Decision%20Analysis

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Slides Prepared by JOHN S. LOUCKS St. Edwards
University
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Statistical Methods for Quality Control

• Statistical Process Control
• Acceptance Sampling

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Quality Terminology

• Quality is the totality of features and
  characteristics of a product or service that
  bears on its ability to satisfy given needs.

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Quality Terminology

• Quality assurance refers to the entire system of
  policies, procedures, and guidelines established
  by an organization to achieve and maintain
  quality.
• The objective of quality engineering is to
  include quality in the design of products and
  processes and to identify potential quality
  problems prior to production.
• Quality control consists of making a series of
  inspections and measurements to determine whether
  quality standards are being met.

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Statistical Process Control (SPC)

• The goal of SPC is to determine whether the
  process can be continued or whether it should be
  adjusted to achieve a desired quality level.
• If the variation in the quality of the production
  output is due to assignable causes (operator
  error, worn-out tooling, bad raw material, . . .
  ) the process should be adjusted or corrected as
  soon as possible.
• If the variation in output is due to common
  causes (variation in materials, humidity,
  temperature, . . . ) which the manager cannot
  control, the process does not need to be adjusted.

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SPC Hypotheses

• SPC procedures are based on hypothesis-testing
  methodology.
• The null hypothesis H0 is formulated in terms of
  the production process being in control.
• The alternative hypothesis Ha is formulated in
  terms of the process being out of control.
• As with other hypothesis-testing procedures, both
  a Type I error (adjusting an in-control process)
  and a Type II error (allowing an out-of-control
  process to continue) are possible.

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Decisions and State of the Process

• Type I and Type II Errors
• State of Production Process
• Decision

H0 True In Control
Ha True Out of Control
Correct Decision
Type II Error Allow out-of-control process to
continue
Continue Process
Correct Decision
Type I Error Adjust in-control process
Adjust Process
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Control Charts

• SPC uses graphical displays known as control
  charts to monitor a production process.
• Control charts provide a basis for deciding
  whether the variation in the output is due to
  common causes (in control) or assignable causes
  (out of control).

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Control Charts

• Two important lines on a control chart are the
  upper control limit (UCL) and lower control limit
  (LCL).
• These lines are chosen so that when the process
  is in control, there will be a high probability
  that the sample finding will be between the two
  lines.
• Values outside of the control limits provide
  strong evidence that the process is out of
  control.

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Types of Control Charts

• An x chart is used if the quality of the output
  is measured in terms of a variable such as
  length, weight, temperature, and so on.
• x represents the mean value found in a sample of
  the output.
• An R chart is used to monitor the range of the
  measurements in the sample.
• A p chart is used to monitor the proportion
  defective in the sample.
• An np chart is used to monitor the number of
  defective items in the sample.

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x Chart Structure
x
UCL
Center Line
Process Mean When in Control
LCL
Time
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Control Limits for an x Chart

• Process Mean and Standard Deviation Known

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Example Granite Rock Co.

• Control Limits for an x Chart Process Mean
• and Standard Deviation Known
• The weight of bags of cement filled by
  Granites packaging process is normally
  distributed with a mean of 50 pounds and a
  standard deviation of 1.5 pounds.
• What should be the control limits for samples
  of 9 bags?

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Example Granite Rock Co.

• Control Limits for an x Chart Process Mean
• and Standard Deviation Known
• ????? 50, ? 1.5, n 9
• UCL 50 3(.5) 51.5
• LCL 50 - 3(.5) 48.5

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Control Limits for an x Chart

• Process Mean and Standard Deviation Unknown
• where
• x overall sample mean
• R average range
• A2 a constant that depends on n taken
  from
• Factors for Control Charts table

_
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Factors for x and R Control Charts

• Factors Table (Partial)

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Control Limits for an R Chart

• UCL RD4
• LCL RD3
• where
• R average range
• D3, D4 constants that depend on n
  found in Factors for Control
  Charts table

_
_
_
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Factors for x and R Control Charts

• Factors Table (Partial)

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Example Granite Rock Co.

• Control Limits for x and R Charts Process Mean
• and Standard Deviation Unknown
• Suppose Granite does not know the true mean and
  standard deviation for its bag filling process.
  It wants to develop x and R charts based on
  twenty samples of 5 bags each.
• The twenty samples resulted in an overall
  sample mean of 50.01 pounds and an average range
  of .322 pounds.

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Example Granite Rock Co.

• Control Limits for R Chart Process Mean
• and Standard Deviation Unknown
• x 50.01, R .322, n 5
• UCL RD4 .322(2.114) .681
• LCL RD3 .322(0) 0

_

_
_
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Example Granite Rock Co.

• R Chart

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Example Granite Rock Co.

• Control Limits for x Chart Process Mean
• and Standard Deviation Unknown
• x 50.01, R .322, n 5
• UCL x A2R 50.01 .577(.322) 50.196
• LCL x - A2R 50.01 - .577(.322) 49.824

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Example Granite Rock Co.

