Class 2 Outline
1. Monte-Carlo:
• Framework/Definition
• Algorithm
• Examples
2. Random Number Generation
Jérémie Gallien
Monte-Carlo Framework
Estimate q = E[ h(X) ]
where X = {X1,…,Xm} is a random vector in Rm,
h(.) is a function Rm -> R,
and E[ | h(X) | ] < ¥
Jérémie Gallien
Monte-Carlo Algorithm
1. Generate n samples of X:
X1, … , Xn
2. Compute h(X1), h(X2), …, h(Xn)
3. Estimate q = E[ h(X) ] with
q = [h(X1)+h(X2)+…+h(Xn)]/n
• Why is q a good estimator?
Jérémie Gallien
A Sales Incentive Plan
A company has a sales incentive plan with the
following structure: In the months when sales are
lower than $10,000, a salesrep get a base salary of
$2000. If the monthly sales achieved are between
$10,000 and $15,000, then he/she is paid 20% of the
sales. Finally, if the monthly sales figure is larger than
$15,000, the salesrep get 30% of the sales.
Assuming the monthly sales achieved by a salesman
follows an exponential distribution with mean $15,000,
what is the mean and standard deviation of the
salesman’s monthly salary under this compensation
plan?
Jérémie Gallien
Product Reliability
• Consider a product reliability model given by:
B
A
E
C
F
D
• Component life lengths: XA ~ exp(5.1); XB ~ N(4.5,1); XC ~ exp(6.5)
XD ~ exp(5.4); XE ~ exp(6.4); XF ~ N(5.5,0.8)
• What is the probability that the product will function for at least 3
years (warranty period)?
Jérémie Gallien
DreamCast Launch Project
Task
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Description
Market Study
External Developer Focus Group
Feature Selection
Hardware Engineering
Operating System Devlpt.
Advertising Campaign
Supplier Selection & Negotiation
Component Inventory Buildup
Assembly Facility Setup
Finished Good Inventory Buildup
Library & Programmer Toolkit Devlpt.
External Development Support Setup
Internal Game Devlpt.
External Game Devlpt.
Platform Promotion
Publisher Selection & Negotiation
Website Setup
Release Promotion Material Design
Distribution Channels Devlpt. & Negotiation
Carrier Selection & Negotiation
Launch Event Organization & PR
Hardware & Software Shipment
Launch!
Duration (weeks)
14
10
4
25
16
20
20
45
18
7
12
5
30
32
6
5
20
3
9
4
5
1
0
Precedence
A,B
C
D,B
D,M,N
D
G
D
I,H
E
K
K
L,O
K
M
M
D,M,N
R
S
R
J,T,P,N
V,U,Q,F
Jérémie Gallien
Setup Factory
FG Inv Build
I
18
G
14
Features
C
4
B
10
Ext Dev
Focus Grp
V
7
1
45
D
25
20
Plan Launch
Event
HW Eng
E
16
O/S Dev
GO!
Setup
Comps Website
H Inv
Q
20
A
0
Ship HW
& SW
J
Supp Sel & Neg
Mkt Study
W
U
Int Dev
SW Dev Kit M
K
5
30
Publisher
Sel & Neg
12
Dev Supp
O
6
Platform
Mktg
Distr Neg
R
L
5
5
P
3
N
32
Ext Dev
S
9
T
4
Release Carrier Neg
Promo
F
20
Advert Plan
Jérémie Gallien
Critical Path Algorithm
2 , 16
A
1. A task Early Start Date
is the largest of all
Early Finish Dates of
its predecessors
14
5 , 15
B
ES, EF
D
LS, LF
10
3
5 , 10
C
LS, LF
5
ES, EF
A
LS, LF
2. A task Late Finish Date
is the earliest of all
Late Start Dates of its
successors
14
ES, EF
A
14
D
3
LS, LF
20, 34
ES, EF
C
LS, LF
ES, EF
B
10
15, 25
5
14, 19
Jérémie Gallien
129, 129
W
43, 61
I
18
