Statistical_Quality_Control.ppt

 W. Edward Deming advocated the implementation of a statistical quality management
approach.
 His philosophy behind this approach is ‘reduce variation’- fundamental to
 the principle of continuous improvement and
 the achievement of consistency, reliability, and uniformity.
 It helps in trustworthiness, competitive position, and success.
 Statistical Quality Control
 Statistics: data sufficient enough to obtain a reliable result.
 Quality: relative term and can be defined as totality of features and characteristics of a
product or service that bear on its ability to satisfy stated or implied need (ISO).
 Control: The operational techniques and activities (a system for measuring and checking)
used to fulfill the requirements for quality. It
 incorporates a feedback mechanism system to explore the causes of poor quality
or unsatisfactory performance and
 takes corrective actions.
 also suggests when to inspect, how often to inspect, and how much to inspect.
Basic Concept

 A quality control system using statistical techniques to control quality by
performing
 inspection,
 testing and
 analysis
to conclude whether the product is as stated or designed quality standard.
 Relying on the probability theory, SQC
 evaluates batch quality and
 controls the quality of processes or products
 It makes the inspection more reliable and less costly.
 The basis of the measurement is the performance indicator, either individual,
group or departmental calculated over time (hourly, daily, or weekly).
 These performance measures are plotted on a chart.
 Pattern obtained from plotting these measures are basis of taking
appropriate actions so that
 The process variation in minimized and
 Major problems are prevented in future.
 The timing and type of, and responsibility for, these actions depends on
whether the causes of variation is controlled or uncontrolled
Basic Concept ….Cont’d

 In repetitive manufacture of a product, even with refined machinery,
skilled operator, and selected material, variations are inevitable in the
quality of units produced due to interactions of various causes.
 Variation may be due to
 Common or random causes of variation (as a result of normal
variation in material, method, and so on that causes natural variation
in product or process quality) resulting in stable pattern of variation.
 Special causes (changes in men, machine, materials or tools, jigs and
fixture and so on) resulting in a shift from the stable pattern of
variation.
 SQC assists in timely identification and elimination of the problem with
an object of reducing variations in process or product.
The application of statistical method of collecting and analyzing
inspection and other data for setting the economic standards of
product quality and maintaining adherence to the standards so that
variation in product quality can be controlled
Basic Concept ….Cont’d

Statistical quality control (SQC) is the term used to describe the
set of statistical tools used by quality professionals
SQC encompasses three broad categories of;
 Descriptive statistics
 used to describe quality characteristics and relationships.
 the mean, standard deviation, and range.
 Acceptance sampling used to randomly inspect a batch of products to
determine acceptance or rejection of entire lot based on the results.
 Does not help to identify and catch the in-process problems
 Statistical process control (SPC)
 Involves inspecting the output from a process
 Quality characteristics are measured and charted
 Helpful in identifying in-process variations
Three SQC Categories

Variability: Sources of Variation
 Variation exists in all processes.
 Variation can be categorized as either;
 Common or Random causes of variation, or
Random causes that cannot be identified
Unavoidable: inherent in the process
Normal variation in process variables such as material,
environment, method and so on.
 Can be reduced almost to zero only through improvements in
the process variables.
 Assignable causes of variation
Causes can be identified and eliminated
e.g. poor employee training, worn tool, machine needing repair
 Can be controlled by operator but it needs attention of
management.

Traditional Statistical Tools
Descriptive Statistics include
Measure of accuracy (centering)
 Measure of central tendency indicating the central
position of the series.
 A measure of the central value is necessary to estimate
the accuracy or centering of a process.
 The Mean- simply the average of a set of data
 Sum of all the measurements/data divided by the
number of observations.
 The Median- simply the value of middle item if the
data are arranged in ascending or descending order.
 Applies directly if the number in the series is odd.
 It lies between two middle numbers if the number
of the series is even.
 The Mode- value that repeat itself maximum number
of times in the series.
Shape of Distribution of Observed Data
 A measure of distribution of data
 Normal or bell shaped
 Skewed
n
x
x
n
1
i
i



1
K
j
j
X
K






Distribution of Data
 Also a measure of quality
characteristics.
 Symmetric distribution - same
number of data are observed above
and below the mean.
This is what we see only when
normal variation is present in the
data
 Skewed distribution – a
disproportionate number of data are
observed either above or below the
mean.
Mean and median fall at different
points in the distribution
Centre of gravity is shifted to
oneside or other.

Traditional Statistical Tools …cont’d
Measure of Precision or Spread
 Reveals the extent to which numerical data
tend to spread about the mean value.
 The Range- the simplest possible measure
of dispersion.
 Difference between largest and smallest
observations in a set of data.
o Depends on sample size and it tends
to increase as sample size increases.
o Remains the same despite changes
in values lying between two extreme
values.
 Standard Deviation- a measure deviation
of the values from the mean.
 Small values >> data are closely
clustered around the mean
 Large values >> data are spread out
around the mean.
 
1
n
X
x
σ
n
1
i
2
i






Statistical Process Control
Process Control
 Refers to procedures or techniques adopted to evaluate, maintain and
improve the quality standard in various stages of manufacture.
 A process is considered satisfactory as long as it produces items within
designed specification.
Process should be continuously monitored to ensure that the
process behaves as it is expected.
 Salient features of process control
Controling the process at the right level and variability.
Detecting the deviation as quickly as possible so as to take
immediate corrective actions.
Ultimate aim is not only to detect trouble, but also to find out the
cause.
Developing an efficient information system in order to establish an
efficient system of process control.

Statistical Process Control (SPC)
 Statistical evaluation of the output of a process during production.
 Goal is to make the process stable over time and then keep it stable unless the
planned changes are made.
 Statistical description of stability requires that ‘pattern of variation’ remains
stable over time, not that there be no variation in the variable measured.
 In statistical process control language:
 A process that is in control has only common or random cause variation -
an inherent variability of the system.
 When the normal functioning of the prosess is disturbed by some
unpredictable events, special cause variation is added to common cause
variation.
Applying SPC to service
 Nature of defect is different in services
 Service defect is a failure to meet customer requirements
 One way to deal with service quality is to devise quantifiable measurement
of service elements
 Number of complaints received per month,
 Number of telephone rings before call is answered

 Hospitals
 timeliness and quickness of care, staff responses to requests, accuracy of lab
tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance and
checkouts
 Grocery Stores
 waiting time to check out, frequency of out-of-stock items, quality of food
items, cleanliness, customer complaints, checkout register errors
 Airlines
 flight delays, lost luggage and luggage handling, waiting time at ticket counters
and check-in, agent and flight attendant courtesy, accurate flight information,
passenger cabin cleanliness and maintenance
 Fast-Food Restaurants
 waiting time for service, customer complaints, cleanliness, food quality, order
accuracy, employee courtesy
 Catalogue-Order Companies
 order accuracy, operator knowledge and courtesy, packaging, delivery time,
phone order waiting time
 Insurance Companies
 billing accuracy, timeliness of claims processing, agent availability and response
time

Statistical Process Control: Control Chart
Control Chart
 A graphical display of data over time (data are displayed in time sequence in
which they occurred/measured) used to differentiate common cause variation
from special cause variation.
 Control charts combine numerical and graphical description of data with the use
of sampling distribution
 normal distribution is basis for control chart.
 Goal of using this chart is to achieve and mainatain process stability
 A state in which a process has displayed a certain degree of consistency
 Consistency is characterized by a stream of data falling within the
control limits.
Basic Components of a Control Chart
 A control chart always has
 a central line usually mathematical average of
all the samples plotted;
 upper control and lower control limits defining
the constraints of common variations or range
of acceptable variation;
 Performance data plotted over time.
 Lines are determined from historical data.

