introduction to statistical process control 20_raul_soto... · introduction to statistical process...

Introduction to

Statistical Process Control

Raul Soto, MSc, CQEIVT Conference - March 2017

Amsterdam, Netherlands

(c) 2017 Raul Soto 2

The contents of this presentation represent the opinion of the

speaker; and not necessarily that of his present or past employers.

IVT Amsterdam MAR2017

About the Author• 20+ years of experience in the medical devices, pharmaceutical, biotechnology, and consumer electronics industries

• MS Biotechnology, emphasis in Biomedical Engineering

• BS Mechanical Engineering

• ASQ Certified Quality Engineer (CQE)

• I have led validation / qualification efforts in multiple scenarios:

• High-speed, high-volume automated manufacturing and packaging equipment; machine vision systems

• Manufacturing Execution Systems (MES)

• Enterprise resource planning applications (i.e. SAP)

• IT network infrastructure, reporting systems, Enterprise applications

• Laboratory information systems and instruments

• Mobile apps

• Product improvements, material changes, vendor changes

• Contact information:

• Raul Soto [email protected]

3

4

• Introduce and describe the fundamental concepts of Statistical Process Control (SPC)• Control Charts for Attributes Data• Control Charts for Variables Data• Short-Run Charts• Rules to detect Special Cause variation• Average Run Length for Variables Charts• Using Control Charts to determine Process

Capability

• Can’t teach a 1-semester SPC class in 90 min…

What this talk is about

(c) 2017 Raul SotoIVT Amsterdam MAR2017

Part 1

What is Statistical Process Control (SPC)

Benefits / Example

What is SPC?

• Statistical Process Control is a methodology that uses statistical analysis to:

• MONITOR whether a process remains in a state of statistical control

• ALERT when a process is out of statistical control

• Basic assumption:• All processes are subject to variation

IVT Amsterdam MAR2017 © 2017 Raul Soto 6

Use of SPC in Validation

• Regular production: Allows us to monitor stable processes and determine if changes occur due to factors OTHER than random variation (detect if special cause variation is present)

• Validation of changes: Allows us to monitor continuous improvement efforts. When changes are made to a process to reduce variation / defects, SPC can be used to determine if the changes were effective.


Basic Process


1. Run your process

2. Take m subgroups of n units each at regular time intervals

3. Measure the characteristic of interest in all sample units

4. Calculate the control limits using the appropriate formulae for your chart

5. Plot the corresponding values in your chart

6. Any points outside the control limits indicate the presence of special cause variation, lack of statistical control

7. Use the rules to detect the presence of trends that may signal that the process is moving out of control

Part 2

Fundamental Concepts

Which Product Quality Characteristics Should We Monitor?

• Product’s critical quality characteristics

• Most critical dimensions

• Contractual agreements with a client

• Input variables, if known to be sources of variation


SPC as a Hypothesis Test

• SPC is basically a continuous hypothesis test

• We use it to detect as quickly as possible departures from the norm that are statistically significant

• A statistically significant departure from the norm is typically caused by special cause (non random) variation


SPC vs Process Capability

• Statistical control: process does not exhibit non-random (special cause) variation

• Controlled Process:• mean and standard deviation

are consistent

• Capable Process:• consistently meets its

specifications

(c) 2017 Raul Soto 12

Variation: Common vs Special Cause

• Common cause variability (“natural”, “random” variability)

• Inherent in the process• Not controllable by operators

• Examples:• Variations in raw materials• Variations in ambient conditions such

as temperature and humidity

• Confusing common cause with special cause => Overadjustment

• Special cause variability (“assignable”, “non-random” causes)

• Unusual events that, if detected, can be removed or adjusted

• May be driven by one or multiple causes

• Examples:• Tool wear• Major changes in raw materials• Equipment failure

• Confusing special cause with common cause => Underadjustment

1

2

3

4

Rational Subgroups

• Rational method to select samples for a control chart

• We want to select subgroups so that, if assignable causes of variation are present:

• Greater probability of sample-to-sample variability

• Smaller probability of within-sample variability

• TWO main methods:

• METHOD 1: pick a sample of n units every t amount of time

• METHOD 2: Pick a sample of n units from all the product made since the last t

• + effective in detecting shifts between samples

• BUT can make out-of-control process seem in control if time interval is large


Rational Subgroups

• Method 1

• Method 2


Start

t = 0

t = 60 mint = 30 min

Take sample of n units Take sample of n units

Start

t = 0

t = 60 mint = 30 min

Take sample of n units Take sample of n units Take sample of n units

• Control limits ≠ Specification limits

• We can be in spec but out of control

• If a sample is outside the CLs, your process is out of statistical control

• CLs are calculated based on process data (your samples)

