Representing Uncertainty
by Probability and Possibility
‐ What’s the Difference?
Presentation at
Palisade 2011 Risk Conference
Amsterdam, March 29‐30, 2011
Hans Schjær‐Jacobsen
Professor, Director RD&I
Copenhagen University College of Engineering
Ballerup, Denmark
+45 4480 5030
hsj@ihk.dk
www.ihk.dk
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
1
Agenda
1. Why do we need uncertainty management?
2. Alternative representations of uncertainty
3. Some principles of New Budgeting
4. Introducing uncertainty in the cost model
5. Numerical examples
6. Resumé and perspectives
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
2
1. Why do we need uncertainty management?
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
3
Cost overruns and demand shortfalls
in urban rail
• Average cost escalation for urban rail projects is 45% in
constant prices
• For 25% of urban rail projects cost escalations are at least 60%
• Actual ridership is on average 51% lower than forecast
• For 25% of urban rail projects actual ridership is at least 68%
lower than forecast
(Flyvbjerg 2007)
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
4
2. Alternative representations of uncertainty
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
5
Two worlds of risk and uncertainty
Uncertainty
Imprecision
Ignorance
Lack of knowledge
Statistical nature
Randomness
Variability
Palisade 2011 Risk
Conference
World
Representation
and calculation
Possibility
Possibility distributions
[a; …; b]
Interval arithmetic
Global optimisation
Probability
Probability distributions
{µ; σ}
Linear approximation
Monte Carlo simulation
Copenhagen University College of
Engineering
6
Rectangular representation [a; b] and {µ; σ}
Alternative
interpretations
Possibility
distribution [a; b]
Probability
distribution {µ; σ}
1
h
h = 1/(b-a)
μ = (a+b)/2
σ2 = (b-a)2/12
0
Palisade 2011 Risk
Conference
a
Copenhagen University College of
Engineering
b
0
7
Trapezoidal representation [a; c; d; b] and {µ; σ}
Alternative
interpretations
Possibility
distribution [a; c; b]
Probability
distribution {µ; σ}
1
α
0
Palisade 2011 Risk
Conference
h
h = 2/(b-a+d-c)
μ = h((b3-d3)/(b-d)-(c3-a3)/(c-a))/6
σ2 = (3(r+2s+t)4+6(r2+t2)(r+2s+t)2
-(r2-t2)2)/(12(r+2s+t))2
α-cut
r
a
s
c
t
d
Copenhagen University College of
Engineering
b
0
8
Triangular representation [a; c; b] and {µ; σ}
Alternative
interpretations
Possibility
distribution [a; c; b]
Probability
distribution {µ; σ}
1
h
h = 2/(b-a)
μ = (a+b+c)/3
σ2 = (a2+b2+c2-ab-ac-bc)/18
α
0
Palisade 2011 Risk
Conference
α-cut
a
c
Copenhagen University College of
Engineering
b
0
9
Independent stochastic variables
Intervals
Triple estimates
{μ; σ} = {μ1; σ1} # {μ2; σ2}
[a; b] = [a1; b1] # [a2; b2]
[a; c; b] = [a1; c1; b1] # [a2; c2; b2]
μ = μ1 + μ2;
a = a1 + a2;
σ2 = σ12 + σ22
b = b1 + b2
μ = μ1 - μ2;
a = a1 - b2;
σ2 = σ12 + σ22
b = b1 - a2
μ = μ1·μ2;
a = min(a1a2, a1b2, b1a2, b1b2);
σ2 ≅ σ12·μ22 + σ22·μ12
b = max(a1a2, a1b2, b1a2, b1b2)
μ = μ1/μ2;
a = min(a1/b2, a1/a2, b1/b2, b1/a2,);
σ2 ≅ σ12/μ22 + σ22·μ12/μ24,
b = max(a1/b2, a1/a2, b1/b2, b1/a2),
if μ2 ≠ 0
if 0 ∉ [a2; b2]
Addition
Subtraction
Multiplication
Division
a = a1 + a2;
c = c1 + c2;
b = b1 + b2
a = a1 - b2;
c = c 1 - c2 ;
b = b 1 - a2
a = min(a1a2, a1b2, b1a2, b1b2);
c = c1c2;
b = max(a1a2, a1b2, b1a2, b1b2)
a = min(a1/b2, a1/a2, b1/b2, b1/a2,);
c = c1/c2;
b = max(a1/b2, a1/a2, b1/b2, b1/a2),
if 0 ∉ [a2; b2]
Table 1. Formulas for basic calculations with alternative representations of uncertain variables.
