#### Transcript Statistical Process Control - University of Hawaii at Hilo

```Operations
Management
Supplement 6 –
Statistical Process
Control
PowerPoint presentation to accompany
Heizer/Render
Principles of Operations Management, 7e
Operations Management, 9e
S6 – 1
Statistical Process Control
(SPC)
 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
S6 – 2
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”
S6 – 3
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
 Incorporate the good causes
S6 – 4
Samples
To measure the process, we take samples
and analyze the sample statistics following
these 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 – 5
Samples
To measure the process, we take samples
and analyze the sample statistics following
these steps
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 – 6
Samples
To measure the process, we take samples
and analyze the sample statistics following
these steps
Frequency
(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
Figure S6.1
Central tendency
Weight
Variation
Weight
Shape
Weight
S6 – 7
Samples
(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
Frequency
To measure the process, we take samples
and analyze the sample statistics following
these steps
Prediction
Weight
Figure S6.1
S6 – 8
Samples
To measure the process, we take samples
and analyze the sample statistics following
these steps
Frequency
(e) If assignable
causes are
present, the
process output is
not stable over
time and is not
predicable
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Prediction
Weight
Figure S6.1
S6 – 9
Control Charts
Constructed from historical data, the
purpose of control charts is to help
distinguish between natural variations
and variations due to assignable
causes
S6 – 10
Process Control
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
(c) Out of control
Size
(weight, length, speed, etc.)
Figure S6.2
S6 – 11
Types of Data
Variables
 Characteristics that
can take any real
value
 May be in whole or
in fractional
numbers
 Continuous random
variables
Attributes
 Defect-related
characteristics
 Classify products
as either good or
defects
 Categorical or
discrete random
variables
S6 – 12
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 – 13
Population and Sampling
Distributions
Three population
distributions
Distribution of
sample means
Mean of sample means = x
Beta
Standard
s
deviation of
the sample = sx = n
means
Normal
Uniform
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-3sx
-2sx
-1sx
x
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+1sx +2sx +3sx
95.45% fall within ± 2sx
99.73% of all x
fall within ± 3sx
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Figure S6.3
S6 – 14
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 – 15
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 – 16
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 – 17
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
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1 2 3 4 5 6 7 8 9 10 11 12
Sample number
Out of
control
Variation due
to assignable
causes
S6 – 18
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 – 19
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 – 20
Setting Control Limits
Process average x = 12 ounces
Average range R = .25
Sample size n = 5
S6 – 21
Setting Control Limits
Process average x = 12 ounces
Average range R = .25
Sample size n = 5
UCLx
= x + A2R
= 12 + (.577)(.25)
= 12 + .144
= 12.144 ounces
From
Table S6.1
S6 – 22
Setting Control Limits
Process average x = 12 ounces
Average range R = .25
Sample size n = 5
UCLx
LCLx
= x + A2R
= 12 + (.577)(.25)
= 12 + .144
= 12.144 ounces
UCL = 12.144
= x - A2R
= 12 - .144
= 11.857 ounces
LCL = 11.857
Mean = 12
S6 – 23
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 – 24
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 – 25
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 – 26
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
mean)
R-chart
LCL
Figure S6.5
S6 – 27
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 dispersion)
x-chart
LCL
UCL
(R-chart detects
increase in
dispersion)
R-chart
LCL
Figure S6.5
S6 – 28
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
S6 – 29
Manual and Automated
Control Charts
S6 – 30
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 – 31
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^
sp =
^
LCLp = p - zsp^
where
p
z
sp^
n
=
=
=
=
p(1 - p)
n
mean fraction defective in the sample
number of standard deviations
standard deviation of the sampling distribution
sample size
S6 – 32
p-Chart for Data Entry
Sample
Number
1
2
3
4
5
6
7
8
9
10
Number
of Errors
Fraction
Defective
6
5
0
1
4
2
5
3
3
2
.06
.05
.00
.01
.04
.02
.05
.03
.03
.02
80
p = (100)(20) = .04
Sample
Number
Number
of Errors
11
6
12
1
13
8
14
7
15
5
16
4
17
11
18
3
19
0
20
4
Total = 80
sp^ =
Fraction
Defective
.06
.01
.08
.07
.05
.04
.11
.03
.00
.04
(.04)(1 - .04)
= .02
100
S6 – 33
p-Chart for Data Entry
UCLp = p + zsp^ = .04 + 3(.02) = .10
Fraction defective
LCLp = p - zsp^ = .04 - 3(.02) = 0
.11
.10
.09
.08
.07
.06
.05
.04
.03
.02
.01
.00
–
–
–
–
–
–
–
–
–
–
–
–
UCLp = 0.10
p = 0.04
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2
4
6
8
10
12
14
16
18
20
LCLp = 0.00
Sample number
S6 – 34
p-Chart for Data Entry
UCLp = p + zsp^ = .04 + 3(.02) = .10
Fraction defective
Possible
LCLp = p - zsp^ = .04 - 3(.02) =
0
assignable
causes present
.11
.10
.09
.08
.07
.06
.05
.04
.03
.02
.01
.00
–
–
–
–
–
–
–
–
–
–
–
–
UCLp = 0.10
p = 0.04
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2
4
6
8
10
12
14
16
18
20
LCLp = 0.00
Sample number
S6 – 35
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
UCLc = c + 3 c
where
LCLc = c - 3 c
c = mean number defective in the sample
S6 – 36
c-Chart for Cab Company
c = 54 complaints/9 days = 6 complaints/day
LCLc = c - 3 c
=6-3 6
=0
UCLc = 13.35
14 –
12 –
Number defective
UCLc = c + 3 c
=6+3 6
= 13.35
10
8
6
4
2
–
–
–
–
–
c= 6
0 – |
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3
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4
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5
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6
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7
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8
LCLc = 0
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Day
S6 – 37
Managerial Issues and
Control Charts
Three major management decisions:
 Select points in the processes that
need SPC
 Determine the appropriate charting
technique
 Set clear policies and procedures
S6 – 38
Which Control Chart to Use
Variables Data
 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
S6 – 39
Which Control Chart to Use
Attribute 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
S6 – 40
Which Control Chart to Use
Attribute Data
 Using a c-Chart:
 Observations are attributes whose
defects per unit of output can be
counted
 The number counted is 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
S6 – 41
Patterns in Control Charts
Upper control limit
Target
Lower control limit
Figure S6.7
Normal behavior.
Process is “in control.”
S6 – 42
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 – 43
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 – 44
Patterns in Control Charts
Upper control limit
Target
Lower control limit
Figure S6.7
Two plots very near
lower (or upper)
control. Investigate for
cause.
S6 – 45
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 – 46
Patterns in Control Charts
Upper control limit
Target
Lower control limit
Erratic behavior.
Investigate.
Figure S6.7