Post on 04-Apr-2018
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Parul Institute of Engineering & Technology
Subject Code : 171901
Name Of Subject :Operation Research
Name of Unit : Simulation
Topic : Monte Carlo Simulation& Generation of random
Numbers
Name of Faculty : R.N Barot
Name of Students: (i) Shah Devanshu[49]
(ii) Mehta Hardik[50]
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Simulation:A type of model where the computer is used to
imitate the behavior of the system
Monte Carlo simulation: Simulation that makes use of
internally generated (pseudo) random numbers.It is a technique which involves conducting experiments on
the model of the system under study, with some probability
distribution to draw random samples using random numbers.
Monte Carlo Simulation:
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Ways to Study System
Focus of class
System
Experiment w/
actualsystemExperiment w/
modelof
system
Physical
ModelMathematical
Model
Analytical
Model
Simulation
Model
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In a Monte Carlo simulation we attempt to follow the `time
dependence of a model for which change, or growth, does
not proceed in some rigorously predefined fashion (e.g.
according to Newtons equations of motion) but rather in a
stochastic manner which depends on a sequence of randomnumbers which is generated during the simulation.
MC method is often referred to as the method of last
resort, as it is apt to consume large computing resources;
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1) Calculate Probability distribution for variables to beanalysed .
2) Calculate a cumulative probability distribution for each
random variable.
3) Generate random numbers.4) Assign appropriate set of random numbers
5) Conduct simulation experiment using random numbers.
6) Repeat the steps until satisfactory runs are not completed.
7) Design and implement the course of outcome ofsimulation and maintain it.
Steps of Monte Carlo simulation:
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consuming vast computing resourceshave historically had to be executed upon the fastestcomputers available at the timeand employ the most advanced algorithms implemented with substantial programming acumen.
The name "Monte Carlo" comes from the city of MonteCarlo in the principality of Monaco, famous for itsgambling house
Characteristics:
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Monte Carlo methods can help in design and prediction of
behavior of systems in nuclear applications and radiationphysics
The use of MC in the area of nuclear power has undergone
an important evolution. Notable are the extensions to
compute burnup in reactor cores, and full core neutronic
simulations. Random numbers generated by the computer
are used to simulate naturally random processes
many previously intractable thermodynamic and quantummechanics problems have been solved using Monte Carlo
techniques.
Applications:
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Random number generators produce a flow of numbers that
Are observations from a continuous uniform distribution
between 0 and 1
Are independent of each other
Remember the discussion on pseudorandom numbers
Need of RNs in simulation
Generation of random variates to recapture prob. distribution
of stochastic process parameters
Random number generation
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The Midsquare method (von Neumann and Metropolis1940s)
EX: Start with an arbitrary 3-digit #
Z0 = 123 Z02 = (123)2 = 15129
Z1 = 512 U1 = 0.512 Z12 = (512)2 = 262144Z2 = 621 U2 = 0.621 Z22 = (621)2 = 385641
EX: A 4-digit example (see L&K Table 7.1)
ProblemsZero problem 002 004 zero will stay
Cycle length
The one trap 001 001 001
Methods of number generation
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Random-Number Generators
The Midproduct Method
Variation of Midsquare method and improved
EX:Start with two arbitrary 3-digit #s
Z0 = 123 Z1 = 456
Z0Z1 = (123)(456) = 56088 U1 = 0.608 Z2 = 608
Z1Z2 = (456)(608) = 277248 U2 = 0.772 Z3 = 772
Z2Z3 = (608)(772) = 469376 U3 = 0.693 Z4 = 693
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Linear Congruential Generators (LCG)
Problems and Concerns Zis are not really random at all, i.e., Zi can be
determined by m, a, c, and Z0
However, by careful choice of these four parameters
we try to make the corresponding Uis appear to beIID U(0,1) random variates subjected to some tests
Uis are 0, 1/m, 2/m, 3/m, (m-1)/m, impossible toget fractions, say 0.3/m or 1.5/m
i.e., m needs be very large to have desired density
Ex: m109, there are at least a billion possiblevalues, a good approximation to the truecontinuous U(0,1) distribution
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Finding the ith RN Directly from Z0
Also -
Zi can be obtained directly from Z0 withoutgoing through Z1, Z2, , Zi-1,
For i = 1, 2, ...Zi = [a
iZ0 + c(ai - 1)/(a- 1)] (mod m)
This shows Zis are not really random at all.
===> Predictablity
However, Zis are predictable does not meanthey are correlated.
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Linear Congruential Generators (LCG)
Problems and Concerns
Looping (or Repeating cycle):
LCG: m = 16, a = 5, c = 3, Z0= 7 for i = 1, 2, , 19
Zi = (5 Zi-1 + 3) (mod 16) Z0 = 7
Z1 = Z17 = 6, Z2 = Z18 = 1, Z3 = Z19 = 8
(see L&K Table 7.2)
Length of cycle mIf length of cycle = m, then LCG has a full period
If m and loop m, then LCG is adequate
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