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- 1. © 2006 Prentice Hall, Inc. S6 – 1
Operations
Management
Supplement 6 –
Statistical Process Control
© 2006 Prentice Hall, Inc.
PowerPoint presentation to accompany
Heizer/Render
Principles of Operations Management, 6e
Operations Management, 8e
- 2. © 2006 Prentice Hall, Inc. S6 – 2
Outline
Statistical Process Control (SPC)
Control Charts for Variables
The Central Limit Theorem
Setting Mean Chart Limits (x-Charts)
Setting Range Chart Limits (R-Charts)
Using Mean and Range Charts
Control Charts for Attributes
Managerial Issues and Control Charts
- 3. © 2006 Prentice Hall, Inc. S6 – 3
Outline – Continued
Process Capability
Process Capability Ratio (Cp)
Process Capability Index (Cpk )
Acceptance Sampling
Operating Characteristic Curve
Average Outgoing Quality
- 4. © 2006 Prentice Hall, Inc. S6 – 4
Learning Objectives
When you complete this supplement,
you should be able to:
Identify or Define:
Natural and assignable causes of
variation
Central limit theorem
Attribute and variable inspection
Process control
x-charts and R-charts
- 5. © 2006 Prentice Hall, Inc. S6 – 5
Learning Objectives
When you complete this supplement,
you should be able to:
Identify or Define:
LCL and UCL
P-charts and c-charts
Cp and Cpk
Acceptance sampling
OC curve
- 6. © 2006 Prentice Hall, Inc. S6 – 6
Learning Objectives
When you complete this supplement,
you should be able to:
Identify or Define:
AQL and LTPD
AOQ
Producer’s and consumer’s risk
- 7. © 2006 Prentice Hall, Inc. S6 – 7
Learning Objectives
When you complete this supplement,
you should be able to:
Describe or Explain:
The role of statistical quality control
- 8. © 2006 Prentice Hall, Inc. S6 – 8
Variability is inherent in every process
Natural or common causes
Special or assignable causes
Provides a statistical signal when
assignable causes are present
Detect and eliminate assignable
causes of variation
Statistical Process Control
(SPC)
- 9. © 2006 Prentice Hall, Inc. S6 – 9
Natural Variations
Also called common causes
Affect virtually all production processes
Expected amount of variation
Output measures follow a probability
distribution
For any distribution there is a measure
of central tendency and dispersion
If the distribution of outputs falls within
acceptable limits, the process is said to
be “in control”
- 10. © 2006 Prentice Hall, Inc. S6 – 10
Assignable Variations
Also called special causes of variation
Generally this is some change in the process
Variations that can be traced to a specific
reason
The objective is to discover when
assignable causes are present
Eliminate the bad causes
Incorporate the good causes
- 11. © 2006 Prentice Hall, Inc. S6 – 11
Samples
To measure the process, we take samples
and analyze the sample statistics following
these steps
(a) Samples of the
product, say five
boxes of cereal
taken off the filling
machine line, vary
from each other in
weight
Frequency
Weight
#
#
# #
#
#
#
#
#
# # #
# # #
#
# # #
# # #
# # #
#
Each of these
represents one
sample of five
boxes of cereal
Figure S6.1
- 12. © 2006 Prentice Hall, Inc. S6 – 12
Samples
To measure the process, we take samples
and analyze the sample statistics following
these steps
(b) After enough
samples are
taken from a
stable process,
they form a
pattern called a
distribution
The solid line
represents the
distribution
Frequency
Weight
Figure S6.1
- 13. © 2006 Prentice Hall, Inc. S6 – 13
Samples
To measure the process, we take samples
and analyze the sample statistics following
these steps
(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
Weight
Central tendency
Weight
Variation
Weight
Shape
Frequency
Figure S6.1
- 14. © 2006 Prentice Hall, Inc. S6 – 14
Samples
To measure the process, we take samples
and analyze the sample statistics following
these steps
(d) If only natural
causes of
variation are
present, the
output of a
process forms a
distribution that
is stable over
time and is
predictable
Weight
Frequency
Prediction
Figure S6.1
- 15. © 2006 Prentice Hall, Inc. S6 – 15
Samples
To measure the process, we take samples
and analyze the sample statistics following
these steps
(e) If assignable
causes are
present, the
process output is
not stable over
time and is not
predicable
Weight
Frequency
Prediction
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
??