Control Chart …Cont’d
When to use a control chart?
 Controlling ongoing processes by finding and correcting problems as they occur.
 Predicting the expected range of outcomes from a process.
 Determining whether a process is stable (in statistical control).
 Analyzing patterns of process variation from special causes (non-routine events)
or common causes (built into the process).
 Determining whether the quality improvement project should aim to prevent
specific problems or to make fundamental changes to the process.
Control Chart Basic Procedure
 Choose the appropriate control chart for the data.
 Determine the appropriate time period for collecting and plotting data.
 Collect data, construct the chart and analyze the data.
 Look for “out-of-control signals” on the control chart.
When one is identified, mark it on the chart and investigate the cause.
Document how you investigated, what you learned, the cause and how it was
corrected.
 Continue to plot data as they are generated. As each new data point is plotted,
check for new out-of-control signals

 Interpretation of control chart
 Points between control limits are due to
random chance variation
 One or more data points above an UCL or
below a LCL mark statistically significant
changes in the process
 A process is in control if
 No sample points outside limits
 Most points near process average
 About equal number of points above and below centerline
 Points appear randomly distributed
 A process is assumed to be out of control if
 Rule 1: A single point plots outside the control limits;
 Rule 2: Two out of three consecutive points fall outside the two sigma warning limits on
the same side of the center line;
 Rule 3: Four out of five consecutive points fall beyond the 1 sigma limit on the same
side of the center line;
 Rule 4: Nine or more consecutive points fall to one side of the center line;
 Rule 5: There is a run of six or more consecutive points steadily increasing or
decreasing
Time period
Measured
characteristics

Setting Control Limits
 Type I error
 Concluding a process is not in control when it actually is.
 Type II error
 Concluding a process is in control when it is not.
In control Out of control
In control No Error
Type I error
(producers risk)
Out of control
Type II Error
(consumers risk)
No error
Mean
LCL UCL
/2 /2
Probability
of Type I error
Mean
LCL UCL
/2 /2
Probability
of Type I error

General model for a control chart
 UCL = μ + kσ
 CL = μ
 LCL = μ – kσ
where
μ is the mean of the variable
σ is the standard deviation of the variable
UCL=upper control limit; LCL = lower control limit;
CL = center line.
k is the distance of the control limits from the center line,
expressed in terms of standard deviation units.
 When k is set to 3, we speak of 3-sigma control charts.
 Historically, k = 3 has become an accepted standard in
industry.

Suggested Number of Data Points
 More data points means more delay
 Fewer data points means less precision, wider limits
 A tradeoff needs to be made between more delay and less
precision
 Generally 25 data points judged sufficient
Use smaller time periods to have more data points
Fewer cases may be used as approximation
Sample Size
 Attribute charts require larger sample sizes
50 to 100 parts in a sample
 Variable charts require smaller sample sizes
2 to 10 parts in a sample

Types of the control charts
 Variables control charts
Variable data are measured on a continuous scale.
 For example: time, weight, distance or temperature can be
measured in fractions or decimals.
Applied to data with continuous distribution
 Attributes control charts
Attribute data are counted and cannot have fractions or
decimals.
 Attribute data arise when you are determining only the
presence or absence of something:
 success or failure,
 accept or reject,
 correct or not correct.
 For example, a report can have four errors or five errors, but it
cannot have four and a half errors.
Applied to data following discrete distribution

 Variable control charts
 X-bar (mean chart)
 R chart (range chart)
 S chart (sigma chart)
 Individual or run chart
i-chart
Moving range chart
Median chart
EWMA (exponentially weighted moving average chart)
 General formulae for a control chart
 UCL or UAL = μ + kσx k = 3 ; Accepted Standard
 UWL = μ + 2/3 kσx
 CL = μ
 LWL = μ – 2/3 kσx
 LCL or LAL = μ – kσx
 

m
i
i
X
X
m


X
n



 Mean control charts
 Used to detect the
variations in mean of
a process.
 X-bar chart
 Range control charts
 Used to detect the changes in
dispersion or variability
of a process
 R chart
 System can show acceptable central
tendencies but unacceptable variability
or
 System can show acceptable variability
but unacceptable central tendencies
 Use X-bar and R charts together
 Sample size : 2 ~ 10
 Use X-bar and S charts together
 Sample size : > 10
 Use i-chart and Moving range chart
together
 Sample size : 1 or one-at-time data
Interpret the R-chart first:
If R-chart is in control -> interpret the
X-bar chart -> (i) if in control: the
process is in control; (ii) if out of
control: the process average is out of
control
If R-chart is out of control: the
process variation is out of control ->
investigate the cause; no need to
interpret the X-bar chart

 Constructing a X-bar Chart:
A quality control inspector at the Cocoa Fizz soft drink company has taken
three samples with four observations each of the volume of bottles
filled. If the standard deviation of the bottling operation is 0.2 ounces,
use the below data to develop control charts with limits of 3 standard
deviations for the 16 oz. bottling operation.
 Centerline and 3-sigma
control limit formulas
Time 1 Time 2 Time 3
Observation 1 15.8 16.1 16.0
Observation 2 16.0 16.1 15.9
Observation 3 15.8 15.8 15.9
Observation 4 15.9 15.9 15.8
Sample
means (X-
bar)
15.875 15.975 15.9
Sample
ranges (R)
0.2 0.3 0.2
3
X X
UCL X 
 
3
X X
LCL X 
 
X
CL X

m
i
i
X
X
m



X
n


Where,
m: # of sample mean
n: # of observations in each sample

Centerline (x-double bar):
Control limits for±3σ limits:
Control Chart
 Plot the sample mean in the
sequence from which it was
generated and interpret the
pattern in the control chart.
 
 
15.875 15.975 15.9
x 15.92
3
 
    
 
 
 
    
 
 
x x
x x
.2
UCL x zσ 15.92 3 16.22
4
.2
LCL x zσ 15.92 3 15.62
4

Second Method for X-bar Chart using Range and A2 factor
 Use this method when standard deviation for the process
distribution is unknown.
 Control limits solution:
 Center line and 3-sigma
control Fomulas:
1
2
k
i
i
x
n
R
R
k
R R
or
d d n

 


 

;
;&
2
2
2
2
3
3
x
x
x
CL X
R
UCL X X A R
d n
R
LCL X X A R
d n

   
   

OBSERVATIONS(SLIP-RINGDIAMETER,CM)
SAMPLEk 1 2 3 4 5 x R
1 5.02 5.01 4.94 4.99 4.96 4.98 0.08
2 5.01 5.03 5.07 4.95 4.96 5.00 0.12
3 4.99 5.00 4.93 4.92 4.99 4.97 0.08
4 5.03 4.91 5.01 4.98 4.89 4.96 0.14
5 4.95 4.92 5.03 5.05 5.01 4.99 0.13
6 4.97 5.06 5.06 4.96 5.03 5.01 0.10
7 5.05 5.01 5.10 4.96 4.99 5.02 0.14
8 5.09 5.10 5.00 4.99 5.08 5.05 0.11
9 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
50.09 1.15
1 5.02 5.01 4.94 4.99 4.96 4.98 0.08
2 5.01 5.03 5.07 4.95 4.96 5.00 0.12
3 4.99 5.00 4.93 4.92 4.99 4.97 0.08
4 5.03 4.91 5.01 4.98 4.89 4.96 0.14
5 4.95 4.92 5.03 5.05 5.01 4.99 0.13
6 4.97 5.06 5.06 4.96 5.03 5.01 0.10
7 5.05 5.01 5.10 4.96 4.99 5.02 0.14
8 5.09 5.10 5.00 4.99 5.08 5.05 0.11
9 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
50.09 1.15
Calculate limits for X-bar chart using sample range

R-Chart:
 Always look at the Range chart first.
 The control limits on the X-bar chart are
derived from the average range, so if the
Range chart is out of control, then the
control limits on the X-bar chart are
meaningless.
 Look for out of control signal.
 If there are any, then the special causes
must be eliminated.
 There should be more than five distinct
values plotted, and no one value should
appear more than 25% of the time.
 If there are values repeated too often,
then you have inadequate resolution
of your measurements, which will
adversely affect your control limit calculations.
 Once the effect of the out of control points
from the Range chart is removed, look at
the X-bar Chart.
Standard Deviation of Range and
Standard Deviation of the process
is related as:
Centerline and 3-sigma
Control Limit Formulas:
Where
3
3
2
R
d
d R
d
 
 
3 3
4
2 2
3 3
3
2 2
3 1 3
3 1 3
R
R
R
CL R
d d
UCL R R R D R
d d
d d
LCL R R R D R
d d

    
    
