Selecting Input Probability Distributions

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Selecting Input Probability Distributions
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Outline

• Sources of randomness
• Pitfalls in modeling simulation input data
• Choosing a distribution when data are available
• Hypothesizing families of distributions
• Estimation of parameters
• Determining how representative the fitted
  distributions are
• Chi-square test
• Choosing a distribution in the absence of data

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Sources of Randomness

• Almost all real-world systems contain one or more
  sources of randomness. The following are common
  sources of randomness in manufacturing systems
• Interarrival times of parts or raw materials,
• Processing or assembly times of parts,
• Times to failure of a machine,
• Repair times for a machine, and
• Setup times for a machine.
• Failure to choose correct probability
  distributions may drastically affect models
  results.

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Pitfalls in Modeling Simulation Input Data

• 1) Replacing a distribution by its mean
• Example Assume an insurance company with a claim
  department of 3 employees each claim is
  processed by the three employees.
• Insurance claims arrive at the claims department
  every 10 minutes (inter-arrival time) for
  processing.
• When a claim arrives, it takes 1 min. to transfer
  the claim to the first employee. If the first
  employee is not free, the claim waits on his
  desk. When the first employee becomes free, it
  takes 10 min to process the claim. When the first
  employee finishes working on the claim, the claim
  is transferred to the second employee for further
  processing. This transfer takes 1 min.
• Once the second employee is available, it takes
  10 min to complete his portion of the process.
  When the second employee finishes, the claim is
  transferred to the third and final employee. This
  transfer takes 1 min.
• Once the third employee is available, it takes 10
  min to perform his portion of the process. When
  the third employee finishes, the claim is
  complete and is transferred to the mailroom where
  it is sent to the customer with the approval or
  disapproval decision.

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Pitfalls in Modeling Simulation Input Data
Simple graphical representation
Model Input data
Simulation model (Averages)
Run the model
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Pitfalls in Modeling Simulation Input Data
Simulation Output (Averages)

• From the animation and the output
• Queues are not building
• Cycle time is not fluctuating,
• No problems in the system.
• Note This output is similar to using a static
  tool like a spreadsheet or a process map.

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Pitfalls in Modeling Simulation Input Data
Reality

• In reality, the arrival of the claims and
  department operations would never work in perfect
  rhythm, there is variability.
• In reality, variability occurs in every day
  situations and in any business. This is where
  the power of simulation over other methods
  arises.
• Variability and its effect on business operations
  and decision making will be demonstrated in the
  claims department simulation model.

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Pitfalls in Modeling Simulation Input Data
The inter-arrival rate, processing times, and
transfer times used previously in the example
were Averages. Let us go back to
reality! Variability
The Real model Input (Variability -
Distributions)
Real distributions
Press here
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Distributions Used in the Model
Mean 1
Mean 10
Mean 10 s 2
Min 8 Mode 10 Max 12
s Standard deviation
Min 8 Max 12
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Pitfalls in Modeling Simulation Input Data
Simulation model (Distribution)
Run the model
Simulation Output (Distribution)

• From the animation and the output
• Queues are building
• Cycle time is fluctuating,
• There are significant problems in the system.
• The output is not similar to the output based on
  averages.

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Pitfalls in Modeling Simulation Input Data
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Pitfalls in Modeling Simulation Input Data
Averages
Distributions (Variability)
It is evident that using the average only can
have a large impact on simulation output and on
the quality of decisions made with the simulation
results.
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Pitfalls in Modeling Simulation Input Data
(Contd)

• 1) Replacing a distribution by its mean ?
• 2)Selecting the wrong distribution
• In the example Suppose that 200 claims
  processing times are available for the first
  process but their underlying probability
  distribution is unknown. Using some methods
  (described later), The following distributions
  are fit to the observed data
• Normal, Triangular, Lognormal, Beta and Weibull.

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Distributions Used for Process 1
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Pitfalls in Modeling Simulation Input Data
(Contd)

• Then, a simulation run of length 1600 hours is
  made using each of the five distributions. If
  the normal distribution is the best fit for the
  data, the following errors for cycle time are
  observed when using other distributions

It is evident that the choice of probability
distribution can have a large impact on
simulation output and on the quality of decisions
made with the simulation results.
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Choosing a Distribution When Data are Available

• There are three steps in determining what
  probability distribution best represents a set of
  data
• 1. Hypothesize families of distributions,
• 2. Estimate parameters, and
• 3. Determine how representative the fitted
  distributions are.

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Hypothesizing Families of Distributions

• The first step in selecting a particular input
  distribution is to decide what general families
  (e.g., exponential, normal) appear to be
  appropriate on the basis of their shapes.
• Some general techniques used in hypothesizing
  families of distributions include using
• Prior knowledge
• Summary statistics
• Histograms

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Use of Prior Knowledge

• In some situations, prior knowledge about a
  certain random variables role in the system can
  be used to select a distribution or at least to
  rule out some distributions. For example,
• If customers arrive one at a time, at a constant
  rate, so that the numbers of customers arriving
  in disjoint time intervals are independent, the
  interarrival times are probably exponentially
  distributed.
• Service times should (at least in principle) not
  be generated directly from a normal distribution.
• The proportion of defective items in a large
  batch should not be assumed to have a gamma
  distribution, since proportions must be between 0
  and 1 and gamma random variables have no upper
  bounds.

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Use of Summary Statistics

• Summary statistics may be used in some situations
  to suggest an appropriate distribution. Some
  guidelines are
• For a symmetric continuous distribution (e.g.,
  normal) the mean is equal to the median.
• If the coefficient of variation, cv, is close to
  one, it suggests an exponential distribution.
• Skewness is a measure of the symmetry of a
  distribution.
• for symmetric distributions (e.g., normal)
• skewness 0
• if the distribution is skewed to the right
• skewness gt 0
• if the distribution is skewed to the left
• skewness lt 0

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Use of Summary Statistics (Contd)

• For a discrete distribution, the lexis ratio
  plays an important role
• for Poisson lexis ratio 1
• for binomial lexis ratiolt1
• for negative binomial lexis ratiogt 1

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Estimation of Parameters

• Once one or more candidate families of
  distributions have been hypothesized, the values
  of their parameters (i.e., shape, scale, or
  location) must be specified.
• The most popular method for estimation of
  parameters is the method of maximum likelihood.
• For a particular distribution, the method of
  maximum likelihood selects those values for the
  parameters that maximize the likelihood (or
  probability) of having obtained the observed data
  from the distribution.

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Determining How Representative the Fitted
Distributions Are

• After determining one or more probability
  distributions that might fit the observed data,
  the quality of the fitted distributions must be
  evaluated using one or more heuristics.
• Two heuristics used in determining the goodness
  of fit are
• The Chi-square test
• The Kolmogorov-Smirnov test

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Chi-square Test

• The chi-square test measures the error between a
  candidate distributions density function and the
  histogram.
• The test statistic is
• where
• k Number of intervals
• Nj Number of observations in the interval
  aj-1, aj)
• npj Expected number of observations that would
  fall in the jth interval if we were sampling from
  the fitted distribution.
• If , the
  hypothesized distribution is rejected.

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Choosing a Distribution in the Absence of Data

• In some situations it is not possible to collect
  data on the random variables of interest.
  Examples include
• A manufacturing system under study that does not
  currently exist, or
• An existing system where the number of required
  probability distributions is large and the time
  available prohibits necessary data collection and
  analysis.

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Choosing a Distribution in the Absence of Data
(Contd)

• Two heuristic approaches for choosing a
  distribution in the absence of data involve
• Using a triangular distribution
• Using a beta distribution
• The first step in using either heuristic is to
  identify an interval a,b in which it is felt
  that X (for example, the time to perform a task)
  will lie with probability close to 1.
• In order to obtain subjective estimates of a and
  b, experts are asked for their most optimistic
  and pessimistic estimates of the time to perform
  the task.

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Choosing a Distribution in the Absence of Data
(Contd)

• Using a triangular distribution In addition to
  a and b (minimum and maximum values for time to
  perform a task), the experts are asked to specify
  the most likely time to perform the task, denoted
  by m.
• The advantage of this approach is that it is
  simple and it is usually possible to obtain
  estimates for a, b, and m.
• The disadvantage of this approach is that it is
  not flexible and may lead to large errors.