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
10
Modelling by possibility distributions
i.e. intervals, fuzzy intervals, etc.
The actual economic problem is modelled by a function Y of n uncertain
variables Y = Y(X1, X2,…, Xn).
NB: Function can be arranged in different ways.
In case of intervals
Y is calculated by means of interval arithmetic (only applicable in the simple
case) or global optimisation (applicable in the general case).
In case of triple estimates
Extreme values of Y are calculated as above.
In case of fuzzy intervals
As above, for all α‐cuts.
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
11
Modelling by probability distributions
The actual economic problem is modelled by a function Y of n independent
uncertain variables Y = Y(X1, X2,…, Xn).
Linear approximation
Y is approximated by means of a Taylor series
Y ≅ Y(μ1,…, μn) + ∂Y/∂X1∙(X1‐μ1) + ∂Y/∂X2∙(X2‐μ2) + … + ∂Y/∂Xn∙(Xn‐μn),
where ∂Y/∂Xi is the partial derivative of Y with respect to Xi calculated at (μ1,…,
μn).
The expected value is given by
E(Y) = μ = Y(μ1,…, μn).
The variance is approximated by
VAR(Y) = σ2 ≅ (∂Y/∂X1)2∙σ12 +…+ (∂Y/∂Xn)2∙σn2.
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
Monte Carlo
simulation
12
Y = X(1-X), X = [0; 1]
30
25
pdf
20
Monte Carlo simulation
X = RiskUniform(0; 1)
Y = {0,167; 0,075}
15
10
Global optimisation
X = [0; 1]
Y = [0; 0,25]
(Normalised as pdf)
5
0
0,00
0,05
0,10
0,15
0,20
0,25
Y = X(1-X)
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
13
Sum of 10 identical rectangular cost elements [7; 15]
0,06
0,05
Independent variables
Monte Carlo simulation
N{110; 7,3}
pdf
0,04
Fuzzy variables
Fuzzy arithmetic
[70; 150]
(normalized as pdf)
0,03
0,02
0,01
0,00
60
70
80
90
100
110
120
130
140
150
160
Sum
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
14
Sum of 10 identical trapezoidal cost elements [7; 9; 11; 15]
0,08
0,07
Independent variables
Monte Carlo simulation
N{106; 5,4}
0,06
pdf
0,05
Fuzzy variables
Fuzzy arithmetic
[70; 90; 110; 150]
(normalized as pdf)
0,04
0,03
0,02
0,01
0,00
70
80
90
100
110
120
130
140
150
Sum
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
15
Sum of 10 identical triangular cost elements [7; 10; 15]
0,08
0,07
Independent variables
Monte Carlo simulation
N{107; 5,2}
0,06
pdf
0,05
Fuzzy variables
Fuzzy arithmetic
[70; 100; 150]
(normalized as pdf)
0,04
0,03
0,02
0,01
0,00
70
80
90
100
110
120
130
140
150
Sum
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
16
Sum of 10 identical triangular cost elements [7; 10; 15]
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
17
Sum of 10 identical triangular cost elements [7; 10; 15]
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
18
Sum of 10 identical triangular cost elements [7; 10; 15]
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
19
Sum of 10 identical triangular cost elements [7; 10; 15]
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
20
Sum of 10 identical triangular cost elements [7; 10; 15]
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
21
Y = ∑Xi , Xi = [8; 10; 16], i = 1, …,10
0,08
0,07
Monte Carlo simulation
X = RiskTriangular(8; 10; 16)
(Uncorrelated variables)
Y = {113,3; 5,35}
0,06
pdf
0,05
Fuzzy arithmetic
X = [8; 10; 16]
Y = [80; 100; 160]
(Normalised as pdf)
0,04
Monte Carlo simulation
X = RiskTriangular(8; 10; 16)
(100% correlated variables)
Y = {113,3; 17,00}
0,03
0,02
0,01
0,00
80
90
100
110
120
130
140
150
160
Y = sum of Xi, i = 1,…,10
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
22
n
Yn = ∑ Xi, Xi = [90/n;100/n;140/n]