• Each type of control chart has specific formulae for CLs


Control Limits vs Spec Limits

Part 3

Control Charts for Continuous (Variables) Data

- Subgroups-

Run Chart

• Not a CONTROL chart

• No Control Limits, NO alarms

• Simplest type of variables chart, used to monitor

• Steps:

• Take a sample of n units every time period t

• Measure the characteristic of interest for all units

• Calculate the mean and plot it

• Minitab Run chart performs 4 tests for the presence of factors which may indicate special cause variation

• If any of these p-values is < 0.5 there is 95% confidence or higher that there is special cause variation present



Runs about median


Runs

up or down

Run Chart: TestsRandomness Test Observation

(if p<0.05)Interpretation

Number of runs aboutthe median

Significantly more runs observed than expected

Mixture: the may be data from two populations mixed together, or significant change in process conditions

Significantly fewer runs observed than expected

Clustering: indicates non-randomness, may be due to special cause variation (broken tools, process parameter changes, etc.)

Number of runs up or down

Significantly more runs observed than expected

Oscillation: data varied up and down rapidly, indicates process instability

Significantly fewer runs observed than expected

Trending: sustained drift in data, upwards or downwards. Indicates possible presence of special cause variation.


𝑥-R and 𝑥-s charts

• Used in pairs to monitor trends in location ( 𝑥) and dispersion (R, s) in quantitative (variables, continuous) data

• 𝑥 chart monitors between-sample variability (process variability over time)

• S and R charts monitor within-sample variability (instantaneous process variability at a given time)

• When possible use the sample standard deviation (s) instead of range (R).

• s is a better estimator of the true standard deviation σ

• Range is typically used for manual calculations

• R was used widely before the era of calculators and computers



• Variables 𝑥-R and 𝑥-s charts assume a normal distribution

• However, even if the process is not normally distributed, the results will be approximately correct because of the Central Limit Theorem

• Not very efficient at detecting small shifts in the process mean (2𝜎 or smaller)

• To detect smaller shifts, use n = 10 to 25 (ARL… more on this later)

• No “memory”: every point is calculated based on the units taken in that particular sample, does not take into account previous behavior of the chart (previous points)



• Definitions:

• 𝑥 = arithmetic mean of subgroup sample measurements

• R = range = max – min

• S = sample standard deviation

• 𝑋 = grand average = mean of all means

• 𝑅 = mean of range values

• 𝑠 = mean of sample standard deviation values

• m = # of subgroups

• n = # units on each subgroup



• 𝒙 Control limit formulae

• UCL = 𝑋 + A2 𝑅

• CL = 𝑋

• LCL = 𝑋 - A2 𝑅

• R Control limit formulae

• UCL = D4 𝑅

• CL = 𝑅

• LCL = D3 𝑅

• A, B, d, D constants: refer to table

(next slide)

• 𝒙 Control limit formulae

• UCL = 𝑋 + A3 𝑠

• CL = 𝑋

• LCL = 𝑋 - A3 𝑠

• s Control limit formulae

• UCL = B4 𝑠

• CL = 𝑠

• LCL = B3 𝑠

• Estimated standard deviation

• 𝜎 = 𝑅

𝑑2



• Initial CLs are trial limits subject to subsequent revision

• Recalculate:

• Every week / month

• Every 25, 50, or 100 samples



• Example:• One of the critical quality

characteristics of your medical device is pull strength, how much tensile force it takes to separate two components.

• Process runs for 20 hours. You take a sample of n = 4 units every hour (# subgroups m = 20)

• Measure all units, and enter data into a table.




Pick which RULES you want to use



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90

80

70

60

50

Sample

Sa

mp

le M

ea

n

__X=70.89

UCL=88.77

LCL=53.00

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60

45

30

15

0

Sample

Sa

mp

le R

an

ge

_R=24.55

UCL=56.01

LCL=0

Xbar-R Chart of x1, ..., x4

𝑥-R Chart

1. No points outside the Control limits on either chart

2. No trends which may indicate the presence of special cause variation were detected by the rules

Process is in control



191715131197531

90

80

70

60

50

Sample

Sa

mp

le M

ea

n

__X=70.89

UCL=88.41

LCL=53.37

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18

12

6

0

Sample

Sa

mp

le S

tDe

v

_S=10.76

UCL=24.38

LCL=0

Xbar-S Chart of x1, ..., x4

𝑥-s Chart

1. No points outside the Control limits on either chart

2. No trends which may indicate the presence of special cause variation were detected by the rules

Process is in control


• What would happen if one of the subgroups showed an abnormally low or abnormally high value?




