#### Transcript Statistical Process Control

```Operations
Management
Supplement 6 –
Statistical Process Control
PowerPoint presentation to accompany
Heizer/Render
Principles of Operations Management, 6e
Operations Management, 8e
Prentice
Hall, Inc. Hall, Inc.
2006
Prentice
S6 – 1
Outline
 Statistical Process Control (SPC)
 Control Charts for Variables
 Setting Mean Chart Limits (x-Charts)
 Setting Range Chart Limits (R-Charts)
 Process Capability
 Process Capability Ratio (Cp)
 Process Capability Index (Cpk )
S6 – 2
Statistical Process Control
(SPC)
 Variability is inherent in every process
 Natural or common causes
 Special or assignable causes
 SPC charts provide statistical signals
when assignable causes are present
 SPC approach supports the detection
and elimination of assignable causes
of variation
S6 – 3
Inspection
 Inspection is the activity that is done to
ensure that an operation is producing
the results expected (post production
activity).
 Where to inspect
At point of product design
At point of product production (source)
At point of product assembly
At point of product dispatch to customer
At point of product reception by customer
S6 – 4
What to Inspect
 Variables of an entity
Degree of deviation from a target
(continuum scale)
Lifespan of device
Reliability or accuracy of device
 Attributes of an entity
 Classifies attributes into discrete classes
such as good versus bad, or pass or fail
Maximum weight of bag at airport
Minimum height for exit seat
S6 – 5
Data Used for Quality Judgments
Variables
 Characteristics that
can take any real
value
 May be in whole or
in fractional
numbers
 Continuous random
variables, e.g.
weight, length,
duration, etc.
Attributes
 Defect-related
characteristics
 Classify products
as either good or
defects
 Categorical or
discrete random
variables
S6 – 6
How to Inspect: Methods
 Visual inspection
 Manual inspection (weigh, count )
 Mechanical inspection (machine-based)
 Testing of device
S6 – 7
 Most inspections are not done at the source
 Errors are discovered after its too late
 No link between error and the cause of errors
 Errors are often too costly to correct
 As products grow in number more staff are needed
for inspection
 As products grow in number, more time is needed
for inspection
 Most inspections involve the inspection of good
parts as well as bad ones
 CHALLENGE: How could one achieve high product
quality without having to inspect all goods
produced?
S6 – 8
Statistical Process Control
 A statistics-based approach for
monitoring and inspecting results of a
process, through the gathering,
structuring, and analyses of product
variables/attributes, as well as the taking
of corrective action at the source, during
the production process.
S6 – 9
Class Example
 What can we learn from the results of
the bodyguards?
S6 – 10
Common Cause Variations
 Also called natural causes
 Affects virtually all production processes
Generally this requires some change at the
systemic level of an organization
 Managerial action is often necessary
 The objective is to discover avoidable
common causes present in processes
Eliminate (when possible) the root causes of the
common variations, e.g. different arrival times of
suppliers, different arrival times of customers,
weight of products poured in box by machine
S6 – 11
Assignable Variations
 Also called special causes of variation
 Generally this is caused by some change in
the local activity or process
 Variations that can be traced to a specific
reason at a localized activity
 The objective is to discover special causes
that are present
 Eliminate the root causes of the special
variations, e.g. machine wear, material quality,
 Incorporate the good process control
S6 – 12
SPC and Statistical Samples
To conduct inspection using the SPC
approach, one has to compute averages for
several small samples instead of using data
from individual items: Steps
Figure S6.1
Frequency
(a) Samples of the
product, say five
boxes of cereal
taken off the filling
machine line, vary
from each other in
weight
Each of these
represents one
sample of five
boxes of cereal
# #
# # #
# # # #
# # # # # # #
#
# # # # # # # # #
Weight
S6 – 13
SPC Sampling
To measure the process, we take samples of
same size at different times. We plot the
mean of each sample for each point in time
Figure S6.1
Frequency
(b) After enough
samples are
taken from a
stable process,
they form a
pattern called a
distribution
The solid line
represents the
distribution
Weight
S6 – 14
Attributes of Distributions
(c) There are many types of distributions, including
the normal (bell-shaped) distribution, but
distributions do differ in terms of central
tendency (mean), standard deviation or
variance, and shape
Central tendency
Variation
Shape
Frequency
Figure S6.1
Weight
Weight
Weight
S6 – 15
(d) If only common
causes of
variation are
present, the
output of a
process forms a
distribution that
is stable over
time and is
predictable
Frequency
Identifying Presence of
Common Sources of Variation
Prediction
Weight
Figure S6.1
S6 – 16
Identifying Presence of Special
Sources of Variation
Frequency
(e) If assignable
causes are
present, the
process output is
not stable over
time and is not
predicable
?
