Enumerative versus Analytic Studies

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• Enumerative versus Analytic Studies

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Enumerative Versus Analytic Studies

• In 1921 Albert Einstein noted,
• As far as the laws of mathematics refer to
  reality, they are not certain and as far as they
  are certain, they do not refer to reality.
• In 1931 Walter Shewhart noted,
• The fact that the criterion which we happen to
  use has a fine ancestry of mathematical
  statistical theorems does not justify its use.
  Such justification must come from empirical
  evidence that it works.

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Enumerative Versus Analytic Studies

• In 1965 J. T. Davies noted,
• The same uncertainty, stemming from possible
  ignorance of some additional factor, makes the
  application of mathematical probability theory
  logically indefensible in scientific predictions.
• Indeed, mathematical probability, like other
  mathematical deductions, is part of a strictly
  logical system, and is always true in the sense
  that, given the premises (assumptions) and the
  deductive rules to be used, the result has been
  obtained correctly. Truth is thus a word we may
  use of a mathematical or logical deduction, but
  it is not one we should ever use of a scientific
  theory. The latter can best be described as a
  well-tested and precise working hypothesis.

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Enumerative Versus Analytic Studies

• Much of the confusion and misapplication of
  probability and statistical theory to the
  problems of business and industry can be traced
  to a lack of acknowledgment or understanding of
  the important distinction between enumerative and
  analytic studies.
• The ultimate aim of both enumerative and analytic
  studies is to provide data that can be used as a
  rational basis for action.

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Enumerative Versus Analytic Studies

• The enumerative study is focused on obtaining
  information and taking action on specific items
  contained in a frame which is a well-defined
  group of physical items.
• In contrast, the goal of an analytic study is to
  obtain information and take action on the cause
  system producing the frame or frames under study
  with the intent of improving the process
  producing the frames.

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The Shewhart Model of aChance Cause System
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Application to Optimizing a Filling Process

• Consider a production process designed to fill
  cups with a nutritional product for retail sale.
• Let LW denote the label claim for net weight.
• The goal is to fill all cups with net weight at
  or above the label claim at a minimum cost.

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Application to Optimizing a Filling Process
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Application to Optimizing a Filling Process

• The stated goal is achieved when the following
  conditions are met
• 1) the production process is brought into a
  state of
  statistical control
• 2) the process standard deviation ? is
  minimized and
• 3) the process mean is set so that µ - 3? gt
  LW.

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Application to Optimizing a Filling Process

• The key to this strategy is minimizing the
  process standard deviation ? by fundamentally
  changing the cause system of variation.
• This can be accomplished by applying the
  following general strategy which has its roots in
  the scientific method.

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General Optimization Strategy

• Define the performance characteristic of interest
  - fill weight
• Understand the design of the production process

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The Process Diagram
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Identification of FactorsAffecting Variation

• Using the process diagram identify the factors
  that can potentially affect net weight
• 1) Lanes - 4
• 2) Phase - 2
• 3) Unknown factors
• 4) Common causes - noise
• Create a sampling plan to obtain representative
  data from the process

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Sampling Design

• The following sampling design was employed in
  this study.
• 1000 consecutive cups were sampled from each
  lane.
• The net weight, lane, phase, and order of
  production were captured on each cup in the
  sample.

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Analysis of Process Data

• The first step in the analysis was to create a
  histogram for the fill weights.

Histogram for Fill Weights Produced by the
Original Process
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Analysis of Process Data
Lane 1
Lane 2
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Analysis of Process Data
Lane 3
Lane 4
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Analysis of Process Data
Lane 4, Phase 1
Lane 4, Phase 2
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Analysis of Process Data
Lane 1
Lane 2
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Analysis of Process Data
Lane 3
Lane 4
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Analysis of Process Data
Lane 4, Phase 1
Lane 4, Phase 2
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Analysis of Process Data
Histogram for the Fill Weights Produced by the
Original Process
Histogram for the Fill Weights Produced by the
Redesigned Process
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Cost Savings Associated withOptimizing the
Process

• The final analysis performed by the team was to
  determine the cost savings to the organization
  achieved by identifying and removing the pressure
  effect through the redesign the filling process
  piping system.
• Prior to the redesign the process average was set
  at a minimum value of 247 gms.
• After the process redesign, the process average
  was set at 240 gms for a net savings of 7 gms per
  cup.
• The machine was designed to produce 20,000,000
  cups per year, and the average cost of product
  filled by the machine was 0.00068/gm.
• Therefore, the estimated savings in raw material
  were 20,000,000 x 7 gms x 0.00068 95,000 per
  year.

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Confirmation and Maintenanceof Improvement

• Confirmation and maintenance of the improvement
  achieved in this optimization project will only
  come from the implementation of an on-line SPC
  system that monitors the fill weights and
  maintains the process in a state of statistical
  control.
• The longer the process stays in control the
  higher our confidence will be in the results and
  the more significant will be the return on the
  investment in the analysis and design changes.

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Comparison of Shewharts Theory versusClassical
Statistical Theory

• Note that basic statistical concepts and methods
  were used to optimize the filling process.
• None of the standard assumptions, however, were
  necessary including the assumptions of normality,
  independence of observations, or random samples.
• A key to the analysis was uncovering the
  information hidden in the order of production.

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Comparison of Shewharts Theory versusClassical
Statistical Theory

• Also note that the same data were analyzed a
  number of different ways by rationally
  subgrouping the data to answer specific
  questions.
• None of the standard probability calculations,
  such as ? levels or p-values, were necessary or
  even appropriate.
• Confirmation and confidence in the effectiveness
  of the redesign decisions will not come from
  mathematical calculations, but from the
  continuous analysis and reaction to data provided
  by the on-line SPC monitoring system which will
  help ensure that the process remains
  statistically stable over time.

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Comparison of Shewharts Theory versusClassical
Statistical Theory

• And finally, it is highly recommended that you
  research Shewharts and Demings work.
• It is particularly enlightening to read the
  following article and books
• W. Edwards Deming, On Probability as a Basis
  for Action, The American Statistician, v.29,
  pp.146-152, 1975.
• W. Edwards Deming, Out of the Crisis,
  Massachusetts Institute of Technology, Center for
  Advanced Engineering Study, Cambridge, Mass,
  1986.
• Shewhart, Walter A., Economic Control of Quality
  of Manufactured Product, American Society for
  Quality Control, Milwaukee, Wisconsin, 1980.