Production and Operations Management: Manufacturing and Services

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CHASE AQUILANO JACOBS
Operations Management
For Competitive Advantage
Technical Note 7
Process Capability Statistical Quality Control
tenth edition
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Technical Note 7Process Capability and
Statistical Quality Control

• Quality Defined
• Process Variation
• Process Capability
• Process Control Procedures
• Variable data
• Attribute data
• Acceptance Sampling
• Operating Characteristic Curve
• Standard table of sampling plans

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Quality Defined

• Definition of quality
• Durability, Reliability, Long warrantee
• Fitness for use, Degree of conformance
• Maintainability
• Measures of quality
• Grade--measurable characteristics, finish,
• Consistency--Good or bad, predictability
• Conformance--Degree product meets specifications
• Consistency versus conformance

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Two Basic Forms of Variation

• Assignable (special) variation is caused by
  factors that can be clearly identified and
  possibly managed.
• Common (chance or random) variation is inherent
  in the production process.

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Taguchis View of Variation
Exhibits TN7.1 TN7.2
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Process Capability

• Process (control) limits
• Calculated from data gathered from the process
• It is natural tolerance limits
• Defined by /- 3 standard deviation
• Used to determine if process is in statistical
  control
• Tolerance (specification) limits
• Often determined externally, e.g., by customer
• Process may be in control but not within
  specification
• How do the limits relate to one another?

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Process Capability (Cp(USL-LSL)/6?)

• Case 1 Cp gt 1
• USL-LSL gt 6 sigma
• Situation desired
• Process remains in control
• Defacto standard is 1.33

LSL
USL
LNTL
UNTL
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Process Capability (Cp(USL-LSL)/6?)

• Case 2 Cp 1
• USL-LSL 6 sigma
• Approximately 0.27 defective will be made
• Process is unstable

LSL
USL
LNTL
UNTL
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Process Capability (Cp(USL-LSL)/6?)

• Case 1 Cp lt 1
• USL-LSL lt 6 sigma
• Situation is undesirable
• Process is yield sensitive
• Could produce large number of defectives

LNTL
UNTL
USL
LSL
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Process Capability Index, Cpk

• Most widely used capability measure
• Measures design versus specification relative to
  the nominal value
• Based on worst case situation
• Defacto value is 1 and processes with this score
  is capable
• Scores gt 1 indicates 6-sigma subsumed by the
  specification limits
• Scores less than 1 will result in an incapable
  process

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Process Capability Index, Cpk
Capability Index shows how well parts being
produced fit into design limit specifications.
As a production process produces items small
shifts in equipment or systems can cause
differences in production performance from
differing samples.
Shifts in Process Mean
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Types of Statistical Sampling

• Attribute (Go or no-go information)
• Defectives refers to the acceptability of product
  across a range of characteristics.
• Defects refers to the number of defects per unit
  which may be higher than the number of
  defectives.
• p-chart application
• Variable (Continuous)
• Plots specific measurements of a process (e.g.,
  weight)
• Usually measured by the mean and the standard
  deviation.
• X-bar and R chart applications

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UCL
Statistical Process Control (SPC) Charts
Normal Behavior
LCL
Samples over time
1 2 3 4 5
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UCL
Possible problem, investigate
LCL
Samples over time
1 2 3 4 5
6
UCL
Possible problem, investigate
LCL
Samples over time
1 2 3 4 5
6
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Control Limits are based on the Normal Curve
x
m
z
0
1
2
3
-3
-2
-1
Standard deviation units or z units.
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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)
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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 0.0924 LCL -0.0204 (or 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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Example of x-Bar and R Charts Required Data
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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
From Exhibit TN7.7
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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
UCL
LCL
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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).
• Does not necessarily determine quality level
• Results subject to sampling error
• Sampling to determine if the process is within
  acceptable limits (Statistical Process Control)
• Takes steps to increase quality

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Acceptance Sampling

• Purposes
• Make decision about (sentence) a product
• 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

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Acceptance Sampling Single Sampling Plan

• A simple goal
• Determine
• how many units, n, to sample from a lot, and
• the maximum number of defective items, c, that
  can be found in the sample before the lot is
  rejected.

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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.

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Operating Characteristic Curve

• Shows how discriminating a plan is
• Probability of accepting various quality levels
• A perfect discriminator will
• Involve 100 inspection
• High costs
• OC curves are unique for each sampling plan
• They are modeled using binomial distribution

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Operating Characteristic Curve
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Example Acceptance Sampling Problem
Zypercom, a manufacturer of video interfaces,
purchases printed wiring boards from an outside
vender, Procard. Procard has set an acceptable
quality level of 1 and accepts a 5 risk of
rejecting lots at or below this level. Zypercom
considers lots with 3 defectives to be
unacceptable and will assume a 10 risk of
accepting a defective lot. Develop a sampling
plan for Zypercom and determine a rule to be
followed by the receiving inspection personnel.
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Example Step 1. What is given and what is not?
In this problem, AQL is given to be 0.01 and LTDP
is given to be 0.03. We are also given an alpha
of 0.05 and a beta of 0.10.
What you need to determine your sampling plan is
c and n.
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Example Step 2. Determine c
First divide LTPD by AQL.
Then find the value for c by selecting the
value in the TN7.10 n(AQL)column that is equal
to or just greater than the ratio above.
So, c 6.
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Example Step 3. Determine Sample Size

Now given the information below, compute the
sample size in units to generate your sampling
plan.
c 6, from Table n (AQL) 3.286, from Table AQL
.01, given in problem
n(AQL/AQL) 3.286/.01 328.6, or 329 (always
round up)
Sampling Plan Take a random sample of 329 units
from a lot. Reject the lot if more than 6 units
are defective.
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Standard Table of Sampling Plans

• MIL-STD-105D
• For attributes sampling plans
• Needs to know
• The lot size, N
• The inspection level (I, II, III)
• The AQL
• Type of sampling (single, double, multiple)
• Type of inspection (normal, tightened, reduced)
• Find a code letter then read plan from Table

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Standard Table of Sampling PlansSingle Sampling
Plan

• Example If N2000 and AQL.65, find the normal,
  tightened, and reduced single sampling plan using
  inspection level II.

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Standard Table of Sampling PlansDouble Sampling
Plan

• Example If N20,000 and AQL1.5, find the
  normal, tightened, and reduced double sampling
  plan using inspection level I.