PowerPoint presentation to accompany Heizer and Render Operations Management Eleventh Edition Principles of Operations Management Ninth Edition PowerPoint slides by Jeff Heyl 6 2014 Pearson Education Inc ID: 215781
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Slide1
Statistical Process Control
PowerPoint presentation to accompany
Heizer and Render
Operations Management, Eleventh Edition
Principles of Operations Management, Ninth Edition
PowerPoint slides by Jeff Heyl
6
© 2014 Pearson Education, Inc.
SUPPLEMENTSlide2
Outline
Statistical Process Control
Process CapabilityAcceptance SamplingSlide3
Learning Objectives
When you complete this supplement you should be able to :
Explain
the purpose of a control chart
Explain
the role of the central limit theorem in SPC
Build -charts and R-chartsList the five steps involved in building control chartsSlide4
Learning Objectives
When you complete this supplement you should be able to :
Build
p
-charts and
c-chartsExplain process capability and compute Cp and CpkExplain acceptance samplingSlide5
Statistical Process Control
The objective of a process control system is to provide a statistical signal when assignable causes of variation are presentSlide6
Variability is inherent
in every processNatural or common
causesSpecial or assignable
causes
Provides a statistical signal when assignable causes are presentDetect and eliminate assignable causes of variationStatistical Process Control (SPC)Slide7
Natural Variations
Also called common causes
Affect virtually all production processes
Expected amount of variation
Output measures follow a probability distributionFor any distribution there is a measure of central tendency and dispersionIf the distribution of outputs falls within acceptable limits, the process is said to be “in control”Slide8
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 presentEliminate the bad causesIncorporate the good causesSlide9
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
S6.1Slide10
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
S6.1Slide11
Samples
(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
To measure the process, we take samples and analyze the sample statistics following these stepsSlide12
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
Time
Frequency
Prediction
Figure
S6.1Slide13
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
Time
Frequency
Prediction
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Figure
S6.1Slide14
Control Charts
Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causesSlide15
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 controlSlide16
Control Charts for Variables
Characteristics that can take any real value
May be in whole or in fractional numbers
Continuous random variables
x
-chart
tracks changes in the central tendencyR-chart indicates a gain or loss of dispersion
These two charts must be used togetherSlide17
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
The standard deviation of the sampling distribution ( ) will equal the population standard deviation (
s
) divided by the square root of the sample size,
n
The mean of the sampling distribution will be the same as the population mean
mSlide18
Population and Sampling Distributions
Population distributions
Beta
Normal
Uniform
Distribution of sample means
Figure
S6.3
99.73% of all
fall within ±
95.45% fall within ±
| | | | | | |
Standard deviation of the sample means
Mean of sample means =Slide19
Sampling Distribution
=
m
(mean)
Sampling distribution of means
Process distribution of means
Figure
S6.4Slide20
Setting Chart Limits
For
x
-Charts when we know
s
Where
= 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 = population (process) standard deviation n = sample sizeSlide21
Setting Control Limits
Randomly select and weigh nine (n = 9) boxes each hour
WEIGHT OF SAMPLE
WEIGHT OF SAMPLE
WEIGHT OF SAMPLE
HOUR
(AVG.
OF 9 BOXES)
HOUR(AVG. OF 9 BOXES)HOUR(AVG. OF 9 BOXES)
1
16.1516.5
916.3
2
16.8
6
16.4
10
14.8
3
15.5
7
15.2
11
14.2
4
16.5
8
16.4
12
17.3
Average weight in the first sampleSlide22
Setting Control Limits
Average mean of 12 samplesSlide23
Setting Control Limits
Average mean of 12 samplesSlide24
17 = UCL
15 = LCL
16 = Mean
Sample number
| | | | | | | | | | | |
1 2 3 4 5 6 7 8 9 10 11 12
Setting Control Limits
Control Chart for samples of 9 boxes
Variation due to assignable causes
Variation due to assignable causes
Variation due to natural causes
Out of control
Out of controlSlide25
Setting Chart Limits
For
x
-Charts when we don’t know
s
where average range of the samples
A
2 = control chart factor found in Table S6.1 = mean of the sample meansSlide26
Control Chart Factors
TABLE S6.1
Factors for Computing Control Chart Limits (3 sigma)
SAMPLE SIZE,
nMEAN FACTOR,
A2
UPPER RANGE, D4
LOWER RANGE, D32
1.8803.268031.023
2.574
04.729
2.2820
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
0.284Slide27
Setting Control Limits
Process average = 12 ounces
Average range = .25 ounce
Sample size = 5
UCL = 12.144
Mean = 12
LCL = 11.856
From Table S6.1
Super Cola Example
Labeled as “net weight 12 ounces”Slide28
Restaurant Control Limits