• Xbar chart shows that the process is not in statistical control

• Identify source of special-cause variation

• Remove points OOC

• Recalculate CLs

191715131197531

90

80

70

60

50

Sample

Sa

mp

le M

ea

n

__

X=70.5

UCL=87.62

LCL=53.38

191715131197531

24

18

12

6

0

Sample

Sa

mp

le S

tDe

v

_S=10.51

UCL=23.82

LCL=0

1

1

Xbar-S Chart of x1, ..., x4 Test Results for Xbar Chart of

x1, ..., x4

TEST 1. One point more than 3.00 standard

deviations from center line.

Test Failed at points: 6, 20

Multiple Charts

• If a machine makes multiple units at a time (i.e. mold or die with multiple cavities) use separate charts for each cavity.

• To determine if the variation due to differences among cavities is significant to the process => Variance Components Analysis


Part 4

Control Chart Analysis

What is a “1 sigma shift” in the process mean 𝒙?

• Before the shift:• Mean = 70.89

• Standard Deviation = 5

• USL = 90

• Target = 70

• LSL = 50

• Cpk = 1.27

• Cpm = 1.31

• Ppm = 81

• After the shift:• Mean = 75.89 (↑ 1)

• Standard Deviation = 5

• USL = 90

• Target = 70

• LSL = 50

• Cpk = 0.94

• Cpm = 0.86

• Ppm = 2,386 ↑


How quickly can our chart detect a shift in the process mean?

Average Run Length (ARL)

• How quickly can our chart detect the presence of special cause variation?

• Shifts in process mean

• Changes in variability

• ARL: number of sampling points until a control chart signals an out-of-control situation

• ARL is used to measure control chart performance


Average Run Length (ARL) for X-bar charts• ARL1 => Process in control: we want this to be

very large

• This is our rate of false alarms when the process is in control

• ARL1 = 1 / α

• ± 3 sigma process = 99.73% within specs, Cpk = 1

• ± 4 sigma process = 99.9937% within specs, Cpk = 1.33

• For a process within ±3 sigma, α = 0.0027

• ARL = 1 / 0.0027 = 370

• For a process within ±3 sigma, there will be (on the average) one false OOC signal every 370 samples.


Average Run Length (ARL) for X-bar charts• ARL0 => Out of control process: we want

this to be very small

• We WANT to detect an OOC condition as soon as possible

• 𝑨𝑹𝑳 =𝟏

𝟏−𝜷

• 𝜷 = 𝝓 𝟑 − 𝒌 𝒏 − 𝝓(−𝟑 − 𝒌 𝒏)

• β = probability of detecting OOC condition

• k = shift in process mean, in sigmas

• n = sample size

• Φ = standard normal cumulative distribution function [ Excel: NORM.S.DIST(x,1) ]


Average Run Length (ARL) for X-bar charts


• To detect a shift in the mean of ± 1 sigma, with a sample size of n = 5 it will take (on the average) ARL0 = 4.5 => 5 sampling points after the shift

• If we sample every 30 minutes, that means our control chart will detect the process shift 2 ½ hours after it occurs

Average Run Length (ARL) for X-bar charts

• We can run the equation for various values of n and k, and create this table

• For a ±1 sigma shift n should be 12 or larger if we want to detect the shift immediately

• To improve this performance we need to increase the sample size, increase the sampling frequency, or both

• Shewhart charts are not very sensitive to small changes in the process mean



• Control limits are typically ± 3𝜎 from the average• Underlying assumption of normality

• If the underlying distribution is normal, 99.73% of the data will fall within the CLs

• If an observation falls outside the CLs, the probability that it is caused by common cause variation is only 0.27%

• When a point falls outside the CLs, there is a high probability that the process has changed – special cause variation

• Process is out of statistical control


Western Electric Rules

• Rules to determine if special cause variation is present in the process

1. One (1) point more that 3𝜎 from the centerline (on either side)

2. Two (2) out of three (3) points more than 2𝜎 from the centerline (same side)

3. Four (4) out of five (5) points more than 1𝜎 from the centerline (same side)

4. Nine (9) points in a row more than 1𝜎 from the centerline (either side)


AIAG (Automotive Industry Action Group) Alert Rules

• Rules to determine if special cause variation is present in the process

1. One (1) point more that 3𝜎 from the centerline (on either side)

2. Nine (9) points in a row on the same side of the centerline

3. Six (6) points in a row, all increasing or all decreasing

4. Fourteen (14) points in a row, alternating up and down

5. Two (2) out of three (3) points more than 2𝜎 from the centerline (same side)

6. Four (4) out of five (5) points more than 1𝜎 from the centerline (same side)

7. Fifteen (15) points in a row within 1𝜎 from the centerline (either side)

8. Eight (8) points in a row more than 1𝜎 from the centerline (either side)


Part 5

Control Charts for Discrete (Attributes) Data

p-Chart: % Items Non-Conforming

• Used to monitor the proportion nonconforming (% defective) in a process• # bad units / total # units

