MBA8452

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MBA 8452 Systems and Operations Management
Statistical Quality Control
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Objective Quality Analysis

• Process Variation
• Be able to explain Taguchis View of the cost of
  variation.
• Statistical Process Control Charts and Process
  Capability
• Be able to develop and interpret SPC charts.
• Be able to calculate and interpret Cp and Cpk
• Be able to explain the difference between process
  control and process capability
• Sample Size
• Be able to explain the importance of sample size

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Statistical Quality Control Approaches

• Statistical Process Control (SPC)
• Sampling to determine if the process is within
  acceptable limits (under control)
• Acceptance Sampling
• Inspects a random sample of a product
• to determine if the lot is acceptable

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Line graph shows plot of dataand variation from
the average
Process target or average
Sample number
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Why Statistical Quality Control?

• Variations in Manufacturing/Service Processes
• Any process has some variations common and/or
  special
• Variations are causes for quality problems
• If a process is stable (no special variation), it
  is able to produce product/service consistently
• As variation is reduced, quality is improved
• Statistics is the only science that is dedicated
  to dealing with variations.
• Measure, monitor, and reduce variations in the
  process

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Types of Variation

• Exogenous to process
• Not random
• Controllable
• Preventable
• Examples
• tool wear
• human factors (fatigue)
• poor maintenance

• Inherent to process
• Random
• Cannot be controlled
• Cannot be prevented
• Examples
• weather
• accuracy of measurements
• capability of machine

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Cost of Variation Traditional vs. Taguchis
View
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Statistical Process Control

• On-line quality control tool used when the
  product/service is being produced
• Purpose prevent systematic quality problems
• Procedure
• Take periodic random samples from a process
• Plot the sample statistics on control chart(s)
• Determine if the process is under control
• If the process is under control, do nothing
• If the process is out of control, investigate and
  fix the cause

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Statistical Process Control Types Of Data

• Attribute data (discrete values)
• Quality characteristic evaluated about whether it
  meets the required specifications
• Good/bad, yes/no
• Variable data (continuous values)
• Quality characteristic that can be measured
• Length, size, weight, height, time, velocity

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Statistical Process Control Control Charts

• Charts for attributes
• p-chart (for proportions)
• c-chart (for counts)
• Charts for variables
• R-chart (for ranges)
• -chart (for means)

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Control Chart General Structure
Upper control limit (UCL)
Process target or average
Lower control limit (LCL)
Sample number
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A Process Is In Control If ...

• No sample points outside control limits
• Most points near the process average
• About an equal points above below the
  centerline
• Points appear randomly distributed

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Common Out-of-control Signs
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Issues In Building Control Charts

• Number of samples around 25
• Size of each sample large (?100) for attributes
  and small (?25) for variables
• Frequency of sampling depends
• Control limits typically 3-sigma away from the
  process mean

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Control Limits The Normal Distribution
X
If we establish control limits at /- 3 standard
deviations (s), then we would expect 99.74 of
observations (X) to fall within these limits.
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Control Limits General Formulas

• UCL mean z (stand dev)
• LCL mean z (stand dev)
• z is the of standard deviations
• z 3.00 is the most commonly used value with
  99.7 confidence level
• Other z values can be used (e.g. z2 for 95
  confidence and z2.58 for 99 confidence)

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Control Charts for Attributes p-charts
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p-Chart Example

• 20 Samples of 100 pairs of jeans each were
  randomly selected from the Western Jean Companys
  production line.

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p-Chart Example
0.2
UCL
0.18
0.16
0.14
0.12
p
0.1
Proportion defective
0.08
0.06
0.04
0.02
LCL
0
0
2
4
6
8
10
12
14
16
18
20
.
.
Sample number
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Control Charts For VariablesX-bar charts and
R-charts
Where X average of sample means ?Xi / m R
average of sample ranges ?Ri / m Xi mean of
sample i, i 1,2,,m Ri range of sample i, i
1,2,,m m total number of samples A2, D3,
and D4 are constants from Exhibit TN7.7
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Example

• If a company makes jeans, there are a
  specifications that must be met.
• The back pockets of the jeans cant be too small
  or too large.
• The control chart can be established to monitor
  the measurements of the back pocket
• Given 15 samples with 5 observations each, we can
  determine the Upper and Lower control limits for
  the range and x-bar charts.

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X-bar and R Charts Example
(Ri)
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X-bar and R Charts Example
Exhibit TN7.7
Since n5, from Exhibit TN7.7 (also right table),
we find A20.58 D30 D42.11
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X-bar and R Charts Example
R chart
R
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X-bar and R ChartsExample
X-bar chart
X
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Process Capability

• The ability of a process to meet product
  design/technical specifications
• Design specifications for products (Tolerances)
• upper and lower specification limits (USL, LSL)
• Process variability in production process
• natural variation in process (?3? from the mean)
• Process may not be capable of meeting
  specifications if natural variation in a process
  exceeds allowable variation (tolerances)

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Process Capability Illustrations
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Process CapabilityFurther Illustrations
Target
Target
LSL
USL
LSL
USL
Process variation
Tolerance variation
Highly capable process

• Capable process

Process not capable
Process not capable
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Specification Limits ? Control Limits

• Specification limits are pre-established for
  products before production
• Control limits are used to monitor the actual
  production process performance
• It is possible that a process is under control,
  but not capable to meet specifications
• It is also possible that a process that is within
  specifications is out-of-control

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Control Limits Vs. Specification
LimitsIllustrations
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Process Capability IndexCp -- Measure of
Potential Capability
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Process Capability IndexCpk -- Measure of
Actual Capability
Cpk considers both process variation (?) and
process location (X)
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Process Capability IndexExample

• A manufacturing process produces a certain part
  with a mean diameter of 2 inches and a standard
  deviation of 0.03 inches. The lower and upper
  engineering specification limits are 1.90 inches
  and 2.05 inches.

Therefore, the process is not capable (the
variation is too much and the process mean is not
on the target)
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Impact of Process Location on Process Capability
Cp 2.0 Cpk 2.0
Cp 2.0 Cpk 1.5
Cp 2.0 Cpk 1.0
Cp 2.0 Cpk 0
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Acceptance Sampling

• Determines whether to accept or reject an entire
  lot of goods based on sample results
• Measures quality in percent defective
• Usually applied to incoming raw materials or
  outgoing finished goods

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Sampling Plan

• Guidelines for accepting or rejecting a lot
• Single sampling plan
• N lot size
• n sample size
• c max acceptance number of defects
• d number of defective items in sample
• If d lt c, accept lot else reject
• Sampling plan is developed based on the tradeoff
  between producers risk and consumers risk

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Producers Consumers Risk

• Producers Risks
• reject a good lot (TYPE I ERROR)
• a producers risk P(reject good lot)
• 5 is common
• Consumers Risks
• accept a bad lot (TYPE II ERROR)
• b consumers risk P(accept bad lot)
• 10 is typical

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Quality Definitions

• Acceptable quality level (AQL)
• Acceptable proportion of defects on average
• good lot the proportion of defects of the lot
  is less than or equal to AQL
• Lot tolerance percent defective (LTPD)
• Maximum proportion of defects in a lot
• bad lot the proportion of defects of the lot
  is greater than LTPD

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Operating Characteristic Curve
Percent defective in a lot