

Process Analysis with control charts

Process Improvement with control charts

Page 180: Part 4: Statistical Process Control Control Charts

221

ijE ),( 2

eoN sm ),( 2

doN sm

tD

Variation within groups Variation between groups

Distribution in

t = 1 (m1, sd)Distribution in

t = 2 (m2, sd)

mk

Distribution in

t = k (mk, sd)

Dk

xkj

Ek

m

Global Distribution

(m, s)

t = 1

Time

t = 2 t = k

Assuming that the process is only unstable in what concerns location.

Process Analysis with control charts

(2) Process Improvement with control charts

Page 181: Part 4: Statistical Process Control Control Charts

222

tjttjttj EDEx mm

t: index related to the instante when the sample was taken t (t = 1, 2, ..., K)

j: index related to the position of one observation within each sample (j = 1, 2, ..., N)

xtj: j observation of the sample taken at time t

mt: expected value of the distribution at time t

Etj: error (deviation in relation to mt) associated to the j-ésima observation of the sample

taken at time t

m: global expected value

Dt: deviation in relation to m of the expected value of the distribution at time t

ANOVA Model (1 factor, fixed effects)

Process Analysis with control charts

(2) ) Process Improvement with control charts

Assuming that the process is only unstable in what concerns location.

Page 182: Part 4: Statistical Process Control Control Charts

223

)1(

)1(1222

NK

Ned sss

)( 22 sEs : represents the expected value of the global sampling variance

(variância amostral global) (for all data)

)ˆ( 22dd E ss : represents the expected value of the variance within groups, that,

in the case of R charts is given by 2

2dR

)ˆ( 22ee E ss : represents the expected value of the variance between groups

that, in the case of xbar charts is given by )1/()(ˆ 2

1

2

kxxK

t

txs

K>> N 222

ed sss

Process Analysis with control charts

(2) Process Improvement with control charts

Page 183: Part 4: Statistical Process Control Control Charts

225

222

ed sss

is related with the variation that occurs within groups (assuming that

it is homogeneous), due to common causes of variation

results from a change in the process location, possibly due to

assignable (or special) causes of variation

2

ds

2

es

In a process completely in control, = 0 2

es 22

dss

N

dex

222 ˆ

ˆˆs

ss The variability in the xbar chart depends on both

components:

the variability between and within groups

Process Analysis with control charts

(2) Process Improvement with control charts

Page 184: Part 4: Statistical Process Control Control Charts

226

222

ed sss

estimate based on orR s

Ndxe222

ˆˆˆ sss estimate based on

(variance of the observations on the

xbar chart)

Process Analysis with control charts

(2) Process Improvement with control charts

Page 185: Part 4: Statistical Process Control Control Charts

227

Tabela– Pesos (em gramas) de detergente em pó

Lote no. x1 x2 x3 x4x A

1 504.9 506.0 505.2 504.7 505.2 1.32 501.2 499.5 501.1 500.8 500.7 1.73 506.6 503.4 505.6 503.7 504.8 3.24 503.2 502.5 503.2 503.5 503.1 1.05 498.5 500.2 499.1 500.7 499.6 2.2

6 504.0 503.1 502.9 504.4 503.6 1.57 503.7 502.6 502.9 502.9 503.0 1.18 502.7 503.7 503.7 504.5 503.7 1.89 505.0 504.2 506.7 504.9 505.2 2.510 504.9 504.4 505.2 504.0 504.6 1.211 506.3 505.9 504.7 507.0 506.0 2.312 504.2 503.5 504.2 502.5 503.6 1.713 500.3 501.7 500.3 502.6 501.2 2.3

14 501.6 504.0 501.3 502.7 502.4 2.715 502.5 503.5 502.8 503.3 503.0 1.016 504.0 503.3 504.4 506.5 504.6 3.217 502.5 498.9 501.8 501.3 501.1 3.618 502.3 500.2 503.7 503.6 502.5 3.519 503.8 505.2 504.6 504.7 504.6 1.420 506.4 505.8 508.0 505.2 506.4 2.821 505.2 506.6 504.8 504.1 505.2 2.5

22 505.3 504.6 505.9 506.4 505.6 1.823 506.9 503.7 504.0 505.9 505.1 3.224 504.9 506.1 505.7 504.4 505.3 1.725 509.1 506.8 507.8 507.0 507.7 2.3

503.9x 14.2A

(iii) Create an hierarchy of assignable causes of variation/ factors according

to the magnitude of its effects;

