Chapter 9A Process Capability and Statistical Quality Control

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Chapter 9AProcess Capability and Statistical
Quality Control

• Process Variation
• Process Capability
• Process Control Procedures
• Variable data
• Attribute data
• Acceptance Sampling
• Operating Characteristic Curve

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Basic Causes of Variation

• Assignable causes are factors that can be clearly
  identified and possibly managed.
• Common causes are inherent to the production
  process. In order to reduce variation due to
  common causes, the process must be changed.
• Key Determining which is which!

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Types of Control Charts

• Attribute (Go or no-go information)
• Defectives refers to the acceptability of product
  across a range of characteristics.
• p-chart application
• Variable (Continuous)
• Usually measured by the mean and the standard
  deviation.
• X-bar and R chart applications

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Types of Statistical Quality Control
Statistical
Statistical
Quality Control
Quality Control
Process
Acceptance
Process
Acceptance
Control
Sampling
Control
Sampling
Variables
Attributes
Variables
Attributes
Variables
Attributes
Variables
Attributes
Charts
Charts
Charts
Charts
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Statistical Process Control (SPC) Charts
Look for trends!
Excellent review in exhibit TN8.5.
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Control Limits

• We establish the Upper Control Limits (UCL) and
  the Lower Control Limits (LCL) with plus or minus
  3 standard deviations. Based on this we can
  expect 99.7 of our sample observations to fall
  within these limits.

99.7
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Example of Constructing a p-Chart Required Data
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Statistical Process Control FormulasAttribute
Measurements (p-Chart)
Given
Compute control limits
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Example of Constructing a p-chart Step 1
1. Calculate the sample proportions, p (these
are what can be plotted on the p-chart) for each
sample.
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Example of Constructing a p-chart Steps 23
2. Calculate the average of the sample
proportions.
3. Calculate the standard deviation of the sample
proportion
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Example of Constructing a p-chart Step 4
4. Calculate the control limits.
UCL 0.0930 LCL -0.0197 (0)
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Example of Constructing a p-Chart Step 5
5. Plot the individual sample proportions, the
average of the proportions, and the control
limits
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R Chart

• Type of variables control chart
• Interval or ratio scaled numerical data
• Shows sample ranges over time
• Difference between smallest largest values in
  inspection sample
• Monitors variability in process
• Example Weigh samples of coffee compute ranges
  of samples Plot

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R Chart Control Limits
From Table (function of sample size)
Sample Range in sample i
Samples
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R Chart Example

• Youre manager of a 500-room hotel. You want to
  analyze the time it takes to deliver luggage to
  the room. For 7 days, you collect data on 5
  deliveries per day. Is the process in control?

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R Chart Hotel Data

• Sample
• Day Delivery Time Mean Range
• 1 7.30 4.20 6.10 3.45 5.55 5.32

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R Chart Hotel Data

• Sample
• Day Delivery Time Mean Range
• 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85

Largest
Smallest
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R Chart Hotel Data

• Sample
• Day Delivery Time Mean Range
• 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 2 4.60 8.7
  0 7.60 4.43 7.62 6.59 4.27 3 5.98 2.92 6.20 4.20
  5.10 4.88 3.28 4 7.20 5.10 5.19 6.80 4.21 5.70
  2.99 5 4.00 4.50 5.50 1.89 4.46 4.07 3.61 6 10
  .10 8.10 6.50 5.06 6.94 7.34 5.04 7 6.77 5.08 5.
  90 6.90 9.30 6.79 4.22

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R Chart Control Limits Solution
From Table (n 5)
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R Chart Control Chart Solution
UCL
R-bar
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?X Chart

• Type of variables control chart
• Interval or ratio scaled numerical data
• Shows sample means over time
• Monitors process average
• Example Weigh samples of coffee compute means
  of samples Plot

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?X Chart Control Limits
From Table
Mean of sample i
Range of sample i
Samples
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?X Chart Example

• Youre manager of a 500-room hotel. You want to
  analyze the time it takes to deliver luggage to
  the room. For 7 days, you collect data on 5
  deliveries per day. Is the process in control?

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X Chart Hotel Data

• Sample
• Day Delivery Time Mean Range
• 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 2 4.60 8.7
  0 7.60 4.43 7.62 6.59 4.27 3 5.98 2.92 6.20 4.20
  5.10 4.88 3.28 4 7.20 5.10 5.19 6.80 4.21 5.70
  2.99 5 4.00 4.50 5.50 1.89 4.46 4.07 3.61 6 10
  .10 8.10 6.50 5.06 6.94 7.34 5.04 7 6.77 5.08 5.
  90 6.90 9.30 6.79 4.22

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?X Chart Control Limits Solution
From Table (n 5)
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?X ChartControl Chart Solution
UCL
X-bar
LCL
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X AND R CHART EXAMPLEIN-CLASS EXERCISE

• The following collection of data represents
  samples of the amount of force applied in a
  gluing process
• Determine if the process is in control
• by calculating the appropriate upper and lower
• control limits of the X-bar and R charts.