( )
( )
d
D
d
  3
4
2
1 3
max( , )
d
D
d
  3
3
2
0 1 3

1 5.02 5.01 4.94 4.99 4.96 4.98 0.08
2 5.01 5.03 5.07 4.95 4.96 5.00 0.12
3 4.99 5.00 4.93 4.92 4.99 4.97 0.08
4 5.03 4.91 5.01 4.98 4.89 4.96 0.14
5 4.95 4.92 5.03 5.05 5.01 4.99 0.13
6 4.97 5.06 5.06 4.96 5.03 5.01 0.10
7 5.05 5.01 5.10 4.96 4.99 5.02 0.14
8 5.09 5.10 5.00 4.99 5.08 5.05 0.11
9 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
50.09 1.15
1 5.02 5.01 4.94 4.99 4.96 4.98 0.08
2 5.01 5.03 5.07 4.95 4.96 5.00 0.12
3 4.99 5.00 4.93 4.92 4.99 4.97 0.08
4 5.03 4.91 5.01 4.98 4.89 4.96 0.14
5 4.95 4.92 5.03 5.05 5.01 4.99 0.13
6 4.97 5.06 5.06 4.96 5.03 5.01 0.10
7 5.05 5.01 5.10 4.96 4.99 5.02 0.14
8 5.09 5.10 5.00 4.99 5.08 5.05 0.11
9 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
50.09 1.15
Calculate limits for R-chart

 S-Chart
 The sample standard deviations are
plotted in order to control the
process variability.
 For sample size (n>12),
 With larger samples, the
resulting mean range does not
give a good estimate of
standard deviation
 the S-chart is more efficient
than R-chart.
 For situations where sample size
exceeds 12, the X-bar chart and the
S-chart should be used to check the
process stability.
Centerline and 3-sigma Control Limit
Formulas:
Where
s
s
s
CL S
c
UCL S S B S
c
c
LCL S S B S
c


  

  
2
4
4
4
2
4
3
4
1
3
1
3
max( , )
c
B
c
c
B
c

 

 
2
4
4
4
2
4
3
4
1
1 3
1
0 1 3
( )
&
k
n
j j
i
j
i
j
S
x x
S S
n k



 


 2
1
1
1

Changing Sample Size on the X-bar and R Charts
 In some situations, it may be of interest to know the effect of changing
the sample size on the X-bar and R charts. Needed information:
 = average range for the old sample size
 = average range for the new sample size
 nold = old sample size
 nnew = new sample size
d2(old) = factor d2 for the old sample size
d2(new) = factor d2 for the new sample size
Centerline and 3-sigma Control Limit Formulas:
old
R
new
R
( )
( )
( )
( )
old
old
x chart
d new
UCL x A R
d old
d new
LCL x A R
d old

 
   
 
 
   
 
2
2
2
2
2
2
( )
( )
( )
( )
( )
max ,
( )
old
new old
old
R chart
d new
UCL D R
d old
d new
CL R R
d old
d new
LCL D R
d old

 
  
 
 
   
 
 
 
 
  
 
 
 
 
2
4
2
2
2
2
3
2
0

Control Chart: Interpreting the Patterns
Patterns
 A nonrandom identifiable arrangement of plotted points on the chart.
 Provides sufficient reasons to look for special causes.
 Causes that affect the process intermittently and
 can be due to periodic and persistent disturbances
 Natural pattern
 No identifiable arrangement of the plotted points exists
 No point falls outside the control limit;
 Majority of the points are near the centerline; and
 Few points are close to the control limits
 These patterns are indicative of a process that is in control.
 One point outside the control limits
 Also known as freaks and are caused by external disturbance
 Not difficult to identify the special causes for freaks. However, make sure
that no measurement or calculation error is associated with it,
 Sudden, very short lived power failure,
 Use of new tool for a brief test period or a broken tool,
 incomplete operation, failure of components

Interpreting the Patterns …cont’d
 Sudden shift in process mean
 A sudden change or jump in process mean or average service level.
 Afterward, the process becomes stable.
 This sudden change can occur due to changes- intentional or otherwise in
 Process settings e.g. temperature, pressure or depth of cut
 Number of tellers at the Bank,
 New operator, new equipment, new measurement instruments, new vendor
or new method of processing.
 Gradual shift in the process mean
 Such shift occurs when the process parameters change gradually over a period
of time.
 Afterward, the process stabilizes
 X-bar chart might exhibit such shift due to change in incoming quality of raw
materials or components over time, maintenance program or style of
supervision.
 R-chart might exhibit such shift due to a new operator, decrease in worker skill
due to fatigue or monotoy, or improvement in incoming quality of raw
materials.

 Trending pattern
 Trend represents changes that steadily increases or decreases.
 Trends do not stabilize or settle down
 X-bar chart may exhibit a trend because of tool wear, dirt or chip buildup, aging
of equipment.
 R-chart may exhibit trend because of gradual improvement of skill resulting
from on-the-job-training or a decrease in operator skill due to fatigue.
 Cyclic pattern
 A repetitive periodic behavior in the system.
 A high and low points will appear on the control chart
 X-bar chart may exhibit a cyclic behavior because of a rotation of operator,
periodic changes in temperature and humidity, seasonal variation of incoming
components, periodicity in mechanical or chemical properties of the material
 R-chart might exhibit cyclic pattern because of operator fatigue and subsequent
energization following breaks, a difference between shifts, or periodic
maintenance of equipment.
 Graph will not show cyclic pattern, if the samples are taken too infrequently

Zones for Pattern Test
UCL
LCL
Zone A
Zone B
Zone C
Zone C
Zone B
Zone A
Process
average
3 sigma = x + A2R
=
3 sigma = x - A2R
=
2 sigma = x + (A2R)
= 2
3
2 sigma = x - (A2R)
= 2
3
1 sigma = x + (A2R)
= 1
3
1 sigma = x - (A2R)
= 1
3
x
=
Sample number
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
UCL
LCL
Zone A
Zone B
Zone C
Zone C
Zone B
Zone A
Zone A
Zone B
Zone C
Zone C
Zone B
Zone A
Process
average
3 sigma = x + A2R
=
3 sigma = x - A2R
=
2 sigma = x + (A2R)
= 2
3
2 sigma = x - (A2R)
= 2
3
1 sigma = x + (A2R)
= 1
3
1 sigma = x - (A2R)
= 1
3
x
=
Sample number
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13

Performing a Pattern Test
OBSERVATIONS (SLIP- RING DIAMETER, CM)
SAMPLE k 1 2 3 4 5 x R
1 5.02 5.01 4.94 4.99 4.96 4.98 0.08
2 5.01 5.03 5.07 4.95 4.96 5.00 0.12
3 4.99 5.00 4.93 4.92 4.99 4.97 0.08
4 5.03 4.91 5.01 4.98 4.89 4.96 0.14
5 4.95 4.92 5.03 5.05 5.01 4.99 0.13
6 4.97 5.06 5.06 4.96 5.03 5.01 0.10
7 5.05 5.01 5.10 4.96 4.99 5.02 0.14
8 5.09 5.10 5.00 4.99 5.08 5.05 0.11
9 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.10
50.09 1.15

x
x-
- bar
bar
Chart
Chart
Example
Example
(cont.)
(cont.)
UCL = 5.08
LCL = 4.94
Mean
Sample number
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
5.10 –
5.08 –
5.06 –
5.04 –
5.02 –
5.00 –
4.98 –
4.96 –
4.94 –
4.92 –
x = 5.01
=
UCL = 5.08
LCL = 4.94
Mean
Sample number
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
5.10 –
5.08 –
5.06 –
5.04 –
5.02 –
5.00 –
4.98 –
4.96 –
4.94 –
4.92 –
x = 5.01
=
x = 5.01
=

1
1 4.98
4.98 B
B —
— B
B
2
2 5.00
5.00 B
B U
U C
C
3
3 4.95
4.95 B
B D
D A
A
4
4 4.96
4.96 B
B D
D A
A
5
5 4.99
4.99 B
B U
U C
C
6
6 5.01
5.01 —
— U
U C
C
7
7 5.02
5.02 A
A U
U C
C
8
8 5.05
5.05 A
A U
U B
B
9
9 5.08
5.08 A
A U
U A
A
10
10 5.03
5.03 A
A D
D B
B
SAMPLE
SAMPLE x
x ABOVE/BELOW
ABOVE/BELOW UP/DOWN
UP/DOWN ZONE
ZONE
1
1 4.98
4.98 B
B —
— B
B
2
2 5.00
5.00 B
B U
U C
C
3
3 4.95
4.95 B
B D
D A
A
4
4 4.96
4.96 B
B D
D A
A
5
5 4.99
4.99 B
B U
U C
C
6
6 5.01
5.01 —
— U
U C
C
7
7 5.02
5.02 A
A U
U C
C
8
8 5.05
5.05 A
A U
U B
B
9
9 5.08
5.08 A
A U
U A
A
10
10 5.03
5.03 A
A D
D B
B
SAMPLE
SAMPLE x
x ABOVE/BELOW
ABOVE/BELOW UP/DOWN
UP/DOWN ZONE
ZONE
1
1 4.98
4.98 B
B —
— B
B
2
2 5.00
5.00 B
B U
U C
C
3
3 4.95
4.95 B
B D
D A
A
4
4 4.96
4.96 B
B D
D A
A
5
5 4.99
4.99 B
B U
U C
C
6
6 5.01
5.01 —
— U
U C
C
7
7 5.02
5.02 A
A U
U C
C
8
8 5.05
5.05 A
A U
U B
B
9
9 5.08
5.08 A
A U
U A
A
10
10 5.03
5.03 A
A D
D B
B
SAMPLE
SAMPLE x
x ABOVE/BELOW
ABOVE/BELOW UP/DOWN
UP/DOWN ZONE
ZONE
SAMPLE
SAMPLE x
x ABOVE/BELOW
ABOVE/BELOW UP/DOWN
UP/DOWN ZONE
ZONE