103,121
J
G
A
14
0,14
14, 18
C
4
14,18
0, 10
18, 43
25
18, 43
1
128, 129
H
101, 121
45
76, 121
20
Q
109,129
111, 116
43, 59
16
43, 59
10
4,14
20
56,76
71, 101
59, 71
M
K
30
12
59, 71
79, 109
d
LS, LF
O
6
71, 77
5
124,129
P
5
123,128
111, 120
108, 111
L
71, 77
U
101, 106
71, 76
ES, EF
A
129, 129
V
63, 108
D
E
B
124, 125
7
121,128
43, 63
0, 14
0
108, 115
5
72, 77
R
N
3
112,115
32
77, 109
109, 129
77, 109
S
9
115,124
120, 124
T
4
124,128
F
20
109, 129
Jérémie Gallien
129, 129
W
43, 61
I
18
103,121
J
G
A
14
0,14
14, 18
C
4
14,18
0, 10
18, 43
25
18, 43
1
128, 129
H
101, 121
45
76, 121
20
Q
109,129
111, 116
43, 59
16
43, 59
10
4,14
20
56,76
71, 101
59, 71
M
K
30
12
59, 71
79, 109
d
LS, LF
O
6
71, 77
5
124,129
P
5
123,128
111, 120
108, 111
L
71, 77
U
101, 106
71, 76
ES, EF
A
129, 129
V
63, 108
D
E
B
124, 125
7
121,128
43, 63
0, 14
0
108, 115
5
72, 77
R
N
3
112,115
32
77, 109
109, 129
77, 109
S
9
115,124
120, 124
T
4
124,128
F
20
109, 129
Jérémie Gallien
Critical Path Solution
Task
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Description
Market Study
External Developer Focus Group
Feature Selection
Hardware Engineering
Operating System Devlpt.
Advertising Campaign
Supplier Selection & Negotiation
Component Engineering & Inventory Buildup
Assembly Facility Setup
Finished Good Inventory Buildup
Library & Programmer Toolkit Devlpt.
External Development Support Setup
Internal Game Devlpt.
External Game Devlpt.
Platform Promotion
Publisher Selection & Negotiation
Website Setup
Release Promotion Material Design
Distribution Channels Devlpt. & Negotiation
Carrier Selection & Negotiation
Launch Event Organization & PR
Hardware & Software Shipment
Launch!
Duration (weeks) Precedence LS LF
ES
14
-129 -115 -129
10
-125 -115 -129
4
A,B -115 -111 -115
25
C -111 -86 -111
16
D,B -86 -70 -86
20
D,M,N -20
0 -20
20
D -73 -53 -86
45
G -53
-8 -66
18
D -26
-8 -86
7
I,H
-8
-1 -21
12
E -70 -58 -70
5
K -57 -52 -58
30
K -50 -20 -58
32
L,O -52 -20 -52
6
K -58 -52 -58
5
M
-6
-1 -28
20
M -20
0 -28
3
D,M,N -17 -14 -20
9
R -14
-5 -17
4
S
-5
-1
-8
5
R
-5
0 -17
1
J,T,P,N
-1
0
-4
0
V,U,Q,F
0
0
0
EF Slack
-115
0
-119
4
-111
0
-86
0
-70
0
0
0
-66
13
-21
13
-68
60
-14
13
-58
0
-53
1
-28
8
-20
0
-52
0
-23
22
-8
8
-17
3
-8
3
-4
3
-12
12
-3
3
0
0
Jérémie Gallien
Stochastic Task Durations
Task
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Description
Duration (weeks) Precedence LS
LF
ES EF Slack
Market Study
14
-129 -115 -129 -115
0
External Developer Focus Group
10
-125 -115 -129 -119
4
Feature Selection
4
A,B -115 -111 -115 -111
0
Hardware Engineering
25
C -111 -86 -111 -86
0
Operating System Devlpt.
16
D,B -86 -70 -86 -70
0
Advertising Campaign
20
D,M,N -20
0 -20
0
0
Supplier Selection & Negotiation
20
D -73 -53 -86 -66
13
Component Engineering & Inventory Buildup
45
G -53
-8 -66 -21
13
Assembly Facility Setup
18
D -26
-8 -86 -68
60
Finished Good Inventory Buildup
7
I,H
-8
-1 -21 -14
13
Library & Programmer Toolkit Devlpt.
12
E -70 -58 -70 -58
0
External Development Support Setup
5
K -57 -52 -58 -53
1
Internal Game Devlpt.