?? ??
?
?
?
?
?
?
?
?
?
??
?
?
?
Prediction
Weight
Figure S6.1
S6 – 17
Central Limit Theorem
Regardless of the distribution of the
population, the distribution of sample means
drawn from the population will tend to follow
a normal curve
1. The mean of the sampling
distribution (x) will be the same
as the population mean m
2. The standard deviation of the
sampling distribution (sx) will
equal the population standard
deviation (s) divided by the
square root of the sample size, n
x=m
sx =
s
n
S6 – 18
Interpreting SPC Charts
Frequency
Lower Control Limit
(a) In statistical
control and capable
of producing within
control limits
Upper Control Limit
(b) In statistical
control but not
capable of producing
within control limits
Size
(weight, length, speed, etc.)
(c) Out of statistical
control and incapable of
producing within limits
Figure S6.2
S6 – 19
Population and Sampling
Distributions
Three population
distributions
Distribution of
sample means
Mean of sample means = x
Beta
Standard
deviation of
s
the sample = sx = n
means
Normal
Uniform
|
|
|
|
-3sx
-2sx
-1sx
x
|
|
+1sx +2sx +3sx
95.45% fall within ± 2sx
99.73% of all x
fall within ± 3sx
|
Figure S6.3
S6 – 20
Sampling Distribution
Sampling
distribution
of means
Process
distribution
of means
x=m
(mean)
Figure S6.4
S6 – 21
Steps In Creating Control Charts
1. Take representative sample from output of a
process over a long period of time, e.g. 10 units
every hour for 24 hours.
2. Compute means and ranges for the variables and
calculate the control limits
3. Draw control limits on the control chart
4. Plot a chart for the means and another for the
mean of ranges on the control chart
5. Determine state of process (in or out of control)
6. Investigate possible reasons for out of control
events and take corrective action
7. Continue sampling of process output and reset
the control limits when necessary
S6 – 22
In-Class Exercise : Control Charts
6/15
6/16
8AM
10AM
12AM
2PM
8AM
5
6
8
6.5
6.5
5.5
8
6.5
6
7
6.5
6.5
7.5
6.5
7
8
6.5
6.5
6
6
8
7.5
6.5
7
6.5
Calculate X bar and R’s for new data
Calculate X double bar and R bar figures for new data
Draw X bar chart
Calculate LCL and UCL for X bar chart
Draw lines for LCL and UCL and for X double bar in chart
S6 – 23
Control Charts for Variables
 For variables that have continuous
dimensions
 Weight, speed, length, strength, etc.
