1. MONTE CARLO SIMULATION
MATH 182

3. SIMULATION
Simulation of a process – the examination of any emulating process simpler than that under consideration.
Examples:
System’s Simulation such as simulation of engineering systems, large organizational systems, and governmental systems
Operational Gaming such as military gaming and business gaming
Monte Carlo Simulation
Agent-Based simulation

4. AGENT-BASED SYSTEMS
Using Netlogo

5. MONTE CARLO SIMULATION
The process is similar to gambling in casinos such as using roulette wheels, dice and playing cards.
Monte Carlo is a technique for selecting numbers randomly from a probability distribution (not necessarily from a standard distribution).

6. MONTE CARLO SIMULATION
The computer-generated random numbers are called pseudo- random numbers.

7. A PSEUDO-RANDOM NUMBER GENERATOR ≈FOLLOWING Uniform Distribution [0,1]
Linear Congruential Generator (LCG) Period=m–1 x0 ≠0 is the seed number Example: a=13, b=0, m=31, x0 =8 Hence, u0 =0.2581; x1 =11, u1= Minimal Standard Random Number Generator: a=75, b=0, m=231–1 (a Mersenne prime)

8. EXAMPLE 1 DEMAND RANGE OF RANDOM NUMBER Let rϵ{1,2,…,100} 10 1-20 15
21-50 20
51-100
A simple example:

9. IF(B2<=20,10,IF(B2<=50,15,20))
Example 1
Run
Random Number
Simulated Demand 1 83 20 2 45 15 3 48 4 14 10 5 52 6 7 67 8 42 9 81 73 11 21 12 22 13 77
Sample Formula:
ROUNDUP(RAND()*100,0)
IF(B2<=20,10,IF(B2<=50,15,20))

10. example 1
So the average demand is: Using simulation (15 runs): … Using expected value: 0.2*10+0.3*15+0.5*20=16.5 Notice that Monte Carlo simulation is just a statistical experiment.

11. State1 State2 State3 State4
ANOTHER EXAMPLE:
State1
State2
State3
State4
RAND
RAND
RAND
RAND
RAND

12. Integration by MC Simulation
Generate n random sequence xi є [a,b], then for m clusters of runs
If you want to consolidate your m clusters of runs into just one cluster.

13. RAND()*5
CLUSTER
RUNS 1
RAND
CLUSTER RUNS 2
CLUSTER RUNS 3
3.58
12.81
1.22
1.49
4.29
18.37
0.76
0.58
1.12
1.26
1.46
2.12
4.50
20.22
2.68
7.17
4.58
20.93
4.90
24.02
4.74
22.49
0.33
0.11
3.97
15.77
0.26
0.07
3.89
15.11
3.50
12.25
2.04
4.15
0.80
0.64
5.00
24.96
2.25
5.07
1.58
2.49
2.17
4.72
0.08
0.01
0.73
0.53
2.83
8.00
4.66
21.75
3.31
10.94
1.62
2.62
4.54
20.63
2.64
6.95
Ave
SUM(B2:B11)/10*5
62.98
42.04
39.09
48.04

14. Integration by MC Simulation
Generate n random sequence xi є [a,b], n random sequence yi є [c,d] and n random sequence zi є [e,f] , then for m clusters of runs
If you want to consolidate your m clusters of runs into just one cluster.

15. HIT OR MISS MONTE CARLO
Find the AREA of the set of points (x,y) inside the unit square x,yϵ[0,1] that satisfy:
Area=(length of interval1)x(length of interval2)x(number of hits)/(total count of generated random pairs)
Area≈0.547

16. HIT OR MISS MONTE CARLO
Find the VOLUME:

17. IF(AND(A3^2+B3^2<=C3^2,A3^2+B3^2+(C3-1)^2<=1),1,0)x y z
count
RAND()*2-1
RAND()*2
IF(AND(A3^2+B3^2<=C3^2,A3^2+B3^2+(C3-1)^2<=1),1,0) 1 1
SUM(D3:D502)
205
D503/500*8
3.28

18. MONTE CARLO ERROR