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STATISTICAL QUALITY CONTROL Majid Rafiee Department of Industrial Engineering Sharif University of Technology

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• STATISTICAL QUALITY CONTROL

Majid Rafiee

Department of Industrial Engineering

Sharif University of Technology

[email protected]

• How Statistical Process Control (SPC) works

• Parts (text & figures) of this lecture adopted from:

• Introduction to statistical quality control, 5th edition, by douglas C.

Montgomery , arizona state university

• Https://www.Wikipedia.Org

3

• Outline

4

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

• Outline

5

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

• Introduction

• Statistical quality control

• Monitoring various stages of production

• Monitoring of process

• Means to determine major source of observed variation

• Chance variation

• Inevitable

• Assignable cause

• Detected & corrected by appropriate tools

6

• Outline

7

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

• Chance & Assignable Cause of Quality Variation

8

Causes

Chance causes Assignable causes

• Chance & Assignable Cause of Quality Variation

• Chance causes

• In random fashion

• inevitable

• Assignable causes

• Can be assigned to any particular cause

• Defective materials, defective labour, ..

9

• Chance & Assignable Cause of Quality Variation

• A process is operating with only chance causes of variation present

is said to be in statistical control.

• A process that is operating in the presence of assignable causes is

said to be out of control.

10

• Chance & Assignable Cause of Quality Variation

11

• Outline

12

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

• Basic SPC Tools

13

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

• Basic SPC Tools

14

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

• Histogram or steam-and-leaf plot

• Displaying the relative density and shape of the data

• Giving the reader a quick overview of distribution

• Highlighting outliers

• Finding the mode

15

• Basic SPC Tools

16

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

• Check sheet

17

• It is a form used to collect data (quantitative or qualitative)

• In real time

• At the location where the data is generated

• Check sheet

18

• Check sheet

19

• Basic SPC Tools

20

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

• Pareto chart

• To assess the most frequently

occurring defects

• 80/20 rule

• ABC analysis

21

• Pareto chart

• To assess the most frequently

occurring defects

• 80/20 rule

• ABC analysis

22

• Various examples of Pareto charts

23

• Basic SPC Tools

24

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

• Cause & effect diagram

• Visualization tool

• Categorizing the potential causes of a problem

• The design of the diagram looks like a skeleton of a fish

• Fishbone diagram

• Ishikawa diagram

25

• Cause & effect diagram

26

• How to Construct a Cause & effect diagram

• Define the problem to be analyzed

• Draw the effect box & the center line

• Specify major potential cause categories and join them to the center

line

• Identify possible causes & classify them into categories in previous

step

• Rank order the causes

• Take corrective actions

27

• Basic SPC Tools

28

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

• Defect concentration diagram

• a graphical tool

• It is a drawing of the product

• all relevant views displayed

• Shows locations and

frequencies of various defects

29

• Defect concentration diagram

30

• Basic SPC Tools

31

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

• Scatter diagram

• To identify type of

relationship between two

quantitative variables

32

• Basic SPC Tools

33

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

• Control chart & Specifications

• a statistical process control tool used to determine if a

manufacturing or business process is in a state of control.

• Specifications

• Lower specification limit

• Upper specification limit

• Target or nominal values

34

• Control chart & Specifications

35

• Outline

36

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

• Statistical Basis of Control chart & Specifications

• A point that plots within the control limits indicates the process is in

control

• No action is necessary

• A point that plots outside the control limits is evidence that the process

is out of control

• Investigation and corrective action are required to find and eliminate

assignable cause

• There is a close connection between control charts and hypothesis

testing

37

• Shewhart Control Chart Model

• Let w be a sample statistic that measures some CTQ

• Let L be the “distance” of the control limits from the center line

• Expressed in standard deviation units

38

• Photolithography Example

• Important quality characteristic in hard bake is resist flow width

• Process is monitored by average flow width sample of 5 wafers

• Process mean is 1.5 microns

• Process standard deviation is 0.15 microns

39

• Photolithography Example

𝝈𝒂𝒗𝒆𝒓𝒂𝒈𝒆 𝒐𝒇 𝒙 =𝝈𝒙

𝒏⇒ 𝝈𝒂𝒗𝒆𝒓𝒂𝒈𝒆 𝒐𝒇 𝒙 =

𝟎. 𝟏𝟓

𝟓= 𝟎. 𝟎𝟔𝟕𝟏

To assume x-bar is approximately normally distributed, we would

assume 𝟏𝟎𝟎 𝟏 − 𝜶 % of the sample fall btw 𝟏. 𝟓 + 𝒁𝜶𝟐(𝟎. 𝟎𝟔𝟏𝟕) and

𝟏. 𝟓 − 𝒁𝜶𝟐(𝟎. 𝟎𝟔𝟏𝟕)

