cs433 modeling and simulation lecture 11 monté carlo simulation dr. anis koubâa 05 jan 2009...

CS433Modeling and Simulation

Lecture 11

Monté Carlo Simulation

Dr. Anis Koubâa

http://10.2.230.10:4040/akoubaa/cs433/

05 Jan 2009

Al-Imam Mohammad Ibn Saud UniversityAl-Imam Mohammad Ibn Saud University

Reading

Required

Lemmis Park, Discrete Event Simulation -

A First Course,

Chapter 8: Section8.2, Monte Carlo

Estimation

Goals of Today

Understand the concept of Monté Carlo

Simulation

Learn how to use Monté Carlo Simulation

to make good decisions

Learn how to use Monté Carlo Simulation

for estimating complex integrals

What is Monte Carlo Simulation ?

Monte Carlo methods are a widely used class of computational algorithms for simulating the behavior of various physical and mathematical systems, and for other computations.

Monte Carlo algorithm is often a numerical Monte Carlo method used to find solutions to mathematical problems (which may have many variables) that cannot easily be solved, (e.g. integral calculus, or other numerical methods)

What is Monte Carlo Simulation ?

A Monte Carlo simulation is a statistical simulation technique that provides approximate solutions to problems expressed mathematically. It utilizes a sequence of random numbers to perform the simulation.

This technique can be used in different domains: complex integral computations, economics, making decisions in specific complex problems, …

General Algorithm of Monte Carlo Simulation

In general, Monte Carlo Simulation is roughly composed of five steps:1. Set up probability distributions: what is the

probability distribution that will be considered in the simulation

2. Build cumulative probability distributions3. Establish an interval of random numbers for

each variable4. Generate random numbers: only accept

numbers that satisfies a given condition. 5. Simulate trials

Examples

Example 1 and Example 2: using Monte Carlo simulation for the analysis of real systems

Example 3: using Monte Carlo simulation to evaluate an integral.

Example 1. HERFY Cake Shop


The cake seller HERFY shop sells a random number of

cakes each day. HERFY manager wants to determine

a policy for managing his inventory of cakes (e.g.

how many cakes should he prepare in 10 days).

We can use Monte Carlo simulation to analyze HERFY

inventory…

Demand Frequency Probability for cakes 0 10 0.05 1 20 0.1 2 40 0.2 3 60 0.3 4 40 0.2 5 30 0.15 200 1.00

=10/200

Example inspired from: To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna

© 2006 by Prentice Hall, Inc.

Upper Saddle River, NJ 07458


Step 1: Set up the probability distribution for cake sales.

Using historical data HERFY Shop determined that 5% of the time 0 cakes were demanded, 10% of the time 1 cake was demanded, etc…

P(1) = 10%


Step 2: Build a Cumulative Probability Distribution

15% of the time the demand was 0 or 1 cake P(0) = 5% + P(1) = 10%


Dem

and

Freq

uen

cy

Pro

bab

ility

Cu

mula

tive

Pro

bab

ilit

y Ran

dom

N

um

ber

Inte

rval

0 10 0.05 0.05 01 - 05

1 20 0.10 0.15 06 - 15

2 40 0.20 0.35 16 - 35

3 60 0.30 0.65 36 - 65

4 40 0.20 0.85 66 - 85

5 30 0.15 1.00 86 - 00

Step 3: Establish an interval of random numbers.

Mus

t be

in c

orre

ct p

ropo

rtio

n

Note: 5% of the time 0 cakes are demanded, so the random number interval contains 5% of the numbers between 1 and 100


Step 4: Generate random numbers.

5237826998963350889050274581667430

0663570294526933323048881402830534

5028683690622750183661214601148287

8802284936872195502418623278748201

5374057106491113626985691382279374

3035949978566044578223644974760911

1024033223599534345108486697039646

4703111067238962567454316237333382

9929277589786864623017127445115259

3760792185714839313512734131977894

6674909529721755153680028694591325

9185879021908929408569689899810634

3590929425573430900124009242722832

3273413873010964345584169849003023

0059099769989349519292168427649417

8455257134575044956416465464612301

5717367285314430260949133389133758

0760774976955116148559854042523973


Step 5: Simulate a series of trials.

Using random number table on previous slide, simulated demand for 10 days is:

Random number: 52 06 50 88 53 30 10 47 99 37Simulated demand: 3 1 3 5 3 2 1 3 5 3

Tires Interval ofDemanded Random Numbers

0 01 - 051 06 - 152 16 - 353 36 - 654 66 - 855 86 - 100

1

2

3


Create lookup table using cumulative probability

Generate a random number and look it up in the table

Example 2. SEC Power Company



SEC provides power to Riyadh city. The company is concerned about generator failures

because a breakdown costs about 75 SAR per hour versus a 30 SAR per hour salary for repairpersons who work 24 hours a day, seven days a week.

Management wants to evaluate the service maintenance cost, simulated breakdown cost, and total cost.

We can use Monte Carlo simulation to analyze SEC system costs.








½ 5 0.05 0.05 01 - 051 6 0.06 0.11 06 - 11

1 ½ 16 0.16 0.27 12 - 272 33 0.33 0.60 28 - 60

2 ½ 21 0.21 0.81 81 - 813 19 0.19 1.00 82 - 00

Tim

e B

etw

een

Bre

akd

ow

ns

(Hr s

)

Num

ber

of

Tim

es

Obse

rved

Pro

babili

ty Cum

ula

tive

Pro

babili

ty

Random

N

um

ber

Inte

rval

Steps 1-3: Determine probability, cumulative probability, and random number interval - BREAKDOWNS.

Total 100 1.00





1 28 0.28 0.28 01 - 28

2 52 0.52 0.80 29 - 80

3 20 0.20 1.00 81 - 00

Repair

Tim

e

Requir

ed

(Hours

)

Num

ber

of

Tim

es

Obse

rved

Pro

babili

ty

Cum

ula

tiv

e

Pro

babili

ty

Random

N

um

ber

Inte

rval

Steps 1-3: Determine probability, cumulative probability, and random number interval - REPAIRS.





1 57 2 2:00 2:00 7 1 3:00 1

2 17 1.5 3:30 3:30 60 2 5:30 2

3 36 2 5:30 5:30 77 2 7:30 2

4 72 2.5 8:00 8:00 49 210:0

0 2

5 85 3 11:0011:0

0 76 213:0

0 2

: : : : : : : : :

14 89 3 4:00 6:00 42 2 8:00 4

15 13 1.5 5:30 8:00 52 210:0

0 4.5

Sim

ula

tion

Tri

al

Random

N

um

ber

Tim

e R

epair

Can B

egin

Random

N

um

ber

Tim

e R

epair

Ends

Repair

Tim

e

No. of

hrs

.M

ach

ine is

dow

n

Tim

e b

/tB

reakd

ow

ns

Tim

e o

fB

reakd

ow

n





Cost Analysis

Service maintenance =

34 hrs of worker service X 30 SAR per hr =

1,020 SAR

Simulate machine breakdown costs =

44 total hrs of breakdown X $75 lost per hr of

downtime = 3,300 SAR

Total simulated maintenance cost of the

current system = service cost + breakdown

costs = $1,020 + $3,300

= $4,320

Example 3. Computation of Integrals

Example 3. Computation of Integrals The Monte Carlo method can be used to

numerically approximate the value of an

integral

Pick n randomly distributed points x1, x2, …, xn in the

interval [a,b]

Determine the average value of the function

Compute the approximation to the integral

An estimate for the error is

Where

1

1 n

ii

f f xn

b

a

f x dx b a f

b

a

f x dx

22b aError f f

n

2 2

1

1 n

ii

f f xn

Algorithm . Estimation of = 3.1416

% Matlab Program to Find Pi using Random Numbers % Tom Huber, June 15, 1996 Nrand = input('How Many Random Numbers '); NInside = 0; for nloops=1:Nrand Xrand = rand; % Generate Random XY Point Yrand = rand; Rrand = Xrand^2 + Yrand^2; % Find its distance from origin if (Rrand <= 1) NInside = NInside + 1; end end disp(['Total Generated: ' num2str(Nrand) ' Inside Pts: ' num2str(NInside)]); piapprox = 4*NInside/Nrand; disp([' Approximation to pi = ' num2str(piapprox)]);

cs433 modeling and simulation lecture 11 monté carlo simulation dr. anis koubâa 05 jan 2009...

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