statistical process control
DESCRIPTION
Statistical Process Control (SPC)TRANSCRIPT
Statistical Process ControlOperations Management, 8e
Natural or common causes
Special or assignable causes
Detect and eliminate assignable causes of variation
Points which might be emphasized include:
- Statistical process control measures the performance of a process, it does not help to identify a particular specimen produced as being “good” or “bad,” in or out of tolerance.
- Statistical process control requires the collection and analysis of data - therefore it is not helpful when total production consists of a small number of units
- While statistical process control can not help identify a “good” or “bad” unit, it can enable one to decide whether or not to accept an entire production lot. If a sample of a production lot contains more than a specified number of defective items, statistical process control can give us a basis for rejecting the entire lot. The issue of rejecting a lot which was actually good can be raised here, but is probably better left to later.
© 2006 Prentice Hall, Inc.
These are to be expected
Output measures follow a probability distribution
For any distribution there is a measure of central tendency and dispersion
© 2006 Prentice Hall, Inc.
Assignable Variations
Variations that can be traced to a specific reason (machine wear, misadjusted equipment, fatigued or untrained workers)
The objective is to discover when assignable causes are present and eliminate them
© 2006 Prentice Hall, Inc.
S6 – *
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
Figure S6.1
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#
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Each of these represents one sample of five boxes of cereal
© 2006 Prentice Hall, Inc.
S6 – *
Samples
(b) After enough samples are taken from a stable process, they form a pattern called a distribution
Figure S6.1
Frequency
Weight
S6 – *
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
Figure S6.1
S6 – *
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
Figure S6.1
S6 – *
Samples
(e) If assignable causes are present, the process output is not stable over time and is not predicable
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
Students should understand both the concepts of natural and assignable variation, and the nature of the efforts required to deal with them.
© 2006 Prentice Hall, Inc.
May be in whole or in fractional numbers
Continuous random variables
Classify products as either good or bad or count defects
Categorical or discrete random variables
Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process.
© 2006 Prentice Hall, Inc.
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
© 2006 Prentice Hall, Inc.
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
Hour 1
© 2006 Prentice Hall, Inc.
17 = UCL
15 = LCL
16 = Mean
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
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
© 2006 Prentice Hall, Inc.
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
© 2006 Prentice Hall, Inc.
Average range R = .25
Sample size n = 5
© 2006 Prentice Hall, Inc.
Average range R = .25
Sample size n = 5
Average range R = .25
Sample size n = 5
Difference between smallest and largest values in sample
Monitors process variability
where
D3 and D4 = control chart factors from Table S6.1
© 2006 Prentice Hall, Inc.
Sample size n = 5
UCL = 11.2
Mean = 5.3
LCL = 0
(Sampling mean is shifting upward but range is consistent)
R-chart
UCL
LCL
x-chart
UCL
LCL
UCL
LCL
(b)
x-chart
UCL
LCL
Good/bad, yes/no, acceptable/unacceptable
Charts may measure
Percent defective (p-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
z = number of standard deviations
sp = standard deviation of the sampling distribution
n = sample size
^
Instructors may wish to point out the calculation of the standard deviation reflects the binomial distribution of the population
© 2006 Prentice Hall, Inc.
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)
2 4 6 8 10 12 14 16 18 20
UCLp = p + zsp = .04 + 3(.02) = .10
^
^
2 4 6 8 10 12 14 16 18 20
UCLp = p + zsp = .04 + 3(.02) = .10
^
^
UCLp = 0.10
LCLp = 0.00
p = 0.04
There is always a focus on finding and eliminating problems. But control charts find any process changed, good or bad. The clever company will be looking at Operator 3 and 19 as they reported no errors during this period. The company should find out why (find the assignable cause) and see if there are skills or processes that can be applied to the other operators.
© 2006 Prentice Hall, Inc.
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
Instructors may wish to point out the calculation of the standard deviation reflects the Poisson distribution of the population where the standard deviation equals the square root of the mean
© 2006 Prentice Hall, Inc.
|
Figure S6.7
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
© 2006 Prentice Hall, Inc.
Patterns in Control Charts
One plot out above (or below). Investigate for cause. Process is “out of control.”
Figure S6.7
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
© 2006 Prentice Hall, Inc.
Patterns in Control Charts
Trends in either direction, 5 plots. Investigate for cause of progressive change.
Figure S6.7
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
© 2006 Prentice Hall, Inc.
Patterns in Control Charts
Two plots very near lower (or upper) control. Investigate for cause.
Figure S6.7
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
© 2006 Prentice Hall, Inc.
Patterns in Control Charts
Run of 5 above (or below) central line. Investigate for cause.
Figure S6.7
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
© 2006 Prentice Hall, Inc.
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
© 2006 Prentice Hall, Inc.