 x-charts are to control the central
tendency of the process
 R-charts are to control the dispersion of
the process
 These two charts must be used together
S6 – 24
Setting Chart Limits
For x-Charts when we know s
Upper control limit (UCL) = x + zsx
Lower control limit (LCL) = x - zsx
where
x = mean of the sample means or a target
value set for the process
z = number of normal standard deviations
sx = standard deviation of the sample means
= s/ n
s = population standard deviation
n = sample size
S6 – 25
Setting Control Limits
Hour 1
Sample
Weight of
Number
Oat Flakes
1
17
2
13
3
16
4
18
n=9
5
17
6
16
7
15
8
17
9
16
Mean 16.1
s=
1
Hour
1
2
3
4
5
6
Mean
16.1
16.8
15.5
16.5
16.5
16.4
Hour
7
8
9
10
11
12
Mean
15.2
16.4
16.3
14.8
14.2
17.3
For 99.73% control limits, z = 3
UCLx = x + zsx = 16 + 3(1/3) = 17 ozs
LCLx = x - zsx = 16 - 3(1/3) = 15 ozs
S6 – 26
Setting Control Limits
Control Chart
for sample of
9 boxes
Variation due
to assignable
causes
Out of
control
17 = UCL
Variation due to
natural causes
16 = Mean
15 = LCL
| | | | | | | | | | | |
1 2 3 4 5 6 7 8 9 10 11 12
Sample number
Out of
control
Variation due
to assignable
causes
S6 – 27
Setting Chart Limits
For x-Charts when we don’t know s
Upper control limit (UCL) = x + A2R
Lower control limit (LCL) = x - A2R
where
R = average range of the samples
A2 = control chart factor found in Table S6.1
x = mean of the sample means
S6 – 28
Control Chart Factors
Sample Size
n
Mean Factor
A2
Upper Range
D4
Lower Range
D3
2
3
4
5
6
7
8
9
10
12
1.880
1.023
.729
.577
.483
.419
.373
.337
.308
.266
3.268
2.574
2.282
2.115
2.004
1.924
1.864
1.816
1.777
1.716
0
0
0
0
0
0.076
0.136
0.184
0.223
0.284
Table S6.1
S6 – 29
Setting Control Limits
Process average x = 16.01 ounces
Average range R = .25
Sample size n = 5
S6 – 30
Setting Control Limits
Process average x = 16.01 ounces
Average range R = .25
Sample size n = 5
UCLx
= x + A2R
= 16.01 + (.577)(.25)
= 16.01 + .144
= 16.154 ounces
From
Table S6.1
S6 – 31
Setting Control Limits
Process average x = 16.01 ounces
Average range R = .25
Sample size n = 5
UCLx
LCLx
= x + A2R
= 16.01 + (.577)(.25)
= 16.01 + .144
= 16.154 ounces
UCL = 16.154
= x - A2R
= 16.01 - .144
= 15.866 ounces
LCL = 15.866
Mean = 16.01
S6 – 32
R – Chart
 Type of variables control chart
 Shows sample ranges over time
 Difference between smallest and
largest values in sample
 Monitors process variability
 Independent from process mean
S6 – 33
Setting Chart Limits
For R-Charts
Upper control limit (UCLR) = D4R
Lower control limit (LCLR) = D3R
where
R = average range of the samples
D3 and D4 = control chart factors from Table S6.1
S6 – 34
Setting Control Limits
Average range R = 5.3 pounds
Sample size n = 5
From Table S6.1 D4 = 2.115, D3 = 0
UCLR = D4R
= (2.115)(5.3)
= 11.2 pounds
UCL = 11.2
LCLR
LCL = 0
= D3R
= (0)(5.3)
= 0 pounds
Mean = 5.3
S6 – 35
Mean and Range Charts
(a)
(Sampling mean is
shifting upward but
range is consistent)
These
sampling
distributions
result in the
charts below
UCL
(x-chart detects
shift in central
tendency)
x-chart
LCL
UCL
(R-chart does not
detect change in
R-chart
LCL
Figure S6.5
S6 – 36
Mean and Range Charts
(b)
These
sampling
distributions
result in the
charts below
(Sampling mean
is constant but
dispersion is
increasing)
UCL
(x-chart does not
detect the increase
in mean)
x-chart
LCL
UCL
(R-chart detects
increase in
dispersion)
R-chart
LCL
Figure S6.5
S6 – 37
Automated Control Charts
S6 – 38
Control Charts for Attributes
 For variables that are categorical
acceptable/unacceptable
 Measurement is typically counting
defectives
 Charts may measure
 Percent defective (p-chart)
 Number of defects (c-chart)
S6 – 39
Patterns in Control Charts
Upper control limit
Target
Lower control limit
Figure S6.7
Normal behavior.