𝑼𝑪𝑳 = 𝟏. 𝟓 + 𝟑(𝟎. 𝟎𝟔𝟕𝟏)

𝑪𝑳 = 𝟏. 𝟓

𝑳𝑪𝑳 = 𝟏. 𝟓 − 𝟑(𝟎. 𝟎𝟔𝟕𝟏)

40

• Photolithography Example

• Note that all plotted points

fall inside the control limits

• Process is considered to be

in statistical control

• Called three sigma

41

• Distribution of X-bar vs X

42

• Process improvement using control chart

• Using control charts consequently

• Identify assignable causes

• Eliminate causes

• Reducing variation

• Improving quality

43

• Out of Control Action Plan (OCAP)

44

• a companion to the control chart

• reactions to out-of-control

situations

• Out of Control Action Plan (OCAP)-1

45

• Out of Control Action Plan (OCAP)-2

46

• Types of Control Charts

47

Control Charts

Variables Control Charts Attributes Control Charts

• Review over Classifying Data On Quality Characteristics

• Variables

• Often continuous measurements

• Length, voltage, viscosity

• Following continuous distribution

• Attributes

• Usually discrete data

• Often taking the form of counts

• Following discrete distribution

48

• Types of Control Charts

• Variables Control Charts

• Applied to data following continuous distribution

• Attributes Control Charts

• Applied to data following discrete distribution

49

• Types of Control Charts

50

Control Charts

Variables Control Charts Attributes Control Charts

• Variables Control Charts

51

Variables Control Charts

X-bar chart R chart S chart S**2 chart

• Types of Control Charts

52

Control Charts

Variables Control Charts Attributes Control Charts

• Attributes Control Charts

53

Attributes Control Charts

C chart U chart Np chart P chart

• Control chart design

54

Control chart design

selection of sample size & sampling frequency

control limits

• Control chart design

55

Control chart design

selection of sample size & sampling frequency

control limits

• Types of Process Variability

56

Process Variability

Stationary & uncorrelated

NonstationaryStationary & auto correlated

• Types of Process Variability

57

Process Variability

Stationary & uncorrelated

NonstationaryStationary & auto correlated

• Stationary & uncorrelated Process Variability

• Data vary around a fixed mean

in a stable or predictable

manner

58

• Types of Process Variability

59

Process Variability

Stationary & uncorrelated

NonstationaryStationary & auto correlated

• Stationary & auto correlated Process Variability

• Successive observations are

dependent with tendency to

move in long runs on either

side of mean

60

• Types of Process Variability

61

Process Variability

Stationary & uncorrelated

NonstationaryStationary & auto correlated

• Nonstationary Process Variability

• process drifts without any

sense of a stable or fixed

mean

62

• Types of Errors

63

Error Types

Type I Type II

• Types of Errors

• Type I

• denoted by αlpha

• is the rejection of a true null hypothesis

• Type II

• denoted by Beta

• is the failure to reject a false null hypothesis

64

• Average Run Length (ARL)

• Average # of points that must be plotted before a point indicates an

out-of-control condition

• Let P be the probability that any point exceeds control limits

• If observations are uncorrelated, then

ARL = 1/P

65

• Average Run Length (ARL) in Control

• In 3-sigma

• P=0.0027 (alpha)

• The probability that a single point falls out of limits, when process is in

control

• Then the average run length of x-bar chart when process is in control

is

𝐴𝑅𝐿0 =1

𝑃=

1

0.0027= 370

66

• Average Run Length (ARL) Out of Control

• In our Photolithography Example

• Mean has been shifted to 1.75 instead of 1.5

• Process is out of control

• Probability of single point of x-bar being btw control limits via changed mean

is 0.5 (1-Beta)

• Then the average run length of x-bar chart when process is out of

control is

𝐴𝑅𝐿𝟏 =1

𝑃=

1

0. 𝟓= 𝟐

67

• Average Run Length (ARL) Distribution

• Distribution of run length for a Shewhart control chart is geometric

distribution

• Standard deviation is very large

• Geometric distribution is very skewed

• Mean of distribution (ARL) is not necessarily a very typical value of run length