 A process is assumed to be out of control if
 Rule 1: A single point plots outside the control
limits;
 Rule 2: Two out of three consecutive points fall
outside the two sigma warning limits on the same
side of the center line;
 Rule 3: Four out of five consecutive points fall
beyond the 1 sigma limit on the same side of the
center line;
 Rule 4: Nine or more consecutive points fall to
one side of the center line;
 Rule 5: There is a run of six or more consecutive
points steadily increasing or decreasing

Control Chart for Attributes
Attributes are discrete events: yes/no or pass/fail
Construction and interpretation are same as that of variable control charts.
Attributes control charts
 p chart
 Uses proportion nonconforming (defective) items in a sample
 Based on a binomial distribution
 Can be used for varying sample size.
 np chart
 Uses number of nonconforming items in a sample
 Should not be used when sample size varies
 c chart
 Uses total number of nonconformities or defects in samples of constant
size.
 Occurence of nonconformities follows poisson distribution.
 u chart
 when the sample size varies, the number of nonconformities per unit is
used as a basis for this control chart.

Control Chart: p chart
 Proportion nonconforming or defectives for each sample are plotted on the
p-chart
 The chart is examined to determine whether the process is in control.
 Means to calculate center line and control limits
 No standard or target value of proportion nonconforming is specified
 It must be estimated from sample infromation and
 For each sample, proportion of nonconforming items are
determined as
 The average of these individual sample proportion of
nonconforming items is used as the center line (CLp):
As true value of p is not known,
p-bar is used as an estimate
x
p
n

m m
i
i i
p
p x
CL p
m nm
  
 
( )
( )
p
p
p p
UCL p
n
p p
LCL p
n

 

 
1
3
1
3

20 samples of 100 pairs of jeans
NUMBER OF
NUMBER OF PROPORTION
PROPORTION
SAMPLE
SAMPLE DEFECTIVES
DEFECTIVES DEFECTIVE
DEFECTIVE
1
1 6
6 .06
.06
2
2 0
0 .00
.00
3
3 4
4 .04
.04
:
: :
: :
:
:
: :
: :
:
20
20 18
18 .18
.18
200
200
NUMBER OF
NUMBER OF PROPORTION
PROPORTION
SAMPLE
SAMPLE DEFECTIVES
DEFECTIVES DEFECTIVE
DEFECTIVE
1
1 6
6 .06
.06
2
2 0
0 .00
.00
3
3 4
4 .04
.04
:
: :
: :
:
:
: :
: :
:
20
20 18
18 .18
.18
200
200
UCL = p + z = 0.10 + 3
p(1 - p)
n
0.10(1 - 0.10)
100
UCL = 0.190
LCL = 0.010
LCL = p - z = 0.10 - 3
p(1 - p)
n
0.10(1 - 0.10)
100
= 200 / 20(100) = 0.10
total defectives
total sample observations
p =
UCL = p + z = 0.10 + 3
p(1 - p)
n
0.10(1 - 0.10)
100
UCL = 0.190
UCL = p + z = 0.10 + 3
p(1 - p)
n
0.10(1 - 0.10)
100
UCL = 0.190
LCL = 0.010
LCL = p - z = 0.10 - 3
p(1 - p)
n
0.10(1 - 0.10)
100
LCL = 0.010
LCL = p - z = 0.10 - 3
p(1 - p)
n
0.10(1 - 0.10)
100
LCL = p - z = 0.10 - 3
p(1 - p)
n
0.10(1 - 0.10)
100
= 200 / 20(100) = 0.10
total defectives
p = = 200 / 20(100) = 0.10
total defectives
p = = 200 / 20(100) = 0.10
total defectives
p =
 If the target or standard value is specified
 Center line is selected as that target value i.e.
CLp= po where, po represent a standard value
 Control limits are also based on the target velue.
 If the lower control limit for p is turned out to be negative, LCL is
simply counted as zero.
 Lowest possible value for proportion of nonconformng item is zero
Control Chart: p chart …Cont’d

Control Chart: p chart …Cont’d
 Variable sample size
 Changes in sample size casues the control limits to change, although the center
line remained fixed.
 Control limits can be constructed:
 For individual samples
 If no standard value is given and sample mean proportion nonconforming
is p-bar, control limit for sample i with size ni are
 Using average sample size
Where

 

 
( )
( )
i
i
p p
UCL p
n
p p
LCL p
n
1
3
1
3
( )
( )
p p
UCL p
n
p p
LCL p
n

 

 
1
3
1
3
m
i
i
n
n
m



1

Control Chart: c chart
 No standard given
 Average number of nonconformities per sample unit is found from the sample
observation and is denoted by c-bar.
 The center line and control limits are:
 If lower control limit is found to be less than zero, it is converted to zero.
 Standard given
 if the specified target for the number of nonconformities per sample unit be co..
The center line and control limits are then calculated from:
c
c
c
CL c
UCL c c
LCL c c

 
 
3
3
c o
c o o
o o o
CL c
UCL c c
LCL c c

 
 
3
3

Control Chart: c chart …Cont’d
Number of defects in 15 sample rooms
1 12
1 12
2 8
2 8
3 16
3 16
: :
: :
: :
: :
15 15
15 15
190
190
SAMPLE
SAMPLE
c
c = = 12.67
= = 12.67
190
190
15
15
UCL
UCL =
= c
c +
+ z
z
c
c
= 12.67 + 3 12.67
= 12.67 + 3 12.67
= 23.35
= 23.35
LCL
LCL =
= c
c +
+ z
z
c
c
= 12.67
= 12.67 -
- 3 12.67
3 12.67
= 1.99
= 1.99
NUMBER
OF
DEFECTS
1 12
1 12
2 8
2 8
3 16
3 16
: :
: :
: :
: :
15 15
15 15
190
190
SAMPLE
SAMPLE
c
c = = 12.67
= = 12.67
190
190
15
15
c
c = = 12.67
= = 12.67
190
190
15
15
c
c = = 12.67
= = 12.67
190
190
15
15
190
190
15
15
UCL
UCL =
= c
c +
+ z
z
c
c
= 12.67 + 3 12.67
= 12.67 + 3 12.67
= 23.35
= 23.35
UCL
UCL =
= c
c +
+ z
z
c
c
= 12.67 + 3 12.67
= 12.67 + 3 12.67
= 23.35
= 23.35
LCL
LCL =
= c
c +
+ z
z
c
c
= 12.67
= 12.67 -
- 3 12.67
3 12.67
= 1.99
= 1.99
NUMBER
OF
DEFECTS
3
3
6
6
9
9
12
12
15
15
18
18
21
21
24
24
Number
of
defects
Number
of
defects
Sample number
Sample number
2
2 4
4 6
6 8
8 10
10 12
12 14
14 16
16
UCL = 23.35
LCL = 1.99
c = 12.67
3
3
6
6
9
9
12
12
15
15
18
18
21
21
24
24
Number
of
defects
Number
of
defects
Sample number
Sample number
2
2 4
4 6
6 8
8 10
10 12
12 14
14 16
16
UCL = 23.35
LCL = 1.99
c = 12.67
3
3
6
6
9
9
12
12
15
15
18
18
21
21
24
24
Number
of
defects
Number
of
defects
Sample number
Sample number
2
2 4
4 6
6 8
8 10
10 12
12 14
14 16
16
2
2 4
4 6
6 8
8 10
10 12
12 14
14 16
16
UCL = 23.35
LCL = 1.99
c = 12.67