Process is “in control.”
S6 – 40
Patterns in Control Charts
Upper control limit
Target
Lower control limit
Figure S6.7
One plot out above (or
below). Investigate for
cause. Process is “out
of control.”
S6 – 41
Patterns in Control Charts
Upper control limit
Target
Lower control limit
Figure S6.7
Trends in either
direction, 5 plots.
Investigate for cause of
progressive change.
S6 – 42
Patterns in Control Charts
Upper control limit
Target
Lower control limit
Figure S6.7
Run of 5 above (or
below) central line.
Investigate for cause.
S6 – 43
Process Capability
 The natural variation of a process
should be small enough to produce
products that meet the standards
required
 A process in statistical control does not
necessarily meet the design
specifications
 Process capability is a measure of the
relationship between the natural
variation of the process and the design
specifications
S6 – 44
Process Capability Ratio
Upper Specification - Lower Specification
Cp =
6s
 A capable process must have a Cp of at
least 1.0
 Does not look at how well the process
is centered in the specification range
 Often a target value of Cp = 1.33 is used
to allow for off-center processes
 Six Sigma quality requires a Cp = 2.0
S6 – 45
Process Capability Ratio
Insurance claims process
Process mean x = 210.0 minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes
Upper Specification - Lower Specification
Cp =
6s
S6 – 46
Process Capability Ratio
Insurance claims process
Process mean x = 210.0 minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes
Upper Specification - Lower Specification
Cp =
6s
213 - 207
=
= 1.938
6(.516)
S6 – 47
Process Capability Ratio
Insurance claims process
Process mean x = 210.0 minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes
Upper Specification - Lower Specification
Cp =
6s
213 - 207
=
= 1.938
6(.516)
Process is
capable
S6 – 48
Process Capability Index
Upper
Lower
Cpk = minimum of Specification - x , x - Specification
Limit
Limit
3s
3s
 A capable process must have a Cpk of at
least 1.0
 A capable process is not necessarily in the
center of the specification, but it falls within
the specification limit at both extremes
S6 – 49
Process Capability Index
New Cutting Machine
New process mean x = .250 inches
Process standard deviation s = .0005 inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches
S6 – 50
Process Capability Index
New Cutting Machine
New process mean x = .250 inches
Process standard deviation s = .0005 inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches
(.251) - .250
Cpk = minimum of
,
(3).0005
S6 – 51
Process Capability Index
New Cutting Machine
New process mean x = .250 inches
Process standard deviation s = .0005 inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches
(.251) - .250
.250 - (.249)
Cpk = minimum of
,
(3).0005
(3).0005
Both calculations result in
.001
Cpk =
= 0.67
.0015
New machine is
NOT capable
S6 – 52
Process Capability Comparison
New Cutting Machine
New process mean x = .250 inches
Process standard deviation s = .0005 inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches
Cp =
Upper Specification - Lower Specification
Cp = .251 - .249
0.66 .0030
6s
=
New machine is
NOT capable
S6 – 53
Interpreting Cpk
Cpk = negative number
Cpk = zero
Cpk = between 0 and 1
Cpk = 1
Cpk > 1
Figure S6.8
S6 – 54
Acceptance Sampling
 Form of quality testing used for
incoming materials or finished goods
 Take samples at random from a lot
(shipment) of items
 Inspect each of the items in the sample
 Decide whether to reject the whole lot
based on the inspection results
 Only screens lots; does not drive
quality improvement efforts