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X AND R CHART EXAMPLEIN-CLASS EXERCISE
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Example of x-bar and R charts Step 1. Calculate
sample means, sample ranges, mean of means, and
mean of ranges.
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Example of x-bar and R charts Step 2. Determine
Control Limit Formulas and Necessary Tabled Values
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Example of x-bar and R charts Steps 34.
Calculate x-bar Chart and Plot Values
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Example of x-bar and R charts Steps 56
Calculate R-chart and Plot Values
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SOLUTIONExample of x-bar and R charts

• 1. Is the process in Control?
• 2. If not, what could be the cause for the
  process being out of control?

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Process Capability

• Process limits - actual capabilities of process
  based on historical data
• Tolerance limits - what process design calls
  for- desired performance of process

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Process Capability

• How do the limits relate to one another?
• You want tolerance range gt process range

Two methods of accomplishing this
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Process Capability Measurement
Cp index Tolerance range / Process range What
value(s) would you like for Cp? Þ Larger Cp
indicates a more reliable and predictable process
(less variability) The Cp index is based on the
assumption that the process mean is centered at
the midpoint of the tolerance range
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UTL
LTL
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• While the Cp index provides useful information on
  process variability, it does not give information
  on the process average relative to the tolerance
  limits. Note

UTL
LTL
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Cpk Index
s standard deviation (Unknown but can be
estimated with the average range)
Together, these process capability Indices show
how well parts being produced conform to design
specifications.
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Since Cp and Cpk are different we can conclude
that the process is not centered, however the Cp
index tells us that the process variability is
very low
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An example of the use of process capability
indices
The design specifications for a machined slot is
0.5 .003 inches. Samples have been taken and
the process mean is estimated to be .501. The
process standard deviation is estimated to be
.001. What can you say about the capability of
this process to produce this dimension?
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Process capability
Machined slot (inches)
0.497 inches LTL
0.503 inches UTL
? 0.001 inches
Process mean 0.501 inches
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Sampling Distributions(The Central Limit Theorem)

• Regardless of the underlying distribution, if the
  sample is large enough (gt30), the distribution of
  sample means will be normally distributed around
  the population mean with a standard deviation of
  

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Computing Process Capability Indexes Using
Control Chart Data

• Recall the following info from our in class
  exercise
• Since A2 is calculated on the assumption of three
  sigma limits

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• From the Central Limit Theorem
• So,
• Therefore,

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• Suppose the Design Specs for the Gluing Process
  were 10.7 ? .2, Calculate the Cp and Cpk
  Indexes
• Answer

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Note, multiplying each component of the Cpk
calculation by 3 yields a Z value. You can use
this to predict the of items outside the
tolerance limits From Appendix E we would
expect .008 .036 .044 or 4.4
non-conforming product from this process
.792 3 2.38
.597 3 1.79
.008 or .8 of the curve
.036 or 3.6 of the curve
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Capability Index In Class Exercise

• You are a manufacturer of equipment. A drive
  shaft is purchased from a supplier close by. The
  blueprint for the shaft specs indicate a
  tolerance of 5.5 inches .003 inches. Your
  supplier is reporting a mean of 5.501 inches.
  And a standard deviation of .0015 inches.
• What is the Cpk index for the suppliers process?

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• Your engineering department is sent to the
  suppliers site to help improve the capability on
  the shaft machining process. The result is that
  the process is now centered and the CP index is
  now .75. On a percentage basis, what is the
  improvement on the percentage of shafts which
  will be unusable (outside the tolerance limits)?

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To answer this question we must determine the
percentage of defective shafts before and after
the intervention from our engineering department
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Before
3x.444 1.33s
(3x.88) 2.67s
Total outside Tolerance .089 .004 .093 or
9.3
-4
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After
Since the process is centered then Cpk Cp Cp
UTL-LTL / 6s, so the tolerance limits are .75
x 6s 4.5s apart each 2.25s from the mean
2.25s
2.25s
So outside of Tolerance .012(2) .024 Or
2.4
-4
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So the percentage decrease in defective parts is
1 (2.4/9.3) 74
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Basic Forms of Statistical Sampling for Quality
Control

• Sampling to accept or reject the immediate lot of
  product at hand (Acceptance Sampling).
• Sampling to determine if the process is within
  acceptable limits (Statistical Process Control)

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Acceptance Sampling

• Purposes
• Determine quality level
• Ensure quality is within predetermined level
• Advantages
• Economy
• Less handling damage
• Fewer inspectors
• Upgrading of the inspection job
• Applicability to destructive testing
• Entire lot rejection (motivation for improvement)

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Acceptance Sampling

• Disadvantages
• Risks of accepting bad lots and rejecting
  good lots
• Added planning and documentation
• Sample provides less information than 100-percent
  inspection
• No information is obtained on the process. Just
  sorting good parts from bad parts

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Risk

• Acceptable Quality Level (AQL)
• Max. acceptable percentage of defectives defined
  by producer.
• a (Producers risk)
• The probability of rejecting a good lot.
• Lot Tolerance Percent Defective (LTPD)
• Percentage of defectives that defines consumers
  rejection point.
• ? (Consumers risk)
• The probability of accepting a bad lot.