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STATISTICAL QUALITY CONTROL Majid Rafiee Department of Industrial Engineering Sharif University of Technology [email protected]

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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

  • Copyright Notice

    • 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

    Adopted from: http://www.syque.com/improvement/Location%20Plot.htm

  • 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

    • Management leadership

    • 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.

    • Used to identify value-added versus nonvalue-added activity

    • To eliminate non-value added activities

    92

  • Flow chart

    93

  • Back up

    94