Figure S6.1
- 16. © 2006 Prentice Hall, Inc. S6 – 16
Control Charts
Constructed from historical data, the
purpose of control charts is to help
distinguish between natural variations
and variations due to assignable
causes
- 17. © 2006 Prentice Hall, Inc. S6 – 17
Types of Data
Characteristics that
can take any real
value
May be in whole or
in fractional
numbers
Continuous random
variables
Variables Attributes
Defect-related
characteristics
Classify products
as either good or
bad or count
defects
Categorical or
discrete random
variables
- 18. © 2006 Prentice Hall, Inc. S6 – 18
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
x = m
s
n
sx =
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
- 19. © 2006 Prentice Hall, Inc. S6 – 19
Process Control
Figure S6.2
Frequency
(weight, length, speed, etc.)
Size
Lower control limit Upper control limit
(a) In statistical
control and capable
of producing within
control limits
(b) In statistical
control but not
capable of producing
within control limits
(c) Out of control
- 20. © 2006 Prentice Hall, Inc. S6 – 20
Population and Sampling
Distributions
Three population
distributions
Beta
Normal
Uniform
Distribution of
sample means
Standard
deviation of
the sample
means
= sx =
s
n
Mean of sample means = x
| | | | | | |
-3sx -2sx -1sx x +1sx +2sx +3sx
99.73% of all x
fall within ± 3sx
95.45% fall within ± 2sx
Figure S6.3
- 21. © 2006 Prentice Hall, Inc. S6 – 21
Sampling Distribution
x = m
(mean)
Sampling
distribution
of means
Process
distribution
of means
Figure S6.4
- 22. © 2006 Prentice Hall, Inc. S6 – 22
Steps In Creating Control
Charts
1. Take samples from the population and
compute the appropriate sample statistic
2. Use the sample statistic to calculate control
limits and draw the control chart
3. Plot sample results on the control chart and
determine the state of the process (in or out of
control)
4. Investigate possible assignable causes and
take any indicated actions
5. Continue sampling from the process and reset
the control limits when necessary
- 23. © 2006 Prentice Hall, Inc. 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
- 24. © 2006 Prentice Hall, Inc. 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
- 25. © 2006 Prentice Hall, Inc. S6 – 25
Setting Control Limits
Hour 1
Sample Weight of
Number Oat Flakes
1 17
2 13
3 16
4 18
5 17
6 16
7 15
8 17
9 16
Mean 16.1
s = 1
Hour Mean Hour Mean
1 16.1 7 15.2
2 16.8 8 16.4
3 15.5 9 16.3
4 16.5 10 14.8
5 16.5 11 14.2
6 16.4 12 17.3
n = 9
LCLx = x - zsx = 16 - 3(1/3) = 15 ozs
For 99.73% control limits, z = 3
UCLx = x + zsx = 16 + 3(1/3) = 17 ozs
- 26. © 2006 Prentice Hall, Inc. S6 – 26
17 = UCL
15 = LCL
16 = Mean
Setting Control Limits
Control Chart
for sample of
9 boxes
Sample number
| | | | | | | | | | | |
1 2 3 4 5 6 7 8 9 10 11 12
Variation due
to assignable
causes
Variation due
to assignable
causes
Variation due to
natural causes
Out of
control
Out of
control
- 27. © 2006 Prentice Hall, Inc. S6 – 27
Setting Chart Limits
For x-Charts when we don’t know s
Lower control limit (LCL) = x - A2R
Upper control limit (UCL) = x + A2R
where R = average range of the samples
A2 = control chart factor found in Table S6.1
x = mean of the sample means
- 28. © 2006 Prentice Hall, Inc. S6 – 28
Control Chart Factors
Table S6.1
Sample Size Mean Factor Upper Range Lower Range
n A2 D4 D3
2 1.880 3.268 0
3 1.023 2.574 0
4 .729 2.282 0
5 .577 2.115 0
6 .483 2.004 0
7 .419 1.924 0.076
8 .373 1.864 0.136
9 .337 1.816 0.184
10 .308 1.777 0.223
12 .266 1.716 0.284
- 29. © 2006 Prentice Hall, Inc. S6 – 29
Setting Control Limits
Process average x = 16.01 ounces
Average range R = .25
Sample size n = 5
- 30. © 2006 Prentice Hall, Inc. S6 – 30
Setting Control Limits
UCLx = x + A2R
= 16.01 + (.577)(.25)
= 16.01 + .144
= 16.154 ounces
Process average x = 16.01 ounces
Average range R = .25
Sample size n = 5
From
Table S6.1
- 31. © 2006 Prentice Hall, Inc. S6 – 31
Setting Control Limits
UCLx = x + A2R
= 16.01 + (.577)(.25)
= 16.01 + .144
= 16.154 ounces
LCLx = x - A2R
= 16.01 - .144
= 15.866 ounces
Process average x = 16.01 ounces
Average range R = .25
Sample size n = 5
UCL = 16.154
Mean = 16.01
LCL = 15.866
- 32. © 2006 Prentice Hall, Inc. 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
- 33. © 2006 Prentice Hall, Inc. S6 – 33
Setting Chart Limits
For R-Charts
Lower control limit (LCLR) = D3R
Upper control limit (UCLR) = D4R
where
R = average range of the samples
D3 and D4 = control chart factors from Table S6.1
- 34. © 2006 Prentice Hall, Inc. S6 – 34
Setting Control Limits
UCLR = D4R
= (2.115)(5.3)
= 11.2 pounds
LCLR = D3R
= (0)(5.3)
= 0 pounds
Average range R = 5.3 pounds
Sample size n = 5
From Table S6.1 D4 = 2.115, D3 = 0
UCL = 11.2
Mean = 5.3
LCL = 0
- 35. © 2006 Prentice Hall, Inc. S6 – 35
Mean and Range Charts
(a)
These
sampling
distributions
result in the
charts below
(Sampling mean is
shifting upward but
range is consistent)
R-chart
(R-chart does not
detect change in
mean)
UCL
LCL
Figure S6.5
x-chart
(x-chart detects
shift in central
tendency)
UCL
LCL
- 36. © 2006 Prentice Hall, Inc. S6 – 36
Mean and Range Charts
R-chart
(R-chart detects
increase in
dispersion)
UCL
LCL
Figure S6.5
(b)
These
sampling
distributions
result in the
charts below
(Sampling mean
is constant but
dispersion is
increasing)
x-chart
(x-chart does not
detect the increase
in dispersion)
UCL
LCL
- 38. © 2006 Prentice Hall, Inc. S6 – 38
Control Charts for Attributes
For variables that are categorical
Good/bad, yes/no,
acceptable/unacceptable
Measurement is typically counting
defectives
Charts may measure
Percent defective (p-chart)
Number of defects (c-chart)
- 39. © 2006 Prentice Hall, Inc. S6 – 39
Control Limits for p-Charts
Population will be a binomial distribution,
but applying the Central Limit Theorem
allows us to assume a normal distribution
for the sample statistics
UCLp = p + zsp
^
LCLp = p - zsp
^
where p = mean fraction defective in the sample
z = number of standard deviations
sp = standard deviation of the sampling distribution
n = sample size
^
p(1 - p)
n
sp =
^
- 40. © 2006 Prentice Hall, Inc. S6 – 40
p-Chart for Data Entry
Sample Number Fraction Sample Number Fraction
Number of Errors Defective Number of Errors Defective
1 6 .06 11 6 .06
2 5 .05 12 1 .01
3 0 .00 13 8 .08
4 1 .01 14 7 .07
5 4 .04 15 5 .05
6 2 .02 16 4 .04
7 5 .05 17 11 .11
8 3 .03 18 3 .03
9 3 .03 19 0 .00
10 2 .02 20 4 .04
Total = 80
(.04)(1 - .04)
100
sp = = .02
^
p = = .04
80
(100)(20)
- 41. © 2006 Prentice Hall, Inc. S6 – 41
.11 –
.10 –
.09 –
.08 –
.07 –
.06 –
.05 –
.04 –
.03 –
.02 –
.01 –
.00 –
Sample number
Fraction
defective
| | | | | | | | | |
2 4 6 8 10 12 14 16 18 20
p-Chart for Data Entry
UCLp = p + zsp = .04 + 3(.02) = .10
^
LCLp = p - zsp = .04 - 3(.02) = 0
^
UCLp = 0.10
LCLp = 0.00
p = 0.04
- 42. © 2006 Prentice Hall, Inc. S6 – 42
.11 –
.10 –
.09 –
.08 –
.07 –
.06 –
.05 –
.04 –
.03 –
.02 –
.01 –
.00 –
Sample number
Fraction
defective
| | | | | | | | | |
2 4 6 8 10 12 14 16 18 20
UCLp = p + zsp = .04 + 3(.02) = .10
^
LCLp = p - zsp = .04 - 3(.02) = 0
^
UCLp = 0.10
LCLp = 0.00
p = 0.04
p-Chart for Data Entry
Possible
assignable
causes present
- 43. © 2006 Prentice Hall, Inc. S6 – 43
Control Limits for c-Charts
Population will be a Poisson distribution,
but applying the Central Limit Theorem
allows us to assume a normal distribution
for the sample statistics
where c = mean number defective in the sample
UCLc = c + 3 c LCLc = c - 3 c
- 44. © 2006 Prentice Hall, Inc. S6 – 44
c-Chart for Cab Company
c = 54 complaints/9 days = 6 complaints/day
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
Day
Number
defective
14 –
12 –
10 –
8 –
6 –
4 –
2 –
0 –
UCLc = c + 3 c
= 6 + 3 6
= 13.35
LCLc = c - 3 c
= 3 - 3 6
= 0
UCLc = 13.35
LCLc = 0
c = 6
- 45. © 2006 Prentice Hall, Inc. S6 – 45
Patterns in Control Charts
Normal behavior.
Process is “in control.”
Upper control limit
Target
Lower control limit
Figure S6.7
- 46. © 2006 Prentice Hall, Inc. S6 – 46
Upper control limit
Target
Lower control limit
Patterns in Control Charts
One plot out above (or
below). Investigate for
cause. Process is “out
of control.”
Figure S6.7
- 47. © 2006 Prentice Hall, Inc. S6 – 47
Upper control limit
Target
Lower control limit
Patterns in Control Charts
Trends in either
direction, 5 plots.
Investigate for cause of
progressive change.
Figure S6.7
- 48. © 2006 Prentice Hall, Inc. S6 – 48
Upper control limit
Target
Lower control limit
Patterns in Control Charts
Two plots very near
lower (or upper)
control. Investigate for
cause.
Figure S6.7
- 49. © 2006 Prentice Hall, Inc. S6 – 49
Upper control limit
Target
Lower control limit
Patterns in Control Charts
Run of 5 above (or
below) central line.
Investigate for cause.
Figure S6.7
- 50. © 2006 Prentice Hall, Inc. S6 – 50
Upper control limit
Target
Lower control limit
Patterns in Control Charts
Erratic behavior.
Investigate.
Figure S6.7
- 51. © 2006 Prentice Hall, Inc. S6 – 51
Which Control Chart to Use
Using an x-chart and R-chart:
Observations are variables
Collect 20 - 25 samples of n = 4, or n =
5, or more, each from a stable process
and compute the mean for the x-chart
and range for the R-chart
Track samples of n observations each
Variables Data
- 52. © 2006 Prentice Hall, Inc. S6 – 52
Which Control Chart to Use
Using the p-chart:
Observations are attributes that can
be categorized in two states
We deal with fraction, proportion, or
percent defectives
Have several samples, each with
many observations
Attribute Data
- 53. © 2006 Prentice Hall, Inc. S6 – 53
Which Control Chart to Use
Using a c-Chart:
Observations are attributes whose
defects per unit of output can be
counted
The number counted is often a small
part of the possible occurrences
Defects such as number of blemishes
on a desk, number of typos in a page
of text, flaws in a bolt of cloth
Attribute Data
- 54. © 2006 Prentice Hall, Inc. S6 – 54
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
- 55. © 2006 Prentice Hall, Inc. S6 – 55
Process Capability Ratio
Cp =
Upper Specification - Lower Specification
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
- 56. © 2006 Prentice Hall, Inc. S6 – 56
Process Capability Ratio
Cp =
Upper Specification - Lower Specification
6s
Insurance claims process
Process mean x = 210.0 minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes
- 57. © 2006 Prentice Hall, Inc. S6 – 57
Process Capability Ratio
Cp =
Upper Specification - Lower Specification
6s
Insurance claims process
Process mean x = 210.0 minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes
= = 1.938
213 - 207
6(.516)
- 58. © 2006 Prentice Hall, Inc. S6 – 58
Process Capability Ratio