Variables Data
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
© 2006 Prentice Hall, Inc.
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
Natural or common causes
Special or assignable causes
Detect and eliminate assignable causes of variation
Points which might be emphasized include:
- Statistical process control measures the performance of a process, it does not help to identify a particular specimen produced as being “good” or “bad,” in or out of tolerance.
- Statistical process control requires the collection and analysis of data - therefore it is not helpful when total production consists of a small number of units
- While statistical process control can not help identify a “good” or “bad” unit, it can enable one to decide whether or not to accept an entire production lot. If a sample of a production lot contains more than a specified number of defective items, statistical process control can give us a basis for rejecting the entire lot. The issue of rejecting a lot which was actually good can be raised here, but is probably better left to later.
© 2006 Prentice Hall, Inc.
These are to be expected
Output measures follow a probability distribution
For any distribution there is a measure of central tendency and dispersion
© 2006 Prentice Hall, Inc.
Assignable Variations
Variations that can be traced to a specific reason (machine wear, misadjusted equipment, fatigued or untrained workers)
The objective is to discover when assignable causes are present and eliminate them
© 2006 Prentice Hall, Inc.
S6 – *
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
Figure S6.1
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
Each of these represents one sample of five boxes of cereal
© 2006 Prentice Hall, Inc.
S6 – *
Samples
(b) After enough samples are taken from a stable process, they form a pattern called a distribution
Figure S6.1
Frequency
Weight
S6 – *
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
Figure S6.1
S6 – *
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
Figure S6.1
S6 – *
Samples
(e) If assignable causes are present, the process output is not stable over time and is not predicable
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
Students should understand both the concepts of natural and assignable variation, and the nature of the efforts required to deal with them.
© 2006 Prentice Hall, Inc.
May be in whole or in fractional numbers
Continuous random variables
Classify products as either good or bad or count defects
Categorical or discrete random variables
Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process.
© 2006 Prentice Hall, Inc.
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
© 2006 Prentice Hall, Inc.
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
Hour 1
© 2006 Prentice Hall, Inc.
17 = UCL
15 = LCL
16 = Mean
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
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
© 2006 Prentice Hall, Inc.
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
© 2006 Prentice Hall, Inc.
Average range R = .25
Sample size n = 5
© 2006 Prentice Hall, Inc.
Average range R = .25
Sample size n = 5
Average range R = .25
Sample size n = 5
Difference between smallest and largest values in sample
Monitors process variability
where
D3 and D4 = control chart factors from Table S6.1
© 2006 Prentice Hall, Inc.
Sample size n = 5
UCL = 11.2
Mean = 5.3
LCL = 0
(Sampling mean is shifting upward but range is consistent)
R-chart
UCL
LCL
x-chart
UCL
LCL
UCL
LCL
(b)
x-chart
UCL
LCL
Good/bad, yes/no, acceptable/unacceptable
Charts may measure
Percent defective (p-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
z = number of standard deviations
sp = standard deviation of the sampling distribution
n = sample size
^
Instructors may wish to point out the calculation of the standard deviation reflects the binomial distribution of the population
© 2006 Prentice Hall, Inc.
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)
2 4 6 8 10 12 14 16 18 20
UCLp = p + zsp = .04 + 3(.02) = .10
^
^
2 4 6 8 10 12 14 16 18 20
UCLp = p + zsp = .04 + 3(.02) = .10
^
^
UCLp = 0.10
LCLp = 0.00
p = 0.04
There is always a focus on finding and eliminating problems. But control charts find any process changed, good or bad. The clever company will be looking at Operator 3 and 19 as they reported no errors during this period. The company should find out why (find the assignable cause) and see if there are skills or processes that can be applied to the other operators.
© 2006 Prentice Hall, Inc.
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
Instructors may wish to point out the calculation of the standard deviation reflects the Poisson distribution of the population where the standard deviation equals the square root of the mean
© 2006 Prentice Hall, Inc.
|
Figure S6.7
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
© 2006 Prentice Hall, Inc.
Patterns in Control Charts
One plot out above (or below). Investigate for cause. Process is “out of control.”
Figure S6.7
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
© 2006 Prentice Hall, Inc.
Patterns in Control Charts
Trends in either direction, 5 plots. Investigate for cause of progressive change.
Figure S6.7
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
© 2006 Prentice Hall, Inc.
Patterns in Control Charts
Two plots very near lower (or upper) control. Investigate for cause.
Figure S6.7
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
© 2006 Prentice Hall, Inc.
Patterns in Control Charts
Run of 5 above (or below) central line. Investigate for cause.
Figure S6.7
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
© 2006 Prentice Hall, Inc.
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
© 2006 Prentice Hall, Inc.
Variables Data
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
© 2006 Prentice Hall, Inc.
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