• x Chart

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Control Limits for a p Chart

• where
• assuming
• np gt 5
• n(1-p) gt 5
• Note If computed LCL is negative, set LCL
  0

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Example Norwest Bank

• Every check cashed or deposited at Norwest Bank
  must be encoded with the amount of the check
  before it can begin the Federal Reserve clearing
  process. The accuracy of the check encoding
  process is of utmost importance. If there is any
  discrepancy between the amount a check is made
  out for and the encoded amount, the check is
  defective.

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Example Norwest Bank

• Twenty samples, each consisting of 250 checks,
  were selected and examined when the encoding
  process was known to be operating correctly. The
  number of defective checks found in the samples
  follow.

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Example Norwest Bank

• Control Limits for a p Chart
• Suppose Norwest does not know the proportion of
  defective checks, p, for the encoding process
  when it is in control.
• We will treat the data (20 samples) collected
  as one large sample and compute the average
  number of defective checks for all the data.
  That value can then be used to estimate p.

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Example Norwest Bank

• Control Limits for a p Chart
• Estimated p 80/((20)(250)) 80/5000 .016

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Example Norwest Bank

• p Chart

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Control Limits for an np Chart

• assuming
• np gt 5
• n(1-p) gt 5
• Note If computed LCL is negative, set LCL 0

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Interpretation of Control Charts

• The location and pattern of points in a control
  chart enable us to determine, with a small
  probability of error, whether a process is in
  statistical control.
• A primary indication that a process may be out of
  control is a data point outside the control
  limits.
• Certain patterns of points within the control
  limits can be warning signals of quality
  problems
• Large number of points on one side of center
  line.
• Six or seven points in a row that indicate either
  an increasing or decreasing trend.
• . . . and other patterns.

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Acceptance Sampling

• Acceptance sampling is a statistical method that
  enables us to base the accept-reject decision on
  the inspection of a sample of items from the lot.
• Acceptance sampling has advantages over 100
  inspection including less expensive, less
  product damage, fewer people involved, . . . and
  more.

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Acceptance Sampling Procedure
Lot received
Sample selected
Sampled items inspected for quality
Results compared with specified quality
characteristics
Quality is not satisfactory
Quality is satisfactory
Accept the lot
Reject the lot
Send to production or customer
Decide on disposition of the lot
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Acceptance Sampling

• Acceptance sampling is based on
  hypothesis-testing methodology.
• The hypothesis are
• H0 Good-quality lot
• Ha Poor-quality lot

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The Outcomes of Acceptance Sampling

• Type I and Type II Errors
• State of the Lot
• Decision

H0 True Good-Quality Lot
Ha True Poor-Quality Lot
Correct Decision
Type II Error Consumers Risk
Accept H0 Accept the Lot
Correct Decision
Type I Error Producers Risk
Reject H0 Reject the Lot
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Probability of Accepting a Lot

• Binomial Probability Function for Acceptance
  Sampling
• where
• n sample size
• p proportion of defective items in lot
• x number of defective items in sample
• f(x) probability of x defective items in
  sample

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Example Acceptance Sampling

• An inspector takes a sample of 20 items from a
  lot.
• Her policy is to accept a lot if no more than 2
  defective
• items are found in the sample.
• Assuming that 5 percent of a lot is defective,
  what is
• the probability that she will accept a lot?
  Reject a lot?
• n 20, c 2, and p .05
• P(Accept Lot) f(0) f(1) f(2)
• .3585 .3774 .1887
• .9246
• P(Reject Lot) 1 - .9246
• .0754

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Example Acceptance Sampling

• Using the Tables of Binomial Probabilities

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Selecting an Acceptance Sampling Plan

• In formulating a plan, managers must specify two
  values for the fraction defective in the lot.
• a the probability that a lot with p0 defectives
  will be rejected.
• b the probability that a lot with p1 defectives
  will be accepted.
• Then, the values of n and c are selected that
  result in an acceptance sampling plan that comes
  closest to meeting both the a and b requirements
  specified.

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Operating Characteristic Curve
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Multiple Sampling Plans

• A multiple sampling plan uses two or more stages
  of sampling.
• At each stage the decision possibilities are
• stop sampling and accept the lot,
• stop sampling and reject the lot, or
• continue sampling.
• Multiple sampling plans often result in a smaller
  total sample size than single-sample plans with
  the same Type I error and Type II error
  probabilities.

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A Two-Stage Acceptance Sampling Plan
Inspect n1 items
Find x1 defective items in this sample
Yes
Accept the lot
x1 lt c1 ?
No
Yes
Reject the lot
x1 gt c2 ?
No
Inspect n2 additional items
Find x2 defective items in this sample
Yes
No
x1 x2 lt c3 ?
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End of Chapter