cdf
i =1
n=1
n = 15
n=5
Palisade 2011 Risk
Conference
n = 10
Copenhagen University College of
Engineering
23
Some observations of probability vs. possibility
• With numerically identical input variables probability results
in less numerical output uncertainty than does possibility
• Uniform probability representation is different from interval
possibility representation
• Probability uncertainty decreases with increasing analytical
complexity whereas possibility uncertainty is independent
• Possibility uncertainty corresponds to fully correlated input
probability variables
• Monte Carlo simulation does not generally produce possibility
results
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
24
3. Some principles of New Budgeting
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
25
A Danish governmental initiative
• ”Best realistic budget based on available knowledge”
• Budget control is done by standardised budgets and logging of
follow‐up results
• Risk and uncertainty management is conducted during entire
project
• Estimates of unit prices, quantities and particular risks
• Experience based supplementary budget of one third of 50%
of rough budget is allocated
• Likelihood of event multiplied by impact is not accepted
• Acceptable to incur additional cost to reduce risk and
uncertainty
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
26
The Anchor Budget
• Project with a number of activities A
• Each activity: unit price p and quantity q
• Total cost C of activities at time t = 0
• Subsequently, additional activities and costs may be
introduced
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
27
The Event Impact Matrix
• Risk events E are identified at any time t = τ
• Additional activities may be initiated
• Impacts of Risk Events on all p and q are estimated
• We keep track of accumulated cost impacts for all
individual risk events
• Impacts from interacting (co‐acting) Risk Events are
pooled
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
28
The Risk Budget
• All identified Risk Events are assumed to occur
• Resulting p, q and cost for each activity is calculated
• Total cost for project is calculated
• Deviations from Anchor Budget is calculated
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
29
Risk events modify Anchor Budget
For the i’th activity Ai, we get the modified estimated cost Ciτ at time τ
Ciτ =
= (pi + ∆pi1 + ∆pi2 + … + ∆pij + … + ∆pim) · (qi + ∆qi1 + ∆qi2 + … + ∆qij + … + ∆qim)
= Ci + pi · (∆qi1 + ∆qi2 + … + ∆qij + … + ∆qim) + (∆pi1 + ∆pi2 + … + ∆pij + … + ∆pim) · qi
+ ∆pi1 · (∆qi1 + ∆qi2 + … + ∆qij + … + ∆qim)
+ ∆pi2 · (∆qi1 + ∆qi2 + … + ∆qij + … + ∆qim)
+…
+ ∆pij · (∆qi1 + ∆qi2 + … + ∆qij + … + ∆qim)
+…
+ ∆pim · (∆qi1 + ∆qi2 + … + ∆qij + … + ∆qim)
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
30
Convenient set‐up for calculations
Event Impact Matrix at t = τ
Anchor Budget
at t = 0
Activity
p
q
Cost
E1
∆p
E2
∆q
∆Cost
∆p
E3
∆q
∆Cost
∆p12
∆q12
∆p22
∆q22
∆p32
∆q32
∆C12τ
∆C22τ
∆C32τ
∆p
Interaction
∆Cost
∆q
∆Cost
∆p13
∆q13
∆c1
∆p23
∆q23
∆p33
∆q33
∆C13τ
∆C23τ
∆C33τ
Risk Budget
at t = τ
Sum
∆Cost
p
p1τ
p2τ
p3τ
q1
∆c3τ
∆C1τ
∆C2τ
∆C3τ
Cost
A1
p1
q1
C1
∆p11
∆q11
A2
p2
q2
C2
∆p21
∆q21
A3
p3
q3
C3
∆p31
∆q31
A4
p4
q4
C4
∆p41
∆q41
∆C41τ
∆p42
∆q42
∆C42τ
∆p43
∆q43
∆C43τ
∆c4τ
∆C4τ
p4τ
q4τ
C4τ
A5
p5
q5
C5
∆p51
∆q51
∆C51τ
∆p52
∆q52
∆C52τ
∆p53
∆q53
∆C53τ
∆c5τ
∆C5τ
p5τ
q5τ
C5τ
∆C·3τ
∆cτ
∆Cτ
Sum
C
∆C·1τ
∆C·2τ
τ
q
∆C11τ
∆C21τ
∆C31τ
∆c2τ
τ
C1τ
q2τ
C2τ
q3τ
C3τ
Cτ
Table 1. Convenient set-up for calculations, n = 5, m = 3.