Process Capability Analysis
 Process Variability
 Natural variation in the process.
 Tolerance or specification
 Range of acceptable values established
design engineers or product design specialist.
 For the product to be considered acceptable, its quality characteristics must fall
within this preset range.
 Process Capability
 Process variability relative to specification.
 Relationship between the process variability and the tolerance can be
formulized by the consideration of standard deviation, σ of the process.
 In order to manufacture product within the specification,
the distance between the upper specification limit (USL) and the lower
specification limit (LSL) i.e. (USL-LSL) or 2T must be equal to or greater than
width of the process variability defined by the control limits, i.e. 6σ
Lower
Specification
Upper
Specification
6σ

Process Capability Analysis …Cont’d
 Relationship between the process width and specification
 The relationship between (USL - LSL) and 6σ results in three levels of
situations:
 Process variation is large relative to the specifications.
 A large percentage of the product will fall outside the specification.
 Process is not capable of meeting specifications all the time.
 The process cannot be considered capable regardless of the
process centering.
 Process variation must be reduced drastically
(a) Natural variation
exceeds design
specifications
Design
Specifications
Process

 Process variability closely matches the predefined specification.
 This is the absolute minimum requirements for the process to be
capable of producing the acceptable product.
 Almost all (99.74%) the output falls within the preset specifaction
range.
 Process is capable of meeting specifications most of the time.
 The process must remain well centered for the process capability to be
maintained at a tolerable level.
 Variation must be reduced and this will reduce the defectives per
million, cost of quality and increase profitability.
(b) Design specifications
and natural variation are
same
Design
Specifications
Process

 Process variation is small relative to the specifications.
 The process mean can shift about without causing the process to
degrade its capability
 Process is capable of meeting specifications all the time.
 This will reduce the defects per million (DPM), reduce the cost of
quality (COQ), and hence increase profitability.
Simply setting up control charts to monitor whether a process is in
control does not guarantee process capability.
(c) Design specifications
greater than natural
variation
Design
Specifications
Process

 Process Capability Index
 A measure that relates the actual performance of a process to its
specified performance.
 Calculated when a process is under statistical control (i.e. Only the
random or common causes of variation are present).
 Process capability can be quantified by the calculation of several
indices:
 Relative Precision Index (RPI)
 A quick and simple measure of the process potential.
 Process is considered to be centered over the specification range
 This index only deals with relative spread or variation.
 RPI is based on the ratio of mean sample range with the tolerance band
or width. i.e.
 To avoid production of defective product, specification
width must be greater than the process variation.
 Value of 6/dn is the minimum RPI
T
RPI
R

2
n
n
T
R
T
d
T
d
R




2 6
6
2
2 6

 Cp Index
 A measure of process potential to meet the specification limit.
 It is valuable index in measuring the process capability.
 It is computed as a ratio of specification width to the width of process
variability
when σ is unknown, it is obtained from its
estimates
 Three possible ranges for Cp
 Cp = 1, process variability just meets specifications and hence, process is
minimally capable.
0.26% of the product will not be acceptable.
 Cp < 1, process variability is outside the specification range and hence,
process is not capable of producing within specifications
 Cp > 1, process variability is tighter than the specification range and
process is capable of meeting the specification all the time.
A way to reduce the generation of defective product is to increase
the process capability.
p
USL LSL
C



6 s R
and
c d
 
 
4 2

Computing the Cp Value at Cocoa Fizz: three bottling machines are being evaluated for
possible use at the Fizz plant. The machines must be capable of meeting the design
specification of 15.8-16.2 oz. with at least a process capability index of 1.0 (Cp≥1)
The table below shows the information
gathered from production runs on each
machine. Are they all acceptable?
 Solution:
 Machine A
 Machine B
Cp =
 Machine C
Cp =
Machine σ USL-LSL 6σ
A .05 .4 .3
B .1 .4 .6
C .2 .4 1.2
USL LSL .4
C = 1.33
6σ 6(.05)
p

 

 Cp Index
 It has one shortcoming: Cp index assumes that the process is centered on the
specification range.
 Process may be off-centered and because of this, a proportion of products
will fall outside the specification range.
Using only
the Cp
measure
would lead
to an
incorrect
conclusion

 CpK Index
 Another measure of process capability
 This measure accounts for the location of the process mean and is used when
the process mean is not at the target value, which is assumed to be the
halfway between the specification limits.
 The process capability of each half of the normal distribution is computed and
minimum of the two is used. i.e.
where
µ = the mean of the process
σ = the standard deviation of the
process
min( , )
pk
USL LSL
C
 
 
 

3 3

Net weight specification = 9.0 oz  0.5 oz
Process mean = 8.80 oz
Process standard deviation = 0.12 oz
Cp =
= = 1.39
upper specification limit -
lower specification limit
6
9.5 - 8.5
6(0.12)
Cp =
= = 1.39
6
9.5 - 8.5
6(0.12)
Cp =
= = 1.39
6
6
9.5 - 8.5
6(0.12)
9.5 - 8.5
6(0.12)
Cpk = minimum
= minimum , = 0.83
x - lower specification limit
3
=
upper specification limit - x
3
=
,
8.80 - 8.50
3(0.12)
9.50 - 8.80
3(0.12)
Cpk = minimum
= minimum , = 0.83
3
=
3
=
,
3
=
3
=
,
8.80 - 8.50
3(0.12)
9.50 - 8.80
3(0.12)
8.80 - 8.50
3(0.12)
8.80 - 8.50
3(0.12)
9.50 - 8.80
3(0.12)
9.50 - 8.80
3(0.12)
 The Cp value leads us to conclude that the process is capable. Whereas, the Cpk
value is less than 1, revealing that process is not capable.
 Reasons for the difference in measures is that process is not centered on
specification range.

Process capability ratio, Cp =
specification width
process width
Upper specification – lower specification
6
Cp =
pk
L USL-
C =min or
3 3
LS
 
 

 
 
 
 If the process is centered use Cp
 If the process is not centered use Cpk

 Conclusion: The process is
 centered (Cp = Cpk), and
 of low capability since the indices are
only just greater than 1.
 Conclusion:
 Cp at 1.89 Indicates a potential for
higher capability than the result (i).
 The low Cpk shows that this
potential is not being realized as the
process is not centered.

Acceptance Sampling
Acceptance sampling is concerned with inspection and decision making
regarding the entire lot/production/shipment.
 A form of inspection used to accept or reject entire lot based on the sample
information.
 Not consistent with TQM or Zero Defects philosophy.
 producer and customer agree on the number of acceptable defects
 a means of identifying not preventing poor quality
Typical Application of Acceptance Sampling
 A company receives a shipment of product from a vendor.
 This product is a raw material used in the company’s manufacturing
process.
 A sample is taken from the shipment, and quality characteristic of the products
in the sample is inspected.
 Based on the information in this sample, a decision is made regarding
disposition of the shipment.
 Usually this decision is either to accept or to reject the entire shipment.
 Accepted lots are used into production, and
 Rejected lots are returned to vendor.

 The decision to accept or reject the shipment is based
on the following set standards:
 Lot size = N
 Sample size = n
 Acceptance number = c
 Defective items = d
 If d <= c, accept lot
 If d > c, reject lot
Acceptance Sampling …Cont’d

Three Important Aspects of Acceptance Sampling
 Purpose is to sentence the lot (accept or reject) rather than to estimate the lot
quality.
 Acceptance sampling plan does not provide any direct form of quality control. It
simply makes the decision whether to reject or accept the lot.
 Process controls are used to control and systematically improve quality
 Most effective use of acceptance sampling is
 Not to “inspect quality into the product,” but rather as audit tool to ensure
that ‘output of process conforms to requirements’.
Three Approaches to Lot Sentencing
 Accept with no inspection.
 100% inspection – inspect every item in the lot, remove all defectives
 Defectives – returned to vendor, reworked, replaced or discarded
 Acceptance sampling
Sample is taken from lot, a quality characteristic is inspected; then on the basis
of information in sample, a decision is made regarding lot disposition.

