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CHAPTER 10: QUALITY CONTROLSuman Niranjan
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STATISTICAL PROCESS CONTROL
Quality conformance of a process Does the output of a process conform to the intent of
the design Determine if it is statistically acceptable
Take periodic samples and compare with a standard Use control charts
It is a time ordered plot of samples obtained from a process
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THE STEPS IN CONTROL PROCESS
Effective control requires Define
What is to be controlled? Measure
Counted or measured? Compare
Any standard available? Evaluate
What is out of control? Correct
Corrective action Monitor results
Results analyzed
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VARIATIONS AND CONTROL There are primarily two types of variations
Random variations Assignable variations
Random Variations Natural variation, usually occurred due to combination of many
minor factors Fixing all these little things might be very expensive Also known as common variability
Assignable Variation A variation whose cause can be identified and tied to specific
cause Also known as specific variability
Samples of a process are taken and checked for non-random variability Sample statistics The goal is to determine if a correctable source of variation is
present
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VARIATIONS AND CONTROL
Process distribution Is a collection of data points One data point refers to one individual observation
Count Measurement
Normally distributed due to the central limit theorem Sampling distribution
Is a collection of data points, where each data point is a mean of several data points
Mean of several data points Mean of several counts Mean of several Measurements
Normally distributed due to the central limit theorem
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VARIATIONS AND CONTROL
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CONTROL CHARTS It is a time ordered plot of all the samples used to
distinguish between random and nonrandom variability Upper control limit Lower control limit
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CONTROL CHARTS Type I error
Concluding a process is in not in control when it actually is in control
Producers risk? Type II error
Concluding a process is in control when it is out of control Buyers risk?
Sample number
UCL
LCL1 2 3 4
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TYPE I AND TYPE II ERRORS
OKType I
(Producers Risk)
Type II(Buyers Risk)
OK
And the Conclusion Process is
In Control Out of Control
Out of Control
In Control
If a
Pro
cess is A
ctu
ally
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TYPES OF CONTROL CHARTS: OVERVIEW
Control Charts for Variables: for continuous measures such as temperature, volume, etc. Mean Chart (or -Chart): detects shift of the
mean Range Chart (or R-Chart): detects change in
shape of distribution Control Charts for Attributes: for discrete
measures such as number of complaints, scratches, etc. p-Chart: measures percent defective c-Chart: measures # of defects per sample
X
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FOR VARIABLES: MEAN CONTROL CHART
Approach 1: When the process mean ( ) and standard deviation ( ) are known:
The standard deviation of sample means (X-bar):
nx
:means sample ofdeviation Standard
x
x
zxLCL
zxUCL
:Limit ControlLower
:Limit ControlUpper
x = Standard deviation of distribution of sample meansn = Sample size
= Process Standard deviation
x = Average of sample means
x
Z = Standard normal deviate
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FOR VARIABLES: MEAN CONTROL CHART
Approach 2: Uses the sample range as a measure of process variability, when mean and standard deviation are unknown The appropriate formulas for mean control limits are:
RAxLCL
RAxUCL
2
2
:Limit ControlLower
:Limit ControlUpper
= A factor from three sigma control limits table 10.3 2A
R = Average of sample ranges
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RANGE CONTROL CHART
Range control chart is used to monitor process dispersion
RDLCL
RDUCL
R
3
4
:chart )( Range
The appropriate formulas for range control limits are:
= A factor from three sigma control limits table 10.3 34 ,DD
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EXAMPLE 1 A quality inspector took five samples, each with four observations,
of the length of time for glue to dry. The analyst computed the mean of each sample and then computed the grand mean. All values are in minutes. Use this information to obtain three-sigma (i.e., z = 3) control limits for means of future times. It is known from previous experience that the standard deviation of the process is .02 minute.
What is UCL, LCL?
Answer: UCL =12.14, LCL =12.08
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TABLE 10.3
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EXAMPLE 2
Twenty samples of n = 8 have been taken from a cleaning operation. The average sample range for the 20 samples was .016 minute, and the average mean was 3 minutes. Determine three-sigma control limits for this process.
Answer: UCL = 3.006 min LCL = 2.994 min
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EXAMPLE 3
Twenty-five samples of n = 10 observations have been taken from a milling process. The average sample range was .01 centimeter. Determine upper and lower control limits for sample ranges.
Answer: UCLr = 0.0178 LCLr = 0.0022
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SOLVED EXAMPLE 2 Control charts for means and ranges. Processing new
accounts at a bank is intended to average 10 minutes each. Five samples of four observations each have been taken. Use the sample data in conjunction with Table 10.2 to construct upper and lower control limits for both a mean chart and a range chart. Do the results suggest that the process is in control?
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MEAN AND RANGE CHARTS
UCL
LCL
UCL
LCL
R-chart
x-Chart Detects shift
Does notdetect shift
(process mean is shifting upward)
SamplingDistribution
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x-Chart
UCL
Does notreveal increase
MEAN AND RANGE CHARTS
UCL
LCL
LCL
R-chart Reveals increase
(process variability is increasing)SamplingDistribution
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CONTROL CHART FOR ATTRIBUTES
p-Chart - Control chart used to monitor the proportion of defectives in a process
c-Chart - Control chart used to monitor the number of defects per unit
Attributes generate data that are counted.
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USE OF P-CHARTS
When observations can be placed into two categories. Good or bad Pass or fail Operate or don’t operate
When the data consists of multiple samples of several observations each (e.g. 15 samples of n = 20 observations each )
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USE OF P-CHARTS
p
p
p
zpLCL
zpUCLn
ppp
p
)1( : ofdeviation Standard
nsobservatio # total
defectives of # total :defectivePercent
Construct p-Charts:
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EXAMPLE 4 An inspector counted the number of defective monthly
billing statements of a company telephone in each of 20 samples. Using the following information, construct a control chart that will describe 99.74 percent of the chance variation in the process when the process is in control. Each sample contained 100 statements.
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PLOTTING THE CONTROL LIMITS
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USE OF C-CHARTS
Use only when the number of occurrences per unit of measure can be counted; non-occurrences cannot be counted. Scratches, chips, dents, or errors per item Cracks or faults per unit of distance Breaks or Tears per unit of area Bacteria or pollutants per unit of volume Calls, complaints, failures per unit of time
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USE OF C-CHARTS
c
c
c
zcLCL
zcUCL
cc
c
: of deviation Standard
samples of #
defects of # total :sampleper defects of No.
Construct c-Charts:
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EXAMPLE 5 Rolls of coiled wire are monitored using a c-chart.
Eighteen rolls have been examined, and the number of defects per roll has been recorded in the following table. Is the process in control? Plot the values on a control chart using three standard deviation control limits.
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PLOT OF EXAMPLE 5
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MANAGERIAL CONSIDERATIONS FOR CONTROL CHARTS
At what point in the process to use control charts
What size samples to take What type of control chart to use
Variables Attributes
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NONRANDOM PATTERNS IN CONTROL CHARTS
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PROCESS CAPABILITY
LowerSpecification
UpperSpecification
Process variability matches specifications
LowerSpecification
UpperSpecification
Process variability well within specifications
LowerSpecification
UpperSpecification
Process variability exceeds specifications
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Processmean
Lowerspecification
Upperspecification
1350 ppm 1350 ppm
1.7 ppm 1.7 ppm
+/- 3 Sigma
+/- 6 Sigma
3 Sigma and 6 Sigma Quality
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TOPICS NOT IN SYLLABUS
Problems from Process Variability Supplement chapter 10
Acceptance sampling