Anchor
Budget
Palisade 2011 Risk
Conference
Event Impact
Matrix
Copenhagen University College of
Engineering
Risk
Budget
31
Event Impact Matrix at t = τ
Anchor Budget
at t = 0
Activity
p
q
Cost
A1
100
1,000 100,000
A2
50
10,000 500,000
A3
200
500
100,000
A4
1,000
150
150,000
E1
E2
∆p
∆q
∆Cost
20
100
32,000
∆p
E3
∆q
∆Cost
25
2,500
30
∆Cost
∆Cost
p
500
35,000
120
1,125 135,000
–10,000
50
9,800 490,000
15,000
230
15,000
10
10,000
150
850,000
∆q
–200 –10,000
A5
Sum
∆p
47,000
300
12,500
45,000
35,000
Risk Budget
at t = τ
Interaction
∆Cost
500
Sum
q
Cost
500
115,000
10,000 1,000
160
160,000
45,000
300
45,000
95,000
150
945,000
Table 2. Numerical test example without event and impact uncertainty.
A1 E1: ∆Cost = (p+∆p)·(q+∆q) - Cost = p·∆q + ∆p·q + ∆p·∆q = 100·100 + 20·1,000 + 20·100 = 32,000
A1 E2: ∆Cost = (p+∆p)·(q+∆q) - Cost = p·∆q + ∆p·q + ∆p·∆q = 100·25 + 0·1,000 + 0·25 = 2,500
A1 Interaction: ∆Cost = 20 · 25 = 500
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
32
4. Introducing uncertainty in the cost model
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
33
Uncertain impacts and likelihoods
• Uncertain impact of Risk Events
‐ Uncertainty for all p and q from Anchor Budget
‐ Additional activities may become necessary
‐ How to estimate and represent uncertainty?
‐ Calculate uncertain Risk Budget
• Likelihood of Risk Events occurring
‐ Estimate probabilities of Risk Events occurring
‐ Construct probability distribution of total project cost
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
34
Triangular representation [a; c; b]
Uncertain impact on unit price p = 100: ∆p = [18; 20; 25]
Uncertain impact on quantity q = 1000: ∆q = [95; 100; 125]
Uncertain impact on cost p∙q
∆Cost = (p+∆p)∙(q+∆q) − p∙q
= p∙∆q + ∆p∙q + ∆p∙∆q
∆Cost = [9,500; 10,000; 12,500] + [18,000; 20,000; 25,000] +
[1,710; 2,000; 3,125]
= [29,210; 32,000; 40,625]
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
35
Triangular representation {µ; σ}
Uncertain impact on unit price p = 100: ∆p = {21.0; 1.47}
Uncertain impact on quantity q = 1000: ∆q = {106.7; 6.56}
Uncertain impact on cost p∙q
∆Cost = (p+∆p)∙(q+∆q) − p∙q
= p∙∆q + ∆p∙q + ∆p∙∆q
∆Cost = {33,907; 1,802} (by Monte Carlo simulation)
min = 29,551, max = 39,966
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
36
5. Numerical examples
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
37
Event Impact Matrix at t = τ
Activity
A1
E1
E2
∆p
∆q
∆Cost
[18; 25]
[95; 125]
[29,210; 40,625]
∆p
E3
∆q
∆Cost
[21; 31]
[2,100; 3,100]
∆p
A2
A3
∆q
∆Cost
[-210; -175] [-10,500; -8,750]
[25; 45]
[12,500; 22,500]
A4
[7,000; 16,000]
A5
∆Cost
∆Cost
[378; 775]
[31,688; 44,500]
[-10,500, -8,750]
[7,000; 16,000]
[145; 165]
[41,710; 63,125]
Sum
[12,500; 22,500]
[7; 16]
Sum
Interaction
[9,100; 19,100]
[280; 350] [40,600; 57,750]
[40,600; 57,750]
[30,100; 49,000] [378; 775] [81,288; 132,000]
Table 3. Event Impact Matrix using interval uncertainty representation [a; b]. (Anchor Budget of Table 2).
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
38
Risk Budget at t = τ
Activity
p
q
Cost
A1
[118; 125]
[1,116; 1,156]
[131,688; 144,500]
A2
50
[9,790; 9,825]
[489,500; 491,250]
A3
[225; 245]
500
[112,500; 122,500]
A4
1,000
[157; 166]
[157,000; 166,000]
A5
[145; 165]
[280; 350]
[40,600; 57,750]
Sum
[931,288; 982,000]
Table 4. Risk Budget using interval uncertainty representation [a; b].
(Anchor Budget of Table 2 and Event Impact Matrix of Table 3).
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
39
Risk Budget at t = τ
Activity
p
q
Cost
A1
[118; 120; 125]
[1,116; 1,125; 1,156] [131,688; 135,000; 144,500]
A2
50
[9,790; 9,800; 9,825] [489,500; 490,000; 491,250]
A3
[225; 230; 245]
500
[112,500; 115,000; 122;500]
A4
1,000
[157; 160; 166]
[157,000; 160,000; 166,000]
A5
[145; 150; 165]
[280; 200; 350]
[40,600; 54,000; 57,750]
Sum
[931,288; 945,000; 982,000]
Table 5. Risk Budget using triple estimate uncertainty representation [a; c; b].
(Combination of Table 2 and 4).
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
40
Event Impact Matrix at t = τ
Activity
E1
∆p
A1
∆q
E2
∆Cost
{21.0; 1.47} {106.7; 6.56} {33,907; 1,802}
∆p
E3
∆q
∆Cost
{25.7; 2.05}
{2,567; 205}
∆p
A2
A3
∆q
{195; 7.36}
{33.3; 4.25}
∆Cost
{-9,750; 368}
{16,667; 2,125}
A4
A5
∆Cost
∆Cost
{539; 57.9}
{37,012; 1,851}
{-9,750; 368}
{11,000; 1,871}
{153.3; 4.25} {310; 14.7} {47,534; 2,619}
{50,573; 2,807}
Sum
{16,667; 2,125}
{11.0; 1.87} {11,000; 1,871}
Sum
Interaction
{13,567; 1,884}
{47,534; 2,619}
{37,784; 2,643} {539; 57.9} {102,462; 4,305}
Table 6. Event Impact Matrix using triangular probability input distributions {µ; σ}. (Anchor Budget of Table 2).
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
41
Risk Budget at t = τ
Activity
p
q
Cost
A1
{121.0; 1.47}
{1,132; 6.87}
{137,012; 1,851}
A2
50
{9,805; 7.36}
{490,250; 368}
A3
{233.3; 4.25}
500
{116,667; 2,125}
A4
1,000
{161.0; 1.87}
{161,000; 1,871}
A5
{153.3; 4.25}
{310.0; 14.7}
{47,534; 2,619}
Sum
{952,462; 4,305}
Table 7. Risk Budget using triangular probability input distributions {µ; σ}.
(Anchor Budget of Table 2 and Event Impact Matrix of Table 6).
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
42
Probabilities of
combinations
pr1=0.6 pr1=0.3 pr3=0.2 pdf
cdf
E1
E2
E3
Cost Cτ
Cost Cτ
[a; c; b]
{µ; σ}
850,000
850,000
no
no
no
0.224
0.224
no
yes
no
0.096
0.320
[859,100; 862,500; 869,100] {863,567; 1,882}
no
no
yes
0.056
0,376
[880,100; 885,000; 899,000] {887,783; 2,642}
yes
no
no
0.336
0.712
[891,710; 897,000; 913,125] {900,573; 2,810}
no
yes
yes
0.024
0.736
[889,200; 897,500; 918,100] {901,350; 3,251}
yes
yes
no
0.144
0.880
[901,188; 910,000; 933,000] {914,679; 3,380}
yes
no
yes
0.084
0.964
[921,810; 932,000; 962,125] {938,356; 3,853}
yes
yes
yes
0.036
1.000
[931,288; 945,000; 982,000] {952,462; 4,305}
Table 8. Distributions of Cτ for representations by triple estimates and probabilities.
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
43
Distribution of cost Cτ (triple estimates, Table 8)
1.0
E1, E2, E3
E1, E3
E1, E2
cdf
E2, E3
E1
E2
0.0
800,000
Palisade 2011 Risk
Conference
E3
900,000
Copenhagen University College of
Engineering
Cost
1,000,000
44
Triangular probability Monte Carlo simulation of Risk Budget.
(Anchor Budget of Table 2, Event Impact Matrix of Table 6).
Uncorrelated
input parameters
Uncorrelated
input parameters
Most poss.:
945,000
100% correlated
input parameters
Max: 982,000
Min: 931,288
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
45
6. Resumé and perspectives
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
46
Resumé and perspectives
• Anchor Budget, unit prices, quantities
• Impacts of individual Risk Events
• Impacts of co‐acting Risk Events
• Impacts on individual Activities
• Uncertainty by probabilities and possibilities
• Under‐ or overestimating uncertain impacts
• Research into practical applications
End
of
presentation
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
47
Back to
Table 6
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
48
Back to
Table 6
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
49
Back to
Table 6
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
50
Back to
Table 6
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
51
Back to
Table 6
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
52
Back to
Table 7
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
53
Back to
Table 7
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
54
Back to
Table 7
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
55
Back to
Table 7
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
56
Thank You!
Palisade 2011 Risk
Conference
Copenhagen University College of
Engineering
57