 When is Acceptance Sampling Useful?
 Testing is destructive and time consuming.
 100% inspection is not technologically feasible.
 100% inspection error rate results in higher percentage of defectives being passed
than is inherent to product.
 Cost of 100% inspection extremely high.
 Vender has excellent quality history so reduction from 100% is desired but not high
enough to eliminate inspection altogether.
 Potential for serious product liability risks; program for continuously monitoring
product required.
 When can Acceptance Sampling be Used?
 At any point in production
 The output of one stage is the input of the next
 At the input stage
 Prevents goods that don’t meet standards from entering into the process
 This saves rework time and money
 At the output stage
 Can reduce the risk of bad quality being passed on from the process to a
consumer
 This can prevent the loss of prestige, customers, and money

Advantages of Acceptance Sampling
 It is usually less expensive because there is less inspection.
 There is less handling of the product, hence reduced damage.
 It is applicable to destructive testing.
 Fewer personnel are involved in inspection activities.
 It often greatly reduces the amount of inspection error.
 The rejection of entire lots are opposed to the sample return of defectives often
provides a stronger motivation to the vendor for quality improvements.
Disadvantages of Acceptance Sampling
 Always a risk of accepting “bad” lots and rejecting “good” lots
 Producer’s Risk: chance of rejecting a “good” lot – 
 Consumer’s Risk: chance of accepting a “bad” lot – 
 Less information is generated about the product or the process that
manufactured the product.
 Requires added planning and documentation of the acceptance sampling
procedure – 100% inspection does not.

Risks of Acceptance Sampling
 Producer’s Risk(α)
 Refers to the probability of rejecting a good lot
 Also refered to as Type I error.
 Acceptable Quality Level (AQL)
 The numerical definition of a ‘good lot’.
 ANSI/ASQC describes AQL as:
 The percentage level of defective or nonconforming items at which the
customer is willing to accept a lot as good.
 Consumer’s Risk (β)
 Refers to the probability of accepting a bad lot
 Also refered to as Type II error
 Lot Tolerance Percentage Defective (LTPD)
 Numerical definition of a ‘bad lot’
 ANSI/ASQC describes LTPD as:
 Upper limit on the percentage defectives that a customer is willing to
accept

 Lot Formation
 Lots should be homogeneous
 Units in a lot should be produced by the same:
 machines,
 operators,
 from common raw materials,
 approximately same time
 If lots are not homogeneous – acceptance-sampling scheme
may not function effectively and make it difficult to
eliminate the source of defective products.
 Larger lots are preferred to smaller ones
 Lots should conform to the materials-handling systems in
both the vendor and consumer facilities
 Lots should be packaged to minimize shipping risks and make
selection of sample units easy

 Mr. Smith owns and operates a manufacturing plant.
 He receives a shipment of 1,000 sheets of glass. Of the shipment, Mr. Smith
chooses to sample 50 sheets. If more than 2 are defective, he is sending back
the entire shipment to the supplier. Mr. Smith observes 5 defective sheets
of glass.
 What should Mr. Smith do in reference to the number
of defective items observed????
 Moonlight Jeans
 Moonlight Jeans store receives a shipment of 300 pairs of jeans from its
warehouse. It is common practice for the store to sample 5% of the total
received. The acceptance number under any and all circumstances for
Moonlight Jeans is 10. Of the 15 pairs of jeans observed, 2 were defective.
 What conclusion should the store manager come to based on this
information?
 The store manager has just found out that the clerk who inspected the
samples made a huge mistake… The actual number of defective
pairs of jeans sampled was 12.
 What type of risk is involved with the error made at
Moonlight Jeans?
 How might this error affect the store and their customers?

 Random Sampling
 Important
 Units selected for inspection from lot must be chosen at random
 Should be representative of all units in a lot.
 Watch for Salting
 Vendor may put “good” units on top layer of lot knowing a lax
inspector might only sample from the top layer.
 Suggested Technique
 Assign a number to each unit, or use location of unit in lot.
 Generate / pick a random number for each unit / location in lot.
 Sort on the random number – reordering the lot / location pairs.
 Select first (or last) n items to make sample.

 Acceptance Sampling Plan
 Specifies the lot size, sample size, number of samples and
acceptance/rejection criteria for lot sentencing.
 Unless it is mentioned following convention is practiced
 Sampling is performed without replacement
 Sampling is a simple random sample
 Each item in the lot has equal probability of being in the sample.
 Designing Sampling Plan for Attributes
 Simpliest sample plan
 Based on binomial distribution (if sample is less than 20 units otherwise use
poisson’s distribution)
 Requires large sample size
 Sampling plan involves
 Single sampling plan
 Double sampling plan,
 Multiple sampling plan

 Single Sampling Plan
 Quality characteristic is an attribute, i.e., conforming or nonconforming
 Define:
 N: lot size
 n: sample size, and
 c: Acceptance number
 Procedure:
 Take a sample of size n and inspect each of the items drawn
 If d ≤ c, accept lot; else reject,
 d: number of defective items in sample

 Double Sampling Plan
 Define:
 n1: sample size of the first sample
 c1: acceptance number for the first sample
 n2: sample size of the second sample
 c2: acceptance number for the second sample
 Procedure:
 Take an initial sample of size n1
 If number of defective items, d1 ≤ c1, accept the lot
 If number of defective items, d1 > c2, reject the lot
 If c1 < d1 ≤ c2, take second sample of size, n2
 If the combined number of defective items (d1+d2) ≤ c2, accept the
lot; otherwise reject the lot

 Characterizing attribute sampling plan
 Typically four graphs are used to characterize a sampling plan.
 Operating Characteristic (OC) curve
 The probability of acceptance for a given quality level.
 Average Sample Number (ASN) curve
 The expected number of items we will sample (most applicable to double,
multiple, and sequential samples)
 Average Outgoing Quality (AOQ) curve
 The expected fraction nonconforming after rectifying inspection for a given
quality level.
 Average Total Inspected (ATI) curve
 The expected number of units inspected after rectifying inspection for a
given quality level.

 Operating Characteristics Curve
 OC curves are graphs which show
the probability of accepting a lot for
various proportions of defective
items in the lot.
 X-axis shows percentage of items
that are defective in a lot- “lot
quality”.
 Y-axis shows the probability or
chance of accepting a lot.
 As proportion of defective items
increases, the chance of accepting
lot decreases.
 Example: 90% chance of accepting
a lot with 5% defectives; 10%
chance of accepting a lot with 24%
defectives

 OC curve is typically used to represent
the four parameters (Producers Risk,
Consumers Risk, AQL and LTPD) of the
sampling plan.
 AQL is the small % of defects that
consumers are willing to accept;
order of 1-2%
 LTPD is the upper limit on the
percentage of defective items that
consumers are willing to tolerate.
 Consumer’s Risk (α) is the chance
of accepting a lot that contains a
greater number of defects than
the LTPD limit; Type II error
 Producer’s risk (β) is the chance a
lot containing an acceptable
quality level will be rejected; Type
I error
Probability
Probability
of
of
Acceptance
Acceptance
Percent
Percent
defective
defective
| | | | | | | | |
0
0 1
1 2
2 3
3 4
4 5
5 6
6 7
7 8
8
100
100 –
95
95 –
75
75 –
50
50 –
25
25 –
10
10 –
0
0 –
Probability
Probability
of
of
Acceptance
Acceptance
Percent
Percent
defective
defective
| | | | | | | | |
0
0 1
1 2
2 3
3 4
4 5
5 6
6 7
7 8
8
100
100 –
95
95 –
75
75 –
50
50 –
25
25 –
10
10 –
0
0 –

 =0.05
=0.05 producer
producer’’sriskforAQL
sriskforAQL

 =0.05
=0.05 producer
producer’’sriskforAQL
sriskforAQL

 =0.10
=0.10
Consumer
Consumer’’s
s
riskforLTPD
riskforLTPD

 =0.10
=0.10
Consumer
Consumer’’s
s
riskforLTPD
riskforLTPD
LTPD
LTPD
AQL
AQL LTPD
LTPD
AQL
AQL
Badlots
Badlots
Indifference
Indifference
zone
zone
Good
Good
lots
lots Badlots
Badlots
Indifference
Indifference
zone
zone
Good
Good
lots
lots
Discriminates between ‘good lot’
and ‘bad lot’

 How to Compute the OC Curve Probabilities
For a specified single sampling plan, the OC curve probability can be computed
using a binomial or Poisson approximation.
 Assume that the lot size N is large (infinite)
 d - # defectives ~ Binomial(p,n)
where
 p - fraction defective items in lot
 n - sample size
 Probability of acceptance:
n!
P(d DEFECTIVES ) = ( )
d!(n-d)!
( )
P(d DEFECTIVES ) , , ,
!
d n d
d np
p p
np e
d
d



 
1
0 1 2
!
( ) ( )
!( )!
c
d n d
a
d
n
P P d c p p
d n d


   


0
1
Binomial
Approx.
Poisson
Approx.