30
K -50 -20 -58 -28
8
External Game Devlpt.
32
L,O -52 -20 -52 -20
0
Platform Promotion
6
K -58 -52 -58 -52
0
Publisher Selection & Negotiation
5
M
-6
-1 -28 -23
Duration of development
tasks
E, M and N22
Website Setup
20
M -20
0 -28
-8
8
now
follow
exponential
distributions
with
Release Promotion Material Design
3
D,M,N -17 -14 -20 -17
3
Distribution Channels Devlpt. & Negotiation means 16, 309 and 32 respectively.
R -14
-5 -17
-8
3
Carrier Selection & Negotiation
4
S
-5
-1
-8
-4
3
Launch Event Organization & PR
5
R
-5
0 -17 -12
12
What
is
the
probability
that
the
total
project
Hardware & Software Shipment
1
J,T,P,N
-1
0
-4
-3
3
duration
will
be
more
than
135
weeks?
Launch!
0
V,U,Q,F
0
0
0
0
0
Jérémie Gallien
Stochastic Programming
Distribution
Centers (j)
Plants (i)
X1
Cap1
Capacity
Cij
Cap2
X2
Demand
X3
Transportation
Unit Costs
Stochastic Demand
X1~N(1500,350)
X2~N(2000,600)
X3~N(1500,450)
Jérémie Gallien
Random Number Generation
• The key in a Monte-Carlo simulation is to
generate sample values drawn from a known
probability distribution. How is this done?
Generate Sample Value from Uniform[0,1]
Discrete Distribution
Continuous Distribution
• Inverse transform method
• Rejection method
• Inverse transform method
• Rejection method
• Composition method
• Polar method
Jérémie Gallien
U[0,1] Sample Generation
1. Calculate Z1, Z2,… given by the Linear
Congruential Generator:
Zi = ( aZi-1 + c ) modulo m
where m, a, c and Z0 are non-neg. integers
modulus
seed
2. Take Ui = Zi / m as a proxy to an
independent sequence of U[0,1] random
variates (note: 0 £ Ui £ 1 for all i)
Jérémie Gallien
Linear Congruential Generator
• Example:
Zi = ( 11Zi-1 ) modulo 16
Z0 = 1
Z1 = (11) mod 16 = 11
Z2 = (121) mod 16 = 9
Z3 = (99) mod 16 = ?
Z4 = (33) mod 16 = ?
• Huge literature on LCG periods, parameter choice,
statistical tests for randomness…
• Crystal Ball uses Zi = (630,360,016 Zi-1 ) mod (231-1)
Jérémie Gallien
Discrete Distributions
• We want to simulate a random variable X
defined by
P( X = xj ) = pj, j=0,1,…,m
1. Generate a sample U from U[0,1]
2. if U < p0 then X = x0
if p0 < U < p0 + p1 then X = x1
…..
if p0 + … + pj-1 < U < p0 + … + pj then X = xj
• Why does this work?
Jérémie Gallien
Solution
1.
2.
Generate a sample U from U[0,1]
if U < p0 then X = x0
if p0 < U < p0 + p1 then X = x1
…..
if p0 + … + pj-1 < U < p0 + … + pj then X = xj
• P( X = xj ) = P(p0 + … + pj-1 < U < p0 + … + pj )
= pj
Jérémie Gallien
Continuous Distributions
• We want to simulate a random variable X
defined by
P( X < t ) = F(t) (F inversible)
1. Generate a sample U from U[0,1]
2. Set X = F-1( U )
• Why does this work?
Jérémie Gallien
Solution
1. Generate a sample U from U[0,1]
2. Set X = F-1( U )
• P ( X < t ) = P( F-1(U) < t ) = P( U < F(t) ) = F(t)
F(x)
1
U[0,1]
0
F-1(
U)
x
Jérémie Gallien
Exponential Distribution
• Variable exp(l) has cdf. F(t) = 1 – exp(-lt);
F-1(y) = - log(1-y) / l
1. Generate U from U[0,1]
2. Compute X = - log(1-U ) / l
3. The distribution of X is exp(l) !
Jérémie Gallien
Class 2 Wrap-Up
1. Monte-Carlo Framework, Algorithm &
Examples
2. Crystal Ball Modeling
3. Theory of Random Number Generation
Jérémie Gallien