Representing Uncertainty by Probability and Possibility What s the Difference?

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

[email protected]

www.ihk.dk

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

1. Why do we need uncertainty management?

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)

2.  Alternative representations of uncertainty

Possibility

Probability

Possibility distributions [a; …; b]

Interval arithmetic Global optimisation

Probability distributions {µ; σ}

Linear approximation Monte Carlo simulation

Representation and calculation Uncertainty

Imprecision Ignorance

Lack of knowledge

Statistical nature Randomness

Variability

World

1

Possibility

distribution [a; b]

Probability distribution {µ; σ}

h = 1/(b-a) μ = (a+b)/2 σ ² = (b-a) ² /12

h

Alternative interpretations

Rectangular representation [a; b] