Computer Simulation

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Computer Simulation

• Henry C. Co
• Technology and Operations Management,
• California Polytechnic and State University

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Simulation Model

• Simulation a descriptive technique that enables
  a decision maker to evaluate the behavior of a
  model under various conditions.
• Simulation models complex situations
• Models are simple to use and understand
• Models can play what if experiments
• Extensive software packages available

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• Analytic models values of decision variables are
  the outputs.
• Simulation models values of decision variables
  are the inputs. Investigate the impacts on
  certain parameters when these values change.

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Why Simulation?
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• Analytic models
• May be difficult or impossible to obtain.
• Typically predict only average or steady-state
  behavior.
• Simulation models
• Wide availability of software and more powerful
  PCs make implementation much easier than before.
• More realistic random factors can be
  incorporated.
• Easier to understand.

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Simulation Process
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• Identify the problem
• Develop the simulation model
• Test the model
• Develop the experiments
• Run the simulation and evaluate results
• Repeat until results are satisfactory

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Implementation

• Identify the boundaries of the system of
  interest.
• Identify the random variables, decision
  variables, parameters, and the performance
  measure(s).
• Develop an objective function for the performance
  measure(s) in terms of random variables, decision
  variables, and parameters.
• Use computer to generate the simulated values of
  these random variables.
• Compute the values of the objective function
  using these simulated values of random variables
  and values of decision variables.
• Statistical analysis.

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Monte Carlo Simulation
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• Monte Carlo method Probabilistic simulation
  technique used when a process has a random
  component
• Identify a probability distribution
• Setup intervals of random numbers to match
  probability distribution
• Obtain the random numbers
• Interpret the results

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Major Components of Models
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• Random input factors sales, demand, stock
  prices, interest rates, the length of time
  required to perform a task.
• Random performance measures
• Business profit within a time interval.
• Average waiting time of a customer in a queuing
  system.
• Random input factors ? random performance
  measures.

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An Analog Approach
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• Game Spinner for uniform random variable on the
  interval 0 to 1.

• Every point on the circumference corresponds to a
  number between 0 and 1.
• For example, when the pointer is in the 3 Oclock
  position, it is pointing to the number 0.25.

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Simulating a Discrete Distribution
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• 10 of the interval (0.0 to 0.09999) is mapped
  (assigned) to a demand d 8.
• 20 of the interval (0.1 to 0.29999) is mapped to
  d 9.
• 30 of the interval (0.3 to 0.59999) is mapped to
  d 10.
• etc., etc.

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Excel Functions Useful in Simulation

• RAND() a volatile Excel Function
• Function RAND() generates a uniformly-distributed
  random number between 0 -1.
• VLOOKUP

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Use function RAND() to generate a
uniformly-distributed random number between 0 and
1.
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F4RAND() copy and paste F5F13 G4VLOOKUP(F4,B
4C10,2,1) copy and paste G5G13
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A Machine Breakdown Example
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F4RAND() copy and paste F5F13 G4VLOOKUP(F4,B
4C10,2,1) copy and paste G5G13
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Simulating a Continuous Distribution
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• The inverse transformation method
• To transform this random number into a sample
  value of the random variable.

F(w) is the CDF F(x)Prob. W? x.
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Inverse Transformation Method

• Define F(x)Prob. W? x the probability that
  random variable W is less than or equal to a
  specific value w.
• Denote the 0-1 random number by u and let u
  F(x).
• Use RAND() to generate a value for u, substitute
  it into x F-1(u) which in turn gives a value of
  x.

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EXCEL Implementation

• Exponential Distribution
• u RAND()
• For example, if arrival rate ? 0.05, and
  RAND().75, the observation from the exponential
  distribution is (-1/0.05)ln(1-.75) 23.73.
• Normal Distn Function NORMINV
• For example, NORMINV(RAND(),1000,100) returns a
  normally distributed random number with mean 1000
  and standard deviation 100.

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Using an EXCEL Simulation Model

• Information obtained from a Simulation model
• Summary statistics about the performance measures
• Downside Risk and Upside Risk
• Distribution of outcomes
• Based on the simulation results (Output), several
  alternatives (decisions) can be evaluated.

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How Reliable is the Simulation?
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• The more trials we run, the higher the confidence
  we have in our results (just like any statistical
  analysis with real data sample).
• The confidence intervals about the parameters (or
  any other estimated parameters) can be computed.
• Given sample size and significant level ?
  confidence intervals can be computed, or given
  the half width of the confidence interval and
  significance level ? compute the minimum number
  of replications we have to run.

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Advantages
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• Solves problems that are difficult or impossible
  to solve mathematically
• Allows experimentation without risk to actual
  system
• Compresses time to show long-term effects
• Serves as training tool for decision makers

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Limitations
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• Does not produce optimum solution
• Model development may be difficult
• Computer run time may be substantial
• Monte Carlo simulation only applicable to random
  systems