Cp =
Upper Specification - Lower Specification
6s
Insurance claims process
Process mean x = 210.0 minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes
= = 1.938
213 - 207
6(.516)
Process is
capable
- 59. © 2006 Prentice Hall, Inc. S6 – 59
Process Capability Index
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
Cpk = minimum of ,
Upper
Specification - x
Limit
3s
Lower
x - Specification
Limit
3s
- 60. © 2006 Prentice Hall, Inc. S6 – 60
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
- 61. © 2006 Prentice Hall, Inc. S6 – 61
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
Cpk = minimum of ,
(.251) - .250
(3).0005
- 62. © 2006 Prentice Hall, Inc. S6 – 62
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
Cpk = = 0.67
.001
.0015
New machine is
NOT capable
Cpk = minimum of ,
(.251) - .250
(3).0005
.250 - (.249)
(3).0005
Both calculations result in
- 63. © 2006 Prentice Hall, Inc. S6 – 63
Interpreting Cpk
Cpk = negative number
Cpk = zero
Cpk = between 0 and 1
Cpk = 1
Cpk > 1
Figure S6.8
- 64. © 2006 Prentice Hall, Inc. S6 – 64
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
- 65. © 2006 Prentice Hall, Inc. S6 – 65
Operating Characteristic
Curve
Shows how well a sampling plan
discriminates between good and
bad lots (shipments)
Shows the relationship between
the probability of accepting a lot
and its quality level
- 66. © 2006 Prentice Hall, Inc. S6 – 66
Return whole
shipment
The “Perfect” OC Curve
% Defective in Lot
P(Accept
Whole
Shipment)
100 –
75 –
50 –
25 –
0 –
| | | | | | | | | | |
0 10 20 30 40 50 60 70 80 90 100
Cut-Off
Keep whole
shipment
- 67. © 2006 Prentice Hall, Inc. S6 – 67
AQL and LTPD
Acceptable Quality Level (AQL)
Poorest level of quality we are
willing to accept
Lot Tolerance Percent Defective
(LTPD)
Quality level we consider bad
Consumer (buyer) does not want to
accept lots with more defects than
LTPD
- 68. © 2006 Prentice Hall, Inc. S6 – 68
Producer’s and Consumer’s
Risks
Producer's risk ()
Probability of rejecting a good lot
Probability of rejecting a lot when the
fraction defective is at or above the
AQL
Consumer's risk (b)
Probability of accepting a bad lot
Probability of accepting a lot when
fraction defective is below the LTPD
- 69. © 2006 Prentice Hall, Inc. S6 – 69
An OC Curve
Probability
of
Acceptance
Percent
defective
| | | | | | | | |
0 1 2 3 4 5 6 7 8
100 –
95 –
75 –
50 –
25 –
10 –
0 –
= 0.05 producer’s risk for AQL
b = 0.10
Consumer’s
risk for LTPD
LTPD
AQL
Bad lots
Indifference
zone
Good
lots
Figure S6.9
- 70. © 2006 Prentice Hall, Inc. S6 – 70
OC Curves for Different
Sampling Plans
n = 50, c = 1
n = 100, c = 2
- 71. © 2006 Prentice Hall, Inc. S6 – 71
Average Outgoing Quality
where
Pd = true percent defective of the lot
Pa = probability of accepting the lot
N = number of items in the lot
n = number of items in the sample
AOQ =
(Pd)(Pa)(N - n)
N
- 72. © 2006 Prentice Hall, Inc. S6 – 72
Average Outgoing Quality
1. If a sampling plan replaces all defectives
2. If we know the incoming percent
defective for the lot
We can compute the average outgoing
quality (AOQ) in percent defective
The maximum AOQ is the highest percent
defective or the lowest average quality
and is called the average outgoing quality
level (AOQL)
- 73. © 2006 Prentice Hall, Inc. S6 – 73
SPC and Process Variability
(a) Acceptance
sampling (Some
bad units accepted)
(b) Statistical process
control (Keep the
process in control)
(c) Cpk >1 (Design
a process that
is in control)
Lower
specification
limit
Upper
specification
limit
Process mean, m Figure S6.10