sup06.ppt

© 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

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

Outline – Continued
 Process Capability
 Process Capability Ratio (Cp)
 Process Capability Index (Cpk )
 Acceptance Sampling
 Operating Characteristic Curve
 Average Outgoing Quality

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

Learning Objectives
Identify or Define:
 LCL and UCL
 P-charts and c-charts
 Cp and Cpk
 Acceptance sampling
 OC curve

Learning Objectives
Identify or Define:
 AQL and LTPD
 AOQ
 Producer’s and consumer’s risk

Learning Objectives
Describe or Explain:
 The role of statistical quality control

 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)

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”

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

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

Samples
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

Samples
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

Samples
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

Samples
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

Control Charts
Constructed from historical data, the
purpose of control charts is to help
distinguish between natural variations
and variations due to assignable
causes

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

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

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

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

Sampling Distribution
x = m
(mean)
Sampling
distribution
of means
Process
distribution
of means
Figure S6.4

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

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

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

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

17 = UCL
15 = LCL
16 = Mean
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

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

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

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
Sample size n = 5
From
Table S6.1

UCLx = x + A2R
= 16.01 + (.577)(.25)
= 16.01 + .144
= 16.154 ounces
LCLx = x - A2R
= 16.01 - .144
= 15.866 ounces
Sample size n = 5
UCL = 16.154
Mean = 16.01
LCL = 15.866

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

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

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

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

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

Automated Control Charts

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)

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 =
^

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)

.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

.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
Possible
assignable
causes present

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

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

Patterns in Control Charts
Normal behavior.
Process is “in control.”
Upper control limit
Target
Lower control limit
Figure S6.7

Upper control limit
Target
Lower control limit
One plot out above (or
below). Investigate for
cause. Process is “out
of control.”
Figure S6.7

Upper control limit
Target
Lower control limit
Trends in either
direction, 5 plots.
Investigate for cause of
progressive change.
Figure S6.7

Upper control limit
Target
Lower control limit
Two plots very near
lower (or upper)
control. Investigate for
cause.
Figure S6.7

Upper control limit
Target
Lower control limit
Run of 5 above (or
below) central line.
Investigate for cause.
Figure S6.7

Upper control limit
Target
Lower control limit
Erratic behavior.
Investigate.
Figure S6.7

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

 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

 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

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

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

Cp =
6s
Insurance claims process
Process mean x = 210.0 minutes
Process standard deviation s = .516 minutes
Design specification = 210 ± 3 minutes

Cp =
6s
= = 1.938
213 - 207
6(.516)

Cp =
6s
= = 1.938
213 - 207
6(.516)
Process is
capable

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

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

New Cutting Machine
Cpk = minimum of ,
(.251) - .250
(3).0005

New Cutting Machine
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

Interpreting Cpk
Cpk = negative number
Cpk = zero
Cpk = between 0 and 1
Cpk = 1
Cpk > 1
Figure S6.8

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

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

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

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

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

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

OC Curves for Different
Sampling Plans
n = 50, c = 1
n = 100, c = 2

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

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)

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

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