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DESCRIPTION
System – set of interdependent elements that interact in order to accomplish a one or more final outcomes. Constrained and affected by: Endogenous factors – internal limits on capabilities of elements (resources), interactions, etc. Exogenous factors – outside (“environmental”) demands, supply, prices, etc. Need ways to anticipate behavior of system in order to make informed decisions.
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System Analysis
System Analysis• System – set of interdependent elements that interact in
order to accomplish a one or more final outcomes.
• Constrained and affected by:• Endogenous factors – internal limits on capabilities of
elements (resources), interactions, etc.
• Exogenous factors – outside (“environmental”) demands, supply, prices, etc.
• Need ways to anticipate behavior of system in order to make informed decisions.
System Analysis
System
Experiment w/ Actual System
Experiment w/ Model of System
Physical Model
Mathematical Model
SimulationAnalytical Solution
Adapted from: Law, A. M. (2007), Simulation Modeling and Analysis (4th ed.), McGraw-Hill, New York.
Our focus!
Simulation
Simulation• Analytic Model – mathematical representation of a
physical system.• Discrete or continuous• Dynamic or static• Deterministic or stochastic (random)
• Simulation – “running” analytical model to study operation of physical system.• May use exact model or an approximation• Usually computer driven
Simulation• What are some advantages of simulation?
• Does not affect system itself
• Can be used before system is implemented
• May reveal critical interaction while designing
• Provides solution when model is infeasible or intractable
• Allows for compression/expansion of time
• Possible to vary conditions
• Useful for “selling” idea or concept
• May be inexpensive (or may not)
Simulation• What are some disadvantages of simulation?
• Requires programmers/computers
• May be time consuming
• Can be abused and/or misunderstood
• Garbage In – Garbage Out
Simulation Process
Define Model
Adapted from: Law, A. M. (2007), Simulation Modeling and Analysis (4th ed.), McGraw-Hill, New York.
Collect Data
Assump. Valid?
Verify w/ Test Data
Program Simulation
Sim. Valid?
No
Yes
No
Yes
Design Experiments
Execute Runs
Analyze & Report
Monte Carlo Simulation
Monte Carlo Simulation• Traditionally, uncertain (random) values were often
replaced by their means.• Missed critical consequences of high/low values
• Monte Carlo Simulation – uses (pseudo) random numbers to vary uncertain inputs/assumptions.• Useful for risk analysis• Randomly samples from known distributions • In Excel, we utilize =rand( ) function.
• Generates value in (0,1)• Used as probability w/ CDF
Monte Carlo SimulationSimple example:
Each bulb produced on a line has a 5% chance of being defective.
A single bulb may be simulated in Excel as: =if(rand()<0.05,1,0)
Monte Carlo SimulationSimple example:
10rand()
10
x0.5
Monte Carlo Simulation
Note: your numbers will be different …It’s the nature of random values!
Monte Carlo Simulation
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pbar
Another example:
Monte Carlo Simulation
Weekly demand for bulbs is (approx.) normally distributed with mean of 100 and std. dev. of 20.
A single day’s demand may be simulated in Excel as: =norminv(rand(),100,20)
Note: for integer values, can use =round(norminv(rand(),100,20),0)
Monte Carlo Simulation
Let’s look at a “system”:
Monte Carlo Simulation
Weekly demand for bulbs is distributed as N(100,20).
Each week, we refill stock up to 105 bulbs.
How many do we sell?
# Sold = min{Demand, 105}
Monte Carlo Simulation
And the standard deviation using s.
But also …
Plus, create confidence intervals.
Monte Carlo Simulation
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Monte Carlo Simulation
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Frequency
And another layer:
Monte Carlo Simulation
Weekly demand for bulbs is distributed as N(100,20).Each week, we refill stock up to 105 bulbs.Each bulb has a 5% chance of being defective.How many do we sell?
(We can use the =binom.inv() function to get a random # defects.)
# Sold = min{Demand, 105 – # Defects }
Monte Carlo Simulation
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Frequency
Monte Carlo Simulation• Additional questions:
• What if defect rate is not known w/ certainty?
• What is optimal stocking level?
• What if we have two (or more) types of bulbs to stock in a fixed space, each with its own costs, demands and defect rates?
We will need special software (e.g., Crystal Ball, @Risk) or to learn how to program in VBA!
Optimization
Optimization Process
Source: Gosavi, A. (2003), Simulation Based Optimization, Kluwer Academic Publishers, The Netherlands.
Simulator
• Simulator used to value current choice• Optimizer moves choice in “better” direction
Optimization Example• Consider the machine maintenance decision
• Simulator: given distribution of (random) time between breakdowns, cost of breakdown, current maintenance frequency and cost of maintenance, determines average cost.
• Optimizer: improves choice of maintenance frequency until stopping condition is reached.
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Optimization Process• Example of optimizer step:
Simulator provides estimated values (V) for xn – hn and xn + hn.
Slope is estimated by: V(xn – hn) – V(xn + hn)
2hn
If current choice is xn.
dn =
Choice is improved: xn+1 = xn + andn
(Note: an and hn get smaller as process continues.)
V
xn xn+1
V(xn + hn)V(xn – hn)
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