For salmon filets at Darden Restaurants
Sample Mean
x
Bar Chart
UCL = 11.524
= – 10.959
LCL = – 10.394
| | | | | | | | |
1 3 5 7 9 11 13 15 17
11.5 –
11.0 –
10.5 –
Sample Range
Range Chart
UCL = 0.6943
= 0.2125
LCL = 0
| | | | | | | | |
1 3 5 7 9 11 13 15 17
0.8 –
0.4 –
0.0 –Slide29
R
– Chart
Type of variables control chart
Shows sample ranges over time
Difference between smallest and largest values in sampleMonitors process variabilityIndependent from process meanSlide30
Setting Chart Limits
For
R
-Charts
whereSlide31
Setting Control Limits
Average range = 5.3 pounds
Sample size = 5
From Table S6.1
D4 = 2.115, D3 = 0
UCL = 11.2
Mean = 5.3
LCL = 0Slide32
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
LCLSlide33
Mean and Range Charts
R
-chart
(
R
-chart detects increase in dispersion)
UCL
LCL
(b)
These sampling distributions result in the charts below
(Sampling mean is constant, but dispersion is increasing)
x
-chart
(
x
-chart indicates no change in central tendency)
UCL
LCL
Figure
S6.5Slide34
Steps In Creating Control Charts
Collect 20 to 25 samples, often of
n
= 4 or
n = 5 observations each, from a stable process and compute the mean and range of each
Compute the overall means (
and ), set appropriate control limits, usually at the 99.73% level, and calculate the preliminary upper and lower control limitsIf the process is not currently stable and in control, use the desired mean, m, instead of
to calculate limits. Slide35
Steps In Creating Control Charts
Graph the sample means and ranges on their respective control charts and determine whether they fall outside the acceptable limits
Investigate points or patterns that indicate the process is out of control – try to assign causes for the variation, address the causes, and then resume the process
Collect additional samples and, if necessary, revalidate the control limits using the new dataSlide36
Setting Other Control Limits
TABLE S6.2
Common
z
ValuesDESIRED CONTROL LIMIT (%)
Z
-VALUE (STANDARD DEVIATION REQUIRED FOR DESIRED LEVEL OF CONFIDENCE
90.01.65
95.01.96 95.45
2.00
99.02.58
99.73
3.00Slide37
Control Charts for Attributes
For variables that are categorical
Defective/nondefective, good/bad, yes/no, acceptable/unacceptable
Measurement is typically counting defectives
Charts may measurePercent defective (p-chart)Number of defects (c-chart)Slide38
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
whereSlide39
p
-Chart for Data Entry
SAMPLE NUMBER
NUMBER OF ERRORS
FRACTION DEFECTIVESAMPLE NUMBER
NUMBER OF ERRORS
FRACTION DEFECTIVE
16.06
116.06
2
5.0512
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
80Slide40
p
-Chart for Data Entry
SAMPLE NUMBER
NUMBER OF ERRORS
FRACTION DEFECTIVESAMPLE NUMBER
NUMBER OF ERRORS
FRACTION DEFECTIVE
16.06
116.06
2
5.0512
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
80
(because we cannot have a negative percent defective)Slide41
.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
UCL
p
= 0.10
LCL
p
= 0.00
p
= 0.04Slide42
.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
UCL
p
= 0.10
LCL
p
= 0.00
p
= 0.04
Possible assignable causes presentSlide43
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 statisticsSlide44
c
-Chart for Cab Company
|
1
|
2
|
3
|4|5
|
6
|7
|
8
|
9
Day
Number defective
14 –
12 –
10 –
8 –
6 –
4 –
2 –
0 –
UCL
c
= 13.35
LCL
c
= 0
c
= 6
Cannot be a
negative numberSlide45
Select points in the processes that need SPC
Determine the appropriate charting technique
Set clear policies and procedures
Managerial Issues and
Control Charts
Three major management decisions:Slide46
Which Control Chart to Use
TABLE S6.3
Helping You Decide Which Control Chart to Use
VARIABLE 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 Slide47
Which Control Chart to Use
TABLE S6.3
Helping You Decide Which Control Chart to Use
ATTRIBUTE DATA
USING A P-CHART
Observations are attributes that can be categorized as good or bad (or pass–fail, or functional–broken), that is, in two states
We deal with fraction, proportion, or percent defectivesThere are several samples, with many observations in each
ATTRIBUTE DATAUSING A C-CHART
Observations are attributes whose defects per unit of output can be countedWe deal with the number counted, which is a small part of the possible occurrencesDefects may be: number of blemishes on a desk; crimes in a year; broken seats in a stadium; typos in a chapter of this text; flaws in a bolt of clothSlide48
Patterns in Control Charts
Normal behavior. Process is “in control.”
Upper control limit
Target
Lower control limit
Figure
S6.7Slide49
Patterns in Control Charts
One plot out above (or below). Investigate for cause. Process is “out of control.”
Upper control limit
Target
Lower control limit
Figure
S6.7Slide50
Patterns in Control Charts
Trends in either direction, 5 plots. Investigate for cause of progressive change.
Upper control limit
Target
Lower control limit
Figure
S6.7Slide51
Patterns in Control Charts
Two plots very near lower (or upper) control. Investigate for cause.
Upper control limit
Target
Lower control limit
Figure
S6.7Slide52
Patterns in Control Charts
Run of 5 above (or below) central line. Investigate for cause.