• p, np charts are based on the binomial distribution



• Definitions• n number of items examined (sample size)

• m number of subgroups

• X number of non-conforming items found in a sample n

• p probability that any one item will be non-conforming• % defective for the process

• typically UNKNOWN, must be estimated

• 𝑝 sample fraction nonconforming• % defective for each subgroup

• 𝑝 average fraction nonconforming • Average of all 𝑝



• Control Limits and formulae

• 𝑝 =𝑋

𝑛=# 𝑑𝑒𝑓𝑒𝑐𝑡𝑠 𝑖𝑛 𝑎 𝑠𝑎𝑚𝑝𝑙𝑒

𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒

• 𝑝 = 𝑖=1𝑚 𝑝𝑖

𝑚=

𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑝

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑏𝑔𝑟𝑜𝑢𝑝𝑠

• 𝑈𝐶𝐿 = 𝑝 + 3𝑝(1−𝑝)

𝑛

• 𝐶𝐿 = 𝑝

• 𝐿𝐶𝐿 = 𝑝 − 3𝑝(1−𝑝)

𝑛



• Example:

• Manufacturing process

• 1000 units are inspected from each day of production over a 12 day period

• m = 12

• n = 1000

• # defects tallied for each day



• Example




• 2 points outside of UCL

• Process is not in control

• OOC points must be investigated

• IF it is determined that these points are indeed OOC AND an assignable cause is identifiedTHEN

• Remove the OOC points and

• Recalculate the control limits121110987654321

0.035

0.030

0.025

0.020

0.015

0.010

0.005

0.000

Sample

Pro

po

rtio

n

_P=0.012

UCL=0.02233

LCL=0.00167

1

1

P Chart of Defects


• Chart with OOC points removed

• CL recalculated after removing OOC points



• If your sample sizes are NOT equal for all subgroups:

• Use the average sample size in the formulae instead of n

OR

• Use an np chart


60

np-Chart: # Items Non-Conforming


• Variation of the p chart

• Used to chart the actual number of non-conforming units

• Can accommodate unequal sample sizes, p-chart can’t

61



• Definitions

• n number of items examined (sample size)


• 𝑝 average fraction nonconforming • Average of all 𝑝

62



• Control Limits and formulae

• 𝑝 = 𝑖=1𝑚 𝑋𝑖

𝑚𝑛

• 𝑈𝐶𝐿 = 𝑛 𝑝 + 3 𝑛𝑝(1 − 𝑝)

• 𝐶𝐿 = 𝑛 𝑝

• 𝐿𝐶𝐿 = 𝑛 𝑝 − 3 𝑛𝑝(1 − 𝑝)

63



• Example (same as p-chart):

• Manufacturing process

• A variable number of units are inspected from each day of production over a 12 day period

• m = 12

• # units inspected and # of defective units tallied for each day

64



65



121110987654321

14

12

10

8

6

4

2

0

Sample

Sam

ple

Co

un

t

__NP=5.14

UCL=11.64

LCL=0

1

NP Chart of Defective Units

Tests performed with unequal sample sizes

• 1 point outside of UCL

• Process is not in control

Test Results for NP Chart of Defective Units

TEST 1. One point more than 3.00 standard deviations from center line.

Test Failed at points: 7

66

c-Chart: # Non-Conformities per Item(constant subgroups)


• Nonconformities: c-chart and u-chart are used when you want to monitor # of defects in a unit

• # defects / unit produced

• Example: when you build large medical devices (MRI machines, X-Ray machines, etc)

• c-chart monitors the actual number of nonconformities per unit

• u-chart monitors the average number of nonconformities

• c, u charts are based on the Poisson distribution

67



• Definitions:

• n number of units inspected, sample size (can be n=1 or greater)


• X number of nonconformities per unit inspected or per subgroup

• 𝑐 average number of nonconformities

68



• Formulae:

• 𝑐 = 𝑖=1𝑚 𝑋𝑖

𝑚

• Control limits

• 𝑈𝐶𝐿 = 𝑐 + 3 𝑐

• 𝐶𝐿 = 𝑐

• 𝐿𝐶𝐿 = 𝑐 − 3 𝑐

69



• Example:

• Manufacturing process for a medical device

• 20 units chosen per day for inspection, over a period of 24 days• constant subgroups

• m = 24

• n = 20

70



71



2321191715131197531

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15

10

5

0

Sample

Sam

ple

Co

un

t

_C=10.21

UCL=19.79

LCL=0.62

C Chart of No. of defects

• No points outside CLs

• Process is in control

72

u-Chart: # Non-Conformities per Item(non-constant subgroups)


• Definitions:

• n number of units inspected, sample size (can be n=1 or greater)


• X number of nonconformities per unit inspected or per subgroup

• ui average number of nonconformities for the i-th subgroups

• 𝑢 overall average number of nonconformities per unit

73



• Formulae:

• 𝑢 = 𝑢 𝑖=1𝑚 𝑢𝑖

𝑚

• 𝑢𝑖 =𝑋𝑖

𝑛

• Control limits

• 𝑈𝐶𝐿 = 𝑢 + 3 𝑢

𝑛

• 𝐶𝐿 = 𝑢

• 𝐿𝐶𝐿 = 𝑢 − 3 𝑢

𝑛

74



• Example (same as for c-chart):

• Manufacturing process for a medical device

• n units chosen per day for inspection, over a period of 24 days

• Non-constant subgroups

• m = 24

75



76



• No points outside CLs

• Process is in control

2321191715131197531

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0.8

0.6

0.4

0.2

0.0

Sample

Sam

ple

Co

un

t P

er

Un

it

_U=0.509

UCL=1.001

LCL=0.018

U Chart of No. of defects

Tests performed with unequal sample sizes

Part 6

Control Charts for Continuous (Variables) Data

- Individuals-

Control Charts for Individuals• When to use these:

• When you want to detect small shifts in the process mean

• Automated inspection and measurement technology is used, every manufactured unit is measured

• Process with very slow production rates or very low throughput

• Where measurements are dangerous, expensive, difficult

• When it’s inconvenient, or very expensive, to allow sample sizes n>1

• When repeat measurements on the process differ only because of lab or analysis error (i.e. many chemical processes)

• CuSum and EWMA charts have memory: calculation for each point takes into account previous behavior of the chart


Control Charts for Individuals

• I-MR: Individuals – Moving Range

• CuSum: Cumulative Sums

• MA: Moving Average

• EWMA: Evolutionary Weighted Moving Average


I-MR Charts

• Individuals – Moving Range

• I chart : plots the measured individual value• Subgroups m consist of n = 1 (a single observation)

• Moving range chart : plots the range of two successive values


I-MR Charts

• Definitions:• m = # of subgroups

• n = # units on each subgroup = 1

• 𝑥 = arithmetic mean of all individual measurements

• MR = moving range = 𝑥𝑖 − 𝑥𝑖−1• 𝑀𝑅 = sum of all moving ranges, divided by m-1

• d, D constants: refer to table


I-MR Charts

• Control limits for I chart

• UCL = 𝑥 + 3𝑀𝑅

𝑑2

• CL = 𝑥

• LCL = 𝑥 − 3𝑀𝑅

𝑑2

• Control limits for MR chart

• UCL = 𝐷4 𝑀𝑅

• CL = 𝑀𝑅

• LCL = 𝐷3 𝑀𝑅


I-MR Charts

• Example:• Filling process

• 24 consecutive units are weighed


I-MR Charts


I-MR Charts


2321191715131197531

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20.25

20.00

19.75

19.50

Observation

Ind

ivid

ua

l Va

lue

_X=20.005

UCL=20.587

LCL=19.423

2321191715131197531

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0.75

0.50

0.25

0.00

Observation

Mo

vin

g R

an

ge

__MR=0.451

UCL=0.955

LCL=0

I-MR Chart of Volume

86

CuSum Charts


• Use this chart when you need to detect small shifts in the process mean• Large sudden shifts are not detected as fast as in Shewhart charts

• Not as intuitive of simple as Xbar-R / Xbar-s

• Shows variation in a pattern centered around zero (0). • If the process is in control, the CuSum chart will remain in this pattern

centered around 0

• If the process mean shifts upwards / downwards, the CuSum chart will eventually shift in that same direction

• CuSum chart is based on the cumulative sum of all deviations up to the most recent point.

87

CuSum Charts


• Graph consists of two one-sided CUSUMs. • Upper CUSUM detects upward shifts in the level of the process• Lower CUSUM detects downward shifts. • This chart uses control limits (UCL and LCL) to determine when an out-of-

control situation has occurred.

• More complex to set-up and run than EWMA

• More efficient than EWMA: can detect OOC condition faster

• Patterns and trends in CuSum and EWMA do not necessarily indicate an OOC process• Focus on points outside CL

88

CuSum Charts


Definitions & Formulae:

• 𝐶𝑖 = 𝑗=1𝑖 ( 𝑥𝑗 − 𝜇0)

• Ci : cumulative sum of deviations up to and including the i-th sample

• 𝑥𝑗:mean of the j-th sample

• 𝜇0: target value for the process average

• H: number of 𝜎 between the centerline and the control limits (typical: 3 – 5 )

• Cumulative sum of deviations

• 𝐶1+ = max 0, 𝑥1 − 𝜇0 + 𝐾 + 𝐶0

+

• 𝐶1− = min(0, 𝑥1 − 𝜇0 − 𝐾 + 𝐶0

−)