Process Analysis with control charts

(2) Process Improvement with control charts

Weights (grams) of powder detergent

Page 186: Part 4: Statistical Process Control Control Charts

228

N = 4

With the 100 observations, we can calculate:

For each sample, we can calculate:

x - estimate of the expected value m

s - estimate of the standard deviation s

- estimate of the expected value mt of the weight distribution at time t

- estimate of the standard deviation sd

x

2d

R

Process Analysis with control charts

(2) Process Improvement with control charts

Page 187: Part 4: Statistical Process Control Control Charts

229

You now suspect that the shift (turno) might have a decisive influence on

the final weight of the packages

Process Analysis with control charts

(2) Process Improvement with control charts

Turno A Turno B

Lote no. x1 x2 x3 x4

1 504.9 506.0 505.2 504.7

2 501.2 499.5 501.1 500.8

3 506.6 503.4 505.6 503.7

4 503.2 502.5 503.2 503.5

. . .

Turno A Turno B

Lote no. x1 x2 x3 x4

1 504.9 506.0 505.2 504.7

2 501.2 499.5 501.1 500.8

3 506.6 503.4 505.6 503.74 503.2 502.5 503.2 503.5

. . .

N = 2 N = 4

Shift A Shift B

Lot #

Shift A Shift B

Lot #

N = 2 N = 4

Turno A Turno B

Lote no. x1 x2 x3 x4

1 504.9 506.0 505.2 504.72 501.2 499.5 501.1 500.83 506.6 503.4 505.6 503.74 503.2 502.5 503.2 503.5. . . . .. . . . .. . . . .

Shift A Shift B

Page 188: Part 4: Statistical Process Control Control Charts

230

Components of variation within and between subgroups:

Process Analysis with control charts

(2) Process Improvement with control charts

Grouping by lot and shift

(N = 2)

Grouping by lot

(N = 4)

Causes for

variation within

subgroups

Causes for

variation between

subgroups

• Measurement error

• Sampling error

• Other factors of variation

acting within subgroups

• Measurement error

• Sampling error

• Factor “change of shift”

• Factor “change of raw material”

• Measurement error

• Sampling error

• Factor “change of shift”

• Other factors of variation

acting within subgroups

• Measurement error

• Sampling error

• Factor “change of raw

material”

Page 189: Part 4: Statistical Process Control Control Charts

231

Control charts Average-Range with N = 2

144.ˆ2

11.128

1.29

d

Rd

s

77.1

2

144.1801.3

ˆˆˆ

2

22

N

dxe

sss

Process Analysis with control charts

(2) Process Improvement with control charts

464136312621161161

508

506

504

502

500

Sample

Sa

mp

le M

ea

n

__X=503,903

UCL=506,337

LCL=501,469

464136312621161161

4

3

2

1

0

Sample

Sa

mp

le R

an

ge

_R=1,294

UCL=4,228

LCL=0

1

1

1

1

1

1

11

1

1

1

Detergent powder packing machine

Page 190: Part 4: Statistical Process Control Control Charts

232

847.

04.683.

ˆˆˆ

2

2

1

4

13

2

N

d

xe

sss

04.1ˆ2

d

Rd

s

1,919+1,04Process Analysis with control charts

(2) Process Improvement with control charts

Control charts Average-Range with N = 4

252321191715131197531

508

506

504

502

500

Sample

Sa

mp

le M

ea

n

__

X=503,903

UCL=505,462

LCL=502,344

252321191715131197531

4,8

3,6

2,4

1,2

0,0

Sample

Sa

mp

le R

an

ge

_

R=2,14

UCL=4,882

LCL=0

1

1

1

11

1

1

1

Detergent powder packing machine

Page 191: Part 4: Statistical Process Control Control Charts

233

R control charts with N = 4

286.3ˆ

082.1ˆ

2

2

e

d

s

s

Process Analysis with control charts

(2) Process Improvement with control chartsThink Break

R control charts with N = 2

147.3ˆ

309.1ˆ

2

2

e

d

s

s

Grouping by lot and shift

(N = 2)

Grouping by lot

(N = 4)

Causes for

variation within

subgroups

Causes for

variation between

subgroups

• Measurement error

• Sampling error

• Other factors of variation

acting within subgroups

• Measurement error

• Sampling error

• Factor “change of shift”

• Factor “change of raw material”

• Measurement error

• Sampling error

• Factor “change of shift”