Upper control limit
Target
Lower control limit
Figure
S6.7Slide53
Patterns in Control Charts
Erratic behavior. Investigate.
Upper control limit
Target
Lower control limit
Figure
S6.7Slide54
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 specificationsSlide55
Process Capability Ratio
C
p
=
Upper Specification – Lower Specification
6
s
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 processesSix Sigma quality requires a Cp = 2.0Slide56
Process Capability Ratio
C
p
=
Upper Specification - Lower Specification
6
s
Insurance claims process
Process mean x = 210.0 minutesProcess standard deviation s = .516 minutesDesign specification = 210 ± 3 minutesSlide57
Process Capability Ratio
C
p
=
Upper Specification - Lower Specification
6
s
Insurance claims process
Process mean x = 210.0 minutesProcess standard deviation s = .516 minutesDesign specification = 210 ± 3 minutes
= = 1.938
213 – 207
6(.516)Slide58
Process Capability Ratio
C
p
=
Upper Specification - Lower Specification
6
s
Insurance claims process
Process mean x = 210.0 minutesProcess standard deviation s = .516 minutesDesign specification = 210 ± 3 minutes
= = 1.938
213 – 207
6(.516)Process variance is small enough for capable processSlide59
Process Capability Index
A capable process must have a C
pk
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
C
pk = minimum of , ,
Upper
Specification – xLimit 3s
Lower
x – Specification Limit 3sSlide60
Process Capability Index
New Cutting Machine
New process mean
x
= .250 inches
Process standard deviation
s = .0005 inchesUpper Specification Limit = .251 inchesLower Specification Limit = .249 inchesSlide61
Process Capability Index
New Cutting Machine
New process mean
x
= .250 inches
Process standard deviation
s = .0005 inchesUpper Specification Limit = .251 inchesLower Specification Limit = .249 inches
Cpk = minimum of ,
(.251) - .250(3).0005Slide62
Process Capability Index
New Cutting Machine
New process mean
x
= .250 inches
Process standard deviation
s = .0005 inchesUpper Specification Limit = .251 inchesLower Specification Limit = .249 inches
Cpk = minimum of ,
(.251) - .250(3).0005
.250 - (.249)
(3).0005Slide63
Process Capability Index
New Cutting Machine
New process mean
x
= .250 inches
Process standard deviation
s = .0005 inchesUpper Specification Limit = .251 inchesLower Specification Limit = .249 inches
Cpk = = 0.67
.001.0015
New machine is NOT capable
C
pk = minimum of ,
(.251) - .250
(3).0005
.250 - (.249)
(3).0005
Both calculations result inSlide64
Lower specification limit
Upper specification limit
Interpreting C
pk
C
pk
= negative number
C
pk
= zero
C
pk
= between 0 and 1
C
pk
= 1
C
pk
> 1
Figure
S6.8Slide65
Acceptance Sampling
Form of quality testing used for incoming materials or finished goods
Take samples at random from a lot (shipment) of
items (n)
Inspect each of the items in the sampleReject the whole lot if there are c or more defective items in the sampleThe sampling plan is defined as (n,c)Only screens lots; does not drive quality improvement effortsSlide66
Acceptance Sampling
Form of quality testing used for incoming materials or finished goods
Take samples at random from a lot (shipment) of
items (n)
Inspect each of the items in the sampleReject the whole lot if there are c or more defective items in the sampleThe sampling plan is defined as (n,c)Only screens lots; does not drive quality improvement effortsSlide67
Operating Characteristic Curve
Shows the relationship between the probability of accepting a lot and its quality
level
% defectives in the lot in the x-axis, probability of accepting the lot is y-axis
Shows how well a sampling plan discriminates between good and bad lots (shipments)Slide68
An OC Curve
Probability of Acceptance
Percent defective
| | | | | | | | |
0 1 2 3 4 5 6 7 8Slide69
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 badConsumer (buyer) does not want to accept lots with more defects than LTPDSlide70
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 shipmentSlide71
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 AQLConsumer's risk (b)Probability of accepting a bad lot Probability of accepting a lot when fraction defective is below the LTPDSlide72
An OC Curve
Probability of Acceptance
Percent defective
| | | | | | | | |
0 1 2 3 4 5 6 7 8
= 0.05 producer’s risk for AQL
= 0.10
Consumer’s risk for LTPD
LTPD
AQL
Bad lots
Indifference zone
Good lots
Figure
S6.9Slide73
OC Curves for Different Sampling Plans
n
= 50,
c
= 1
n
= 100,
c = 2Slide74
Average Outgoing Quality
where
P
d = 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)NSlide75
Average Outgoing Quality
If a sampling plan replaces all defectives
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 limit
(AOQL)Slide76
Automated Inspection
Modern technologies allow virtually 100% inspection at minimal costs
Not suitable for all situationsSlide77
SPC and Process Variability
(a) Acceptance sampling (Some bad units accepted; the “lot” is good or bad)
(b) Statistical process control (Keep the process “in control”)
(c) C
pk
> 1 (Design a process that is in within specification)
Lower specification limit
Upper specification limit
Process mean,
m
Figure
S6.10Slide78
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