• 𝐾 =𝜇1−𝜇0

2

• 𝜇0: target mean

• 𝜇1: out of control mean that we want to be able to detect

• If 𝜇1 is unknown, we can use K = k𝜎, where k is the number of standard deviations

• K = 1.5𝜎 means we wish to detect a shift from 𝜇0 to 𝜇0±1.5𝜎

89

CuSum Charts


• Example:

• Characteristic of interest: weight

• Target mean 𝜇0= 20 mg

• From past experience, 𝜎 = 0.2 mg

• If the process deviates by ± half of the process standard deviation, it’s considered out of control

• K = 0.5 𝝈 = 0.1 mg

• Decision interval H = ±5𝜎 = ±5(0.2) = ± 1

• If any of the calculated cumulative sums fall outside the CLs, the process is out of control

90

CuSum Charts


91

CuSum Charts


Test Results for CUSUM Chart of Volume

TEST. One point beyond control limits.

Test Failed at points: 2, 4, 5, 6, 7, 8, 9, 10,

11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,

22, 23, 24

2321191715131197531

0.75

0.50

0.25

0.00

-0.25

-0.50

Sample

Cu

mu

lati

ve S

um

0

UCL=0.2

LCL=-0.2

CUSUM Chart of Volume

92

Moving Average Charts


• Uses moving average of a certain span (not necessarily consecutive observations)

• Unweighted average of the observations

• When it is used:• When you want to detect small shifts in process average

• When data is collected periodically

• When it takes time to produce a single item

• When you want to dampen the effects of over-control

• Use it to monitor stable processes

• Poor predictor when the process has trends

93



• Definitions and Formulae:

• 𝜎 = process standard deviation

• 𝑤 = span of moving average

• MA = moving average

• MA =(𝑥𝑖+𝑥𝑖−1+⋯+𝑥𝑖−𝑤+1)

𝑤

• The more points we have in the span, the less sensitive M becomes to individual x values

• Control limits

• 𝑈𝐶𝐿 = 𝜇0 +3𝜎

𝑤

• 𝐶𝐿 = 𝜇0• 𝐿𝐶𝐿 = 𝜇0 −

3𝜎

𝑤

94



• Example:




• Want to use

• w = 2 (only count last two values)

95



96



2321191715131197531

20.50

20.25

20.00

19.75

19.50

Sample

Mo

vin

g A

vera

ge

__X=20.005

UCL=20.386

LCL=19.623

Moving Average Chart of Volume

97

EWMA Charts


• EWMA : Exponentially-Weighted Moving Average

• Used to detect small shifts in the process mean

• Chart is based on the current observation and the most recent past observations

• Averages the data so that weight of older data points is reduced

• Patterns and trends in CuSum and EWMA do not necessarily indicate an OOC process

• Focus on points outside CL

• Easier to set-up and run than CuSum

• Less efficient than CuSum: take longer to detect OOC condition

EWMA Charts

• 𝝀 (weighting factor) parameter determines the rate at which data points enter the EWMA statistic

• Large 𝜆 : more weight to recent data, less to older data

• Small 𝜆 : more weight to older data, less to recent data

• Typically 0.05 < 𝜆 < 0.25

• 𝝀= 1 means that only the most recent measurements influence the chart (like in Shewhart charts)


99

EWMA Charts


• Definitions and Formulae:

• 𝑧𝑖 = 𝜆𝑥𝑖 + (1 − 𝜆)𝑧𝑖−1

• 𝜆 = weight, between 0 and 1

• [*] In practice 0.05 < 𝜆 < 0.25

• 𝑥𝑖 = current observation

• 𝑧𝑖−1= previous zi statistic

• Initially z0 = 𝜇0

• 𝜇0 = process target mean

• 𝜎 = process standard deviation

• L = width of control limits

• [*] In practice 2 < L < 3

• 𝜆=1 and L=3 : ± 3 𝜎 CLs

• Control limits

• 𝑈𝐶𝐿 = 𝜇0 + 𝐿𝜎𝜆

2−𝜆

• 𝐶𝐿 = 𝜇0

• 𝐿𝐶𝐿 = 𝜇0 − 𝐿𝜎𝜆

2−𝜆

[*] CQE BoK p.516

100

EWMA Charts


• Example:




• Want to use

• 𝜆 = 0.10

• L = 2

101

EWMA Charts


EWMA Charts


103

EWMA Charts


2321191715131197531

20.2

20.1

20.0

19.9

19.8

Sample

EW

MA __

X=20

UCL=20.2000

LCL=19.8000

EWMA Chart of Volume

Part 7

Short-Run Charts

Short-Run Charts

• Control Charts discussed so far are intended for long, continuous production runs

• Short Run Charts can be used for short production runs• Built – to – order

• Quick turnaround products

• Small runs: 10 – 20 units


Short-Run Charts

• Example: Short-Run Attributes c-chart

• 𝑧𝑖 =𝑐𝑖− 𝑐

𝑐where:

• 𝑐𝑖 = # nonconformities in unit i

• 𝑐 = # total of nonconformities / # total of units

• Standardized random variable zi is normally distributed

• Center line of Short-Run chart is zero (0)

• Control limits are ± 3


Short-Run Charts

• n = 10 units


𝒛𝟏 =(𝟒 − 𝟐. 𝟓)

𝟐. 𝟓= 𝟎. 𝟗𝟓

Short-Run Charts


-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

1 2 3 4 5 6 7 8 9 10

Zi

Units

Short-Run Chart• Any points outside the

control limits (± 3) indicate the presence of special cause variation and a process that is not in a state of statistical control

Part 8

Calculation of % Non-Conforming and Process Capability

from Control Charts for Variables

Process Capability from Variable Control Charts

1. Verify that process is STABLE

• No special-cause variation present after 20 to 30 points are plotted

2. Determine if process data follows a normal distribution

• Normal probability plot + “Fat pencil method”

• Histogram

• Normality tests

• If data is not normal, use a transformation

3. If the data is normal, use the normal distribution to estimate process capability

• Use control chart values to estimate µ and

• 𝜇 ⇒ 𝑥

• 𝜎 ⇒ 𝑅

𝑑2

• Calculate estimated % defective

• Calculate estimated Cp, Cpk, Cpm


Example: Process Capability for 𝑥-R chart

• 𝜇 ⇒ 𝑥 = 70.89

• 𝑅 = 24.55

• d2 (for n = 4) = 2.059

• 𝜎 ⇒ 𝑅

𝑑2= 24.55 / 2.059 = 11.92

• USL = 90N

• T = Target = 70N

• LSL = 50N

• 𝑍 =𝑋 − 𝑥

𝜎



• AD test: not significant (p-value < 0.05)=> data can be considered to follow a normal distribution

• Also:FAT PENCIL TEST

Example: Process Capability for 𝑥-R chart

• Fraction within specs

• P(LSL < X < USL)

= 𝑃𝐿𝑆𝐿− 𝜇

𝜎< 𝑍 <

𝑈𝑆𝐿− 𝜇

𝜎

= 𝑃50−70.89

11.92< 𝑍 <

90−70.78

11.92

= P (-1.75 < Z < 1.6)

= P (Z < 1.6) – P (Z < -1.75)

= NORM.S.DIST(1.6,1)-NORM.S.DIST(-1.75,1)

= 0.905 = 90.5 %

• Fraction outside specs

= 1 – 0.905 = 0.095

= 9.5 %

• Ppm = fraction OOS * 1E6

= 94,289

• Not very good


Process Capability from Variable Control Charts

• 𝐶𝑝 =(𝑈𝑆𝐿 −𝐿𝑆𝐿)

6 𝜎= (90-50) / (6*11.92) = 0.56

• 𝐶𝑝𝑘 = min(𝑈𝑆𝐿− 𝑥

3 𝜎, 𝑥−𝐿𝑆𝐿

3 𝜎)

= min [(90 – 70.89)/(3*11.92) , (70.89 – 50)/(3*11.92)] = min [0.534, 0.584] = 0.534

• 𝐶𝑝𝑚 =𝑈𝑆𝐿 −𝐿𝑆𝐿

6√ 𝜎2+( 𝑥−𝑇)2

= (90 – 70) / (6* √(11.92^2 + (70.89-70)^2)) = 0.56

• Process is in control BUT not very capable


Process Capability from Variable Control Charts• In Excel


116

Which Control Chart Should I Use?Shewhart charts


Control Chart Type Use

Xbar – R Variables data. Charts subgroups. Manual calculations (R is easier to calculate than S)

Xbar – s Variables data. Charts subgroups. Use computer or scientific calculator

p chart Attributes data. Percent units non-conforming

np chart Attributes data. Number of units non-conforming

c chart Attributes data. Nonconformities per unit, constant subgroup size.Use this when you tally # nonconformities per unit, for example in a large medical device

u chart Attributes data. Nonconformities per unit, non-constant subgroup sizeUse this when you tally # nonconformities per unit, for example in a large medical device; and the process throughput is not consistent.

117

Which Control Chart Should I Use?Non-Shewhart charts


Control Chart Type Use

I-MR (individualunits – moving range)

Variables data. Charts Individual units. Very slow or low volume processes, processeswhere measurements are very difficult or expensive, destructive testing of expensive units

CuSum Time-weighted Variables data. Detect very small shifts in the process mean. Provides faster alert of OOC condition than EWMA

EWMA Time-weighted Variables data. Detect very small shifts in the process mean. Easier to set up and run than CuSum

Pre-Control Control process average. Can use as set-up for short production runs

Short Run Small runs, small sample sizes. Exploratory and monitoring tool.

118

Q & A


References1. Automotive Industry Action Group (1991). Fundamental Statistical Process Control, AIAG,

Southfield, MI.

2. Bessegato, L.F.. Table of Control Chart Constants. http://www.bessegato.com.br/UFJF/resources/table_of_control_chart_constants_old.pdf

3. Borror, Connie (editor). The Certified Quality Engineer Handbook. American Society for Quality (ASQ) 3rd Ed. 2008.

4. CuSum Control Charts. NIST Engineering Statistics Handbook. Section 6.3.2.3http://www.itl.nist.gov/div898/handbook//pmc/section3/pmc323.htm

5. EWMA Control Charts. NIST Engineering Statistics Handbook. Sections 2.2.2.1.1 , 6.3.2.4

http://www.itl.nist.gov/div898/handbook/mpc/section2/mpc2211.htmhttp://www.itl.nist.gov/div898/handbook//pmc/section3/pmc324.htm

6. Ge, Zhiqiang, and Zhihuan Song. Multivariate Statistical Process Control. Springer-Verlag London. 2013.

7. Huat, N.K. and H. Midi, 2011. The performance of robust control chart for change in variance. Trends Applied Sci. Res., 6: 1172-1184.

8. Juran, Joseph and Godfrey, AB. Juran’s Quality Handbook. Mc. Graw-Hill. 5th Ed. 1999. Section 45 Statistical Process Control

119(c) 2017 Raul SotoIVT Amsterdam MAR2017

http://www.bessegato.com.br/UFJF/resources/table_of_control_chart_constants_old.pdf

http://www.itl.nist.gov/div898/handbook/mpc/section2/mpc2211.htm

http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc324.htm

References9. Keats, JB and Montgomery, Douglas. Statistical Process Control in Manufacturing. ASQ Press.

1991.

10. Montgomery, Douglas. Statistical Quality Control. John Wiley & Sons / ASQC Quality Press. 2nd Ed. 1991.

11. Spanos, Costas. Introduction to Statistical Process Control. UC Berkeley EE290H Lecture 10.https://inst.eecs.berkeley.edu/~ee290h/fa05/Lectures/PDF/lecture%2010%20intro%20to%20SPC.pdf

12. Spanos, Costas. Control Charts for Attributes. UC Berkeley EE290H Lecture 11.https://inst.eecs.berkeley.edu/~ee290h/fa05/Lectures/PDF/lecture%2011%20attribute%20charts.pdf

13. Spanos, Costas. Control Charts for Variables. UC Berkeley EE290H Lecture 12.https://inst.eecs.berkeley.edu/~ee290h/fa03/handouts/lectures/lecture%2012%20variable%20charts.pdf

14. Spanos, Costas. Control Charts and Data Integration. UC Berkeley EE290H Lecture 13.https://inst.eecs.berkeley.edu/~ee290h/fa03/handouts/lectures/lecture%2013%20spc%20and%20data%20integration.pdf

120(c) 2017 Raul SotoIVT Amsterdam MAR2017

https://inst.eecs.berkeley.edu/~ee290h/fa05/Lectures/PDF/lecture 10 intro to SPC.pdf

https://inst.eecs.berkeley.edu/~ee290h/fa05/Lectures/PDF/lecture 11 attribute charts.pdf

https://inst.eecs.berkeley.edu/~ee290h/fa03/handouts/lectures/lecture 12 variable charts.pdf

https://inst.eecs.berkeley.edu/~ee290h/fa03/handouts/lectures/lecture 13 spc and data integration.pdf

References

15. Spanos, Costas. CuSum, MA, and EWMA Control Charts. UC Berkeley EE290H Lecture 14. https://inst.eecs.berkeley.edu/~ee290h/fa05/Lectures/PDF/lecture%2014%20CUSUM%20and%20EWMA.pdf

16. Stapenhurst, Tim. Mastering Statistical Process Control. Elsevier, 2005.

17. Western Electric (1956). Statistical Quality Control Handbook, Western Electric Corporation, Indianapolis, Indiana.

18. Wetherington, Les. Evaluation of Cumulative Sum (CuSum) and Exponentially Weighted Moving Average (EWMA) Control Charts to Detect Change in Underlying Demand Trends of Naval Aviation Spares (Master Thesis). United States Navy, Naval Postgraduate School. September 2010. http://dtic.mil/dtic/tr/fulltext/u2/a531664.pdf


https://inst.eecs.berkeley.edu/~ee290h/fa05/Lectures/PDF/lecture 14 CUSUM and EWMA.pdf

http://dtic.mil/dtic/tr/fulltext/u2/a531664.pdf

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