 Example: Acceptance Probability
 Suppose p =0.02, n = 60, and c =3.
.
.
.
.
( | . ) .
!
( | . ) .
!
( | . ) .
!
( | . ) .
!
( | . ) . . . .
.
e
prob x p
e
prob x p
e
prob x p
e
prob x p
prob x p





   

   

   

   
     

0 1 2
1 1 2
2 1 2
3 1 2
1 2
0 0 02 0 3012
0
1 2
1 0 02 0 3614
1
1 2
2 0 02 0 2169
2
1 2
3 0 02 0 3012
3
3 0 02 0 3012 0 3614 0 2169 0 0867
0 9662 = probability of accepting the lot.

 How to Construct OC Curve
 OC curve is plot of Pa vs p
 Pa = P (Accepting Lot | true proportion of defective is p)
 p = lot fraction defective
 Suppose, n = 89 and c = 2.
 For each value of p, compute the probability of acceptance using a Binomial or
Poisson approximation as given in the table:
p = fraction defective in
lot
Pa = P [Accepting
Lot]
0.005 0.9897
0.010 0.9397
0.015 0.8502
0.020 0.7366
0.025 0.6153
0.030 0.4985
0.035 0.3936
Probability of Acceptance, Pa
0.0
0.2
0.4
0.6
0.8
1.0
0.00 0.02 0.04 0.06 0.08 0.10
Lot fraction defective, p
Pa
n=89
c=2

 Ideal OC Curve
 This characterizes the ideal sampling plan
that discriminates perfectly between good
and bad shipments
 If the supplier’s process average
nonconforming is below the AQL, the
consumer will accept all the shipped
lots.
 If the supplier’s process average
nonconforming is above the AQL, the
consumer will reject all the shipped
lots.
 Both α and β are zero
 it is obtainable by 100% inspection IF
inspection are error free.
 ideal OC curve is unobtainable in
practice

 Effect of n on OC Curve
 Precision with which a sampling plan differentiates between good and bad lots
increases as the sample size increases
 Increasing n (with c proportional) approaches the ideal OC curve.
Operating Characteristic Curve
0.0%
20.0%
40.0%
60.0%
80.0%
100.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0%
Percent nonconforming, p
Probability
of
acceptance,
Pa
n=200, c=4
n=100, c=2
n= 50, c=1

 Effect of c on OC Curve
 Changing acceptance number, c, does not dramatically change slope of OC
curve.
 Increasing c (with n constant) approaches the ideal OC curve.
Operating Characteristic Curve
0.0%
20.0%
40.0%
60.0%
80.0%
100.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0%
Probability
of
acceptance,
Pa
n=100, c=2
n=100, c=1
n=100, c=0

 Balancing the producer and consumer risks
 The true value of p is unknown
 Even if p known, sampling involves randomness, and we can still
 reject a lot even if p < c/n
 accept a lot even if p > c/n
 The OC curve gives an indication of the values of α and β.
 Producer’s risk- α
 Producer wants as many lots accepted by consumer as possible so
 Producer “makes sure” the process produces a level of fraction defective
equal to or less than:
p1 = AQL = Acceptable Quality Level
 α is the probability that a good lot will be rejected by the consumer even
though the lot really has a fraction defective ≤ p1
Lot is rejected given that process
has an acceptable quality level
P

 
  
 
Lot is rejected
P p AQL
  
 
 

 Consumer’s risk- β
 Consumer wants to make sure that no bad lots are accepted
 Consumer says, “I will not accept a lot if percent defective is greater than or
equal to p2”
p2 = LTPD = Lot Tolerance Percent Defective
 β is the probability a bad lot is accepted by the consumer when the lot really
has a fraction defective ≥ p2
 We receive a shipment of 3000 items, AQL = 0.02, LTPD = 0.06, n = 60, c = 3.
α = probability of rejecting a batch with defective rate p=0.02.
= probability that four or more defective item
= prob (x ≥4 | p = 0.02)
= 1- prob (x ≤3 | p = 0.02) = 1-0.9662 = 0.0338
β = probability of acceptance a batch with p=LTPD=0.06
= prob (x ≤ 3 | p=0.06) = 0.5153
Lot accepted given that lot
has unacceptable quality level
P

 
  
 
Lot accepted
P p LTPD
  
 
 

 Designing a Sampling Plan with a Specified OC Curve
 Use a chart called a Binomial Nomograph to design a sampling plan
 Specify:
 p1 = AQL (Acceptable Quality Level)
 p2 = LTPD (Lot Tolerance Percent Defective)
 1 – α = P [Lot is accepted | p = AQL]
 β = P [Lot is accepted | p = LTPD]
 Draw two lines on Nomograph
 Line 1 connects p1 = AQL to (1- α)
 Line 2 connects p2 = LTPD to β
 Pick n and c from the intersection of the lines
 Example: Suppose
p1 = 0.01, α = 0.05, p2 = 0.06, and β = 0.10.
Find the acceptance sampling plan.

 Designing a Sampling Plan using Tables
 For a given producer’s and consumer’s risks, various tables have been
developed for constructing single and double sampling plans.
 Three alternatives for specifying sampling plans
 Producer’s Risk and AQL specified
 Consumer’s Risk and LTPD specified
 All four parameters specified
Excerpt From a Sampling Plan Table with Producers Risk = 0.05 and Consumers Risk = 0.10
C LTPD/AQL n(AQL) n(LTPD)
0 44.89 0.052 2.334
1 10.946 .355 3.886
2 6.509 .818 5.324
3 4.89 1.366 6.68
4 4.057 1.97 7.992
5 3.549 2.613 9.274
6 3.206 3.286 10.535
7 2.957 3.981 11.772
8 2.768 4.695 12.996
9 2.618 5.426 14.205

 Producer’s Risk and AQL specified
 Consumer’s Risk and LTPD specified
 Choose the acceptance number and divide the appropriate column by the
associated parameter to get the sample size.
 Ex I: Given a producer’s risk of 0.05 and an AQL of 0.015 determine a
sampling plan.
 c = 1; n(AQL) = 0.355; n= 0.355/0.015 ≈ 24
 c = 4; n(AQL) = 1.97; n = 1.97/0.015 ≈ 131
 Ex II: Given a consumer’s risk of 0.10 and an LTPD of 0.08 determine a
sampling plan.
 c = 0; n(LTPD) = 2.334 n = 2.334/0.08 ≈ 29
 c = 5; n(LTPD) = 9.274 n = 9.274/0.08 ≈ 116

 All four parameters specified
 We must first find out a value close to the ratio LTPD/AQL in the table.
 Then find out the values of n and c that corresponds to that specified ratio.
 Example: Given producers risk of 0.05, consumers risk of 0.10, LTPD of 4.5%,
and AQL of 1%, determine a sampling plan.
 Since the ratio of LTPD/AQL = 4.5/1 = 4.5 is in between c= 3 and c = 4;
 Using the n(AQL) column the sample sizes suggested are 137 and 197
respectively.
 Note:
 using n(AQL) column will ensure a producers risk of 0.05.
 using the n(LTPD) column will ensure a consumers risk of 0.10
For double sampling plan, use Grubbs’Tables (Table 10-6 and Table
10-7: Amitava Mitra, Fundamentals of Quality Control and
Improvement, 2nd ed., Pages: 445-446)

 Rectifying Sampling
 All known defective units replaced with good ones, that is,
 If lot is rejected, replace all bad units in lot
 If lot is accepted, just replace the bad units in sample
 Such sampling program is known as rectifying inspection program
 Since such inspection activity affects the final quality of the outgoing product,
two questions come to mind :
 How many items are inspected on average after rectifying inspection?
 What is the average outgoing quality after rectifying inspection?

 Average Outgoing Quality: AOQ
 Quality that results from application of rectifying inspection.
 The long-run ratio of expected number of defectives to expected number of
items successfully passing through the inspection plan
 Average value obtained over long sequence of lots from process with fraction
defective p:
 N - Lot size, n = # units in sample that, after inspection, contains no
defectives.
 If the lot is accepted: number of defective units in the lot
 

 a
P p N n
AOQ
N
  
# units
fraction
remaining
defective
in lot
p N n
 
   
     
   
 

 Average Outgoing Quality: AOQ
 Expected number of defective
units:
 Average fraction defective or
Average Outgoing Quality, AOQ:
 The maximum value or the worst
possible value of AOQ over all
possible values of p is called
Average Outgoing Quality Limit
(AOQL).
 If this is too high, then the
sampling plan should be
revised.
   