68

• Average time to signal

• if samples are taken at fixed intervals of time that are h hours apart

ATS = ARL * h

• If the time interval btw samples is h=1 hour, the average time

required to detect the shift is

𝐴𝑇𝑆1 = 𝐴𝑅𝐿1ℎ = 2 1 = 2 ℎ𝑜𝑢𝑟𝑠

69

• Sample Size & Sampling Frequency

• If sampling frequency is fixed, then as sample size increases, out of

control ARL & ATS decreases

• E.g. if n = 10, instead of n=5 :

𝐴𝑅𝐿1=

1

𝑝

=1

0.9

= 1.11

𝐴𝑇𝑆1 = 𝐴𝑅𝐿1ℎ = 1.11 1 = 1.11 ℎ𝑜𝑢𝑟𝑠

70

• Sample Size & Sampling Frequency

• If it became important to detect shift in approximately first hour

after it occurred, two control chart designs would work:

• Design1

• Sampling size: n=5

• Sampling Frequency: every half hour

• Design2

• Sampling size: n=10

• Sampling Frequency: every hour

71

• Control chart design

72

Control chart design

selection of sample size & sampling frequency

control limits

• Choice of Control Limits

• 3-Sigma Control Limits

• Probability of type I error is 0.0027

• Probability Limits

• Type I error probability is chosen directly

• For example, 0.001 gives 3.09-sigma control limits

• Warning Limits

• Typically selected as 2-sigma limits

73

• Choice of Control Limits

74

• rational subgroup

• Subgroups or samples should be selected so that

• If assignable causes are present, chance for differences between

subgroups will be maximized

• Chance for difference due to assignable causes within a subgroup will be

minimized.

• Two general approaches for constructing rational subgroups:

• Consecutive units

• Random sample of all process output over sampling interval

75

• consecutive units

• Sample consists of units produced at the same time

• Primary purpose is to detect process shifts

76

• random sample

• random sample of all process output over sampling interval

• Sample consists of units that are representative of all units

produced since last sample

• Often used to make decisions about acceptance of product

• Effective at detecting shifts to out-of-control state and back into

in-control state between samples

• we can often make any process appear to be in statistical control

just by stretching out the interval between observations in the

sample.

77

• Different Approaches for Sampling

78

• Analysis of Patterns on Control Chart

79

• Pattern is very nonrandom in

appearance

• 19 of 25 points plot below the center

line, while only 6 plot above

• Following 4th point, 5 points in a row

increase in magnitude, a run up

• There is also an unusually long run

down beginning with 18th point

• Cyclic Pattern

80

• how to detect nonrandom patterns?

• Western Electronic Handbook, Set of decision rules

• 1. Any single data point falls outside the 3σ-limit from the centerline

• 2. Two out of three consecutive points fall beyond the 2σ-limit, on the

same side of the centerline

• 3. Four out of five consecutive points fall beyond the 1σ-limit, on the

same side of the centerline

• 4. Nine consecutive points fall on the same side of the centerline

81

• Discussion of Sensitizing Rules for Control Chart

• Suppose that analyst uses k decision rules & that criterion I has type

I error probability 𝛼𝑖 , then the overall type I error based on k test is:

𝛼 = 1 −ෑ

𝑖=1

𝑘

(1 − 𝛼𝑖)

• Q) What are the consequences of increasing overall error type I?

82

• Control Chart Application

83

Control Chart Application

Phase I Phase II

• Control Chart Application

84

Control Chart Application

Phase I Phase II

• Phase I

• Phase I is a retrospective analysis of process data to construct trial

control limits

• Charts are effective at detecting large, sustained shifts in process

parameters, outliers, measurement errors, data entry errors, etc.

• Facilitates identification and removal of assignable causes

85

• Control Chart Application

86

Control Chart Application

Phase I Phase II

• Phase II

• In this phase, the control chart is used to monitor the process

• Process is assumed to be reasonably stable

• Emphasis is on process monitoring, not on bringing an unruly

process into control

87

• Outline

88

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

• Implementing SPC

• Elements of a successful SPC program

• A team approach

• Education of employees at all levels

• Emphasis on reducing variability

• Measuring success in quantitative terms

• A mechanism for communicating successful results

throughout the organization

89

• Outline

90

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

• Application of SPC

• Industrial applications

• Nonmanufacturing applications

• Sometimes require ingenuity

• Mostly do not have a natural measurement system

• The observability of the process may be fairly low

91

• process mapping

• Flow charts & operation process charts are particularly useful in

developing process definition & process understanding.

• This is sometimes called process mapping.