Control Charts

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Control Charts

• Chapter 5

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Introduction

• In industrial processes, goals are to
• - .
• We need to ___________ an engineering process to
  ensure these goals are met.
• Examples
• - monitor process for filling 16 oz cans
• of pepsi
• - maintain correct outside diameter of
• ball point pens

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Example

• A 10-oz packaging line for a popular cereal has
  a target net weight of 10.5oz. Historically,
  process standard deviation is 0.1oz. The company
  monitors net weights by a random sample of 5
  boxes of cereal each hour. The mean net weights
  for 30 such subgroups is

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Data

• 10.56, 10.56, 10.46,10.46, 10.49, 10.52,
• 10.53,10.47, 10.44, 10.54, 10.49, 10.52,
• 10.53,10.50, 10.50, 10.48, 10.55, 10.68,
• 10.67,10.62, 10.45, 10.47, 10.40, 10.43,
• 10.47,10.50, 10.38, 10.43, 10.50, 10.51

• The target net mean weight is µ010.5oz.

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• Let yi the weight of the ith box of cereal
  (i1,,5) at hour t. (t1,,30).
• Let µ(t) be the process mean (the true mean
  weight at this time).
• A model we can use is
• ei is the random error associated with this box
  of cereal.
• We assume here that the process variance s2 is
  constant.

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Factors that may influence process

• .
• .
• .
• .
• These factors change over time, which
• will influence the true mean weight of
• the cereal boxes.

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Typical Drift in a Process Mean over time

• µ(t)

µ0
t
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Dealing with Process Changes

• Do
• Inspect ________ item and remove the unacceptable
  items. Very expensive, not always 100
  effective.
• Produce batches of products, use statistics to
  determine if entire batch unacceptable.
• Constantly monitor true mean weight, and adjust
  process whenever it begins to drift. Can use
  control charts here.
• Create a process that cannot produce an
  unacceptable product. What we strive for.

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Control Charts

• Are used to
• A random sample (a subgroup) is taken from
  process on a
• As soon as data collected

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A control chart for a process mean

• Is equivalent to carrying out a hypothesis test
  at each time point. In cereal example
• H0
• H1
• If we reject the null hypothesis at a specified
  level of significance (e.g. 5), we conclude the
  true mean weight of cereal boxes has moved away
  from the target value of 10.5 oz at that time
  point.

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Control Limits of an . Chart

• Upper Control Limit UCL
• Lower Control Limit LCL

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Cereal Example X-bar Chart
Out of control
UCL10.63
In control
µ0 10.5
In control
LCL10.37
Out of control
time
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.. Chart of Cereal Box Weights
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Conclusions for Cereal Example

• The control chart suggests the process is
• After this it is within control limits, but
  appears to be a downward trend.
• Process

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The np control chart

• Instead of monitoring the mean of the process,
  for example, average weight of cereal packages,
  we may be interested in the
• Here, we also use a control chart, but a slightly
  different one.

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Brick Example

• A study looked at the number of non-conforming
  bricks from a manufacturing process. It is known
  from historic data that the target proportion of
  non-conforming bricks is 0.06625. Process
  monitored by taking a random sample of 200 bricks
  every day for 16 days.
• Each brick was classified as either
• - conforming
• - non-conforming.

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• Let p0 be the target proportion of nonconforming
  bricks.
• Let yi be the number of nonconforming bricks on
  day i.
• Expected value Eyinp0
• Variance Varyinp0(1-p0).
• As process stays in control, with same sample
  size each day, the expected value and variance
  stay constant.

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An np control chart

• Examines the
• Is equivalent to carrying out a hypothesis test
  at each time point.
• n is the number of items collected

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In Brick Example

• Our true hypotheses are
• H0
• H1
• Our control limits take the form

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In Brick Example

• The control limits are

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Brick Example

• Check data and Figure 5.22 on page 330 in text.

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Class Exercise

• Airplanes approaching the runway for landing are
  required to stay within a certain distance (left
  and right) of the runway. Deviations are called
  exceedences. Each day, one airline randomly
  selects 200 flights and records the number in
  exceedence. The following data presents the
  number of flights in exceedence on each of 30
  days in a row.

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• 16 25 23 16 15 23 24 15 23 13
• 21 19 23 28 30 19 18 15 18 10
• 24 22 20 27 21 26 30 25 26 30
• It is known historically that the
• exceedence rate is 0.0985 for this
• airline. Calculate the appropriate control
• limits and plot the control chart.

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• Here n200

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