• Other factors of variation

acting within subgroups

• Measurement error

• Sampling error

• Factor “change of raw

material”

Page 192: Part 4: Statistical Process Control Control Charts

234

%6.70309.1147.3

147.3

ˆ

ˆ2

2

total

e

s

s

70.6% of the variation is explained by the

factors operating between groups: “Change of

Shift”; “Change of Raw Material”;

N = 2

N = 4 %2.751.0823.286

3.286

ˆ

ˆ2

2

total

e

s

s

75.2% of the variation is explained by the

factors operating between groups: “Change of

Raw Material”;

Conclusion: A better process capability index does not depend on particular working

methods of any of the two shifts, but seems to be related with problems related to

raw materials (and different suppliers) used in detergent manufacturing.

Process Analysis with control charts

(2) Process Improvement with control charts

What is the relative importance of each factor?

Page 193: Part 4: Statistical Process Control Control Charts

What is the relative importance of each factor?

Process Analysis with control charts

(2) Process Improvement with control charts

247.0090.2

039.1

ˆ

ˆ2

2

2

2

t

d

OverallStDev

WithinStDev

s

s

3011.0090.2

147.12

2

2

2

t

d

all)StDev(Over

in)StDev(With

s

s

0541.0

N=4 Between Groups: Raw Materials N=2 Between groups: Raw Materials and

Change of Shift (Turno)

Page 194: Part 4: Statistical Process Control Control Charts

248

Shewhart Control Charts

Control charts are used in two diferent ways:

1) As a retrospective process analysis tool, with the aim of

identifying the presence (or absence) of special causes of

variation and provide their elimination;

2) As a monitoring and surveillance online process tool,

functioning as an early warning system for the occurrence of

uncontrollable situations.

Page 195: Part 4: Statistical Process Control Control Charts

249

Guidelines for implementing control charts

Process Analysis.

Characterization of variation.

Process improvement.

Fixing the definitive control limits.

Phase 1(historical data, no standard given)

Page 196: Part 4: Statistical Process Control Control Charts

250

Guidelines for implementing control charts

Determine which process characteristics to control and evaluation of the measurement process;

Identify possible assignable causes of variation and collect the data accordingly. Verify/ensure the independence of observations.

Calculate the trial control limits; confirm or reject the previous hypotheses and rank the assignable causes of variation (i.e. factors) according to the magnitude of its effects.

Further analize and eliminate the root causes of the relevant factors of variation (process improvement).

Confirm the improvement and calculate the control limits for future production.

Phase 1: initial setup

Page 197: Part 4: Statistical Process Control Control Charts

251

Guidelines for implementing control charts

Before you can install a control chart on the line and expect

positive results, you must first understand process variation, how

to sample to quantify that variation and how to use SPC to guide the

reduction of variation.

Quoting Lynne Hare (Quality Progress, August 2001)

SPC is not a tool to steer the process; it is a means of gaining

process understanding to aid in process improvement.

Phase 1

Page 198: Part 4: Statistical Process Control Control Charts

252

Guidelines for implementing control charts

Phase 2(stable process, standard given)

Control and process monitoring.

Periodic re-estimation of the

control limits.

Page 199: Part 4: Statistical Process Control Control Charts

253

Guidelines for implementing control charts

Initially, use the control limits previously calculated in Phase 1.

Keep the rational subgroup approach to sampling, in order to

better identify assignable causes.

The sampling frequency should be set taking into account the

advantages of maximizing the chance of occurring assignable

causes of variation between samples and the need to obtain

independent observations within the samples.

Control limits should be recalculated periodically.

Phase 2 – process control and monitoring

Page 200: Part 4: Statistical Process Control Control Charts

254

Please note that in Phase 2, even with reasonable

sized samples, the Shewhart control charts should

only be used in processes with a good or even really

good process capability (Cp = Cpk > 1,3).

Always remembre that there are better alternatives!

E

The CUSUM and EWMA control charts!

Guidelines for implementing control charts

Page 201: Part 4: Statistical Process Control Control Charts

256

Unfortunately, in practice, what normally occurs is that:

Ignore its usefulness in Phase 1

Make a bad use of them in Phase 2

Conclusion

Page 202: Part 4: Statistical Process Control Control Charts

257

Page 203: Part 4: Statistical Process Control Control Charts

Strategy for process improvement (ISO/FDIS 7870-2)

258

Page 204: Part 4: Statistical Process Control Control Charts

Systems approach to the construction of variables control charts

(ISO/FDIS 7870-2)

259