Lot # defective
Prob
accepted units in lot
a
P p N n
   
     
   
 
a
P p N n
AOQ
N


Average Outgoing Quality Curve
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
7.0%
0.0% 20.0% 40.0% 60.0% 80.0% 100.0%
Average
fraction
nonconforming,
outgoing
lots
The AOQ curve for
N = 150, n = 20, c = 2
AOQL

 Average Total Inspected: ATI
 Total number of inspection per lot
required for sampling plan.
 If the lot is fully conforming,
p=0.0 (Pa=1.0), then no lot will
be rejected and hence, inspect
only the sample.
 If the lot is totally
nonconforming, p=1.0
(Pa=0.0), then do 100%
inspection on the lot.
 If the lot quality is 0 < p < 1,
the average total number of
inspection per lot:
 p = fraction defectives;
 Pa = Probability of
accepting a lot.
  
1 a
ATI n P N n
   
Average Total Inspection Curve
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
0.0% 20.0% 40.0% 60.0% 80.0% 100.0%
Average
total
inspection
(ATI)
The ATI curve for
N = 150, n = 20, c = 2

Design of Experiment
 Why Experiment?
 To increase knowledge about a process
 What good is increased knowledge?
 Increase the output of the process
 Reduce variability
 Produce a robust product that is not
influenced by environmental factors
 Why Experimental Design?
 To obtain valid conclusions from an
experiment
 To yield the maximum amount of
useful information with the least
amount of experimentation
(experiments cost money)

Design of Experiment ….Cont’d
 Terminology Used
 Design: Specification of the number of experiments, the factor level and the
number of replications for each experiment.
 Factorial design: An experimental design where all combinations of factor
levels under consideration are tested at each replicate.
 Response Variable: Outcome of an experiment, which you are to measure
during experiment e.g., throughput, response time
 Factors: Variables that have effects on the response variable
 Also called predictor variables or predictors
 Primary Factors: The factors whose effects need to be quantified.
 Secondary Factors: Factors whose impact need not be quantified e.g., the
workloads and user educational level.
 Levels: Specific value of the factor
 Also called treatment
 Replication: Repetition of all or some experiments.
 Experimental Unit: Any entity that is used for experiments
 Interaction: Two factors interact if one shows dependence upon another.

 Full Factorial Design
 Consists of all possible combinations of the factor levels.
 Design Matrix is the complete specification of the experimental test runs.
 Treatment Combination is a specific combination of the factor levels.
Speed Octane Tire Pressure
55 85 30
60 85 30
55 90 30
60 90 30
55 85 35
60 85 35
55 90 35
60 90 35
Design
Matrix
A treatment
combination
Speed Octane Tire Pressure
55 85 30
60 85 30
55 90 30
60 90 30
55 85 35
60 85 35
55 90 35
60 90 35
Design
Matrix
A treatment
combination

 What makes up an experiment?
 Response Variable(s)
 Factors
 Randomization
 Repetition and Replication
 Response Variable
 The variable that is measured and the object of the characterization or
optimization (the Y).
 Defining the response variable can be difficult.
 Often selected due to ease of measurement.
 Factors
 A variable which is controlled or varied in a systematic way during the
experiment (the X)
 Tested at 2 or more levels to observe its effect on the response variable(s)
(Ys)

 Randomization
 Randomization can be done in several ways :
 Run the treatment combinations in random order
 Assign experimental units to treatment combinations randomly
 An experimental unit is the entity to which a specific treatment
combination is applied
 Advantage of randomization is to “average out” the effects of extraneous
factors (called noise)
 that may be present
 but are not controlled or measured during the experiment
 Repetition and Replication
 Repetition : Running several samples during one experimental setup (short-
term variability)
 Replication : Repeating the entire experiment (long-term variability)
 Repetition and Replication provide an estimate of the experimental error,
which is used to determine if observed differences are statistically significant.

 Steps in Design of Experiment
 Five key steps in designing an experiment include:
1. Identify factors of interest and a response variable.
2. Determine appropriate levels for each explanatory variable.
3. Determine a design structure/matrix.
4. Randomize the order in which each set of conditions is run and collect
the data.
5. Organize the results in order to draw appropriate conclusions.
How to organize and draw conclusions for a
specific type of design structure, the
full factorial design.

 Steps in Design of Experiment
 Problem Statement: A soft drink bottler is interested in obtaining more
uniform heights in the bottles produced by his manufacturing process. The
filling machine theoretically fills each bottle to the correct target height, but
in practice, there is variation around this target, and the bottler would like to
understand better the sources of this variability and eventually reduce it.
 The engineer can control three input variables during the filling process
(each at two levels):
 Factor A: Carbonation with Levels: 10% and 12%
 Factor B: Operating Pressure in the filler with Levels: 25 and 30 psi
 Factor C: Line Speed with Levels: 200 and 250 bottles produced per
minute (bpm)
 Unit: Each bottle
 Response Variable: Deviation from the target fill height
 The steps to design this experiment include:
1. Identify factors of interest and a response variable.
2. Determine appropriate levels for each explanatory variable.

3. Determine a design structure
 Design structures can be very complicated.
 One of the most basic structures is called the full factorial design.
 This design tests every combination of factor levels an equal amount of
times.
 To ensure each factor combination exactly once:
 1st Column - alternate every (20) row
 2nd Column - alternate every 2 (21) rows
 3rd Column - alternate every 4 (=22) rows
 This is called a 23 full factorial design (i.e. 3 factors
at 2 levels each will need 8 runs).
 Each row in this table gives a specific test run.
For example, the first row represents a specific test
in which the manufacturing process runs with A set
at 10% carbonation, B set at 25 psi, and line speed,
C, is set at 200 bpm.

4. Randomize the order in which each set of test
conditions is run and collect the data.
 In this example the tests are run in the
following order: 7, 4, 1, 6, 8, 5, 2, 3.
 Data are the results obtained from each of
these test conditions.
5. Organize the results in order to draw appropriate
conclusions.
 Results are the observed deviation from the
target fill height for each set of these test
conditions.
 To start organizing the results, computing all
the averages at low and high levels.
 To determine the effect of change in factor A
on the results, calculate the average values
of the test results for A- and A+.

 Main Effects:
 A Main Effect is the difference between the factor average and the grand mean.
 From the table,
 Overall average of the test results (grand mean) is 2.
 Mean values for A- (factor A run at low level) and A+ (factor A run at a high
level) are -1 and 5 respectively
 Likewise, calculate mean values of the test results for factors B and C
 Effect sizes determine which factors have the most significant impact on the
results.
 ANOVA determine the significance of each factor based on these effect
calculations.

 Main Effects Plots:
 Effects plots are a quick and efficient way to
visualize effect size.
 The grand mean is plotted as a horizontal
line.
 The average result is represented by dots
for each factor level.
 The Y axis is always the same for each factor
in Main Effects Plots.
 Factors with steeper slopes have larger
effects and thus
 larger impacts on the results.
 This graph shows that
 A+ has a higher mean fill height than A-
 B+ and C+ also have higher means than B-
and C- respectively, and
 The effect size of factor A, Carbonation, is
larger than the other factor effects.

 Interaction Effects:
 It is often critical to identify how multiple factors
interact in affecting the results.
 An interaction occurs when one factor affects
the results differently depending on a second
factor.
 To find the AB interaction effect, first calculate the
average result for each of the four level
combinations of A and B:
 Calculate the average when factors A and B
are both at the low level (-4 + -1) / 2 = -2.5
 Calculate the mean when factors A and B are
both at the high level (5 + 11) / 2 = 8
 Also calculate the average result for each of
the levels of AC and BC.

 Interaction Effect plots:
 Interaction plots are used to determine the
effect size of interactions.
 For example, the AB plot below shows that
the effect of B is larger when A is 12%.
 AB plot also shows that when the data is
restricted to A+, the B effect is more
steeper than when we restrict our data to
A-.
 The bottom right plot shows the interaction (or
2-way effects) of all three factors:
 When the lines are parallel, no interaction
effects between the factors.
 The more different the slopes, the more
influence the interaction effect has on the
results.
 This graph shows that the AB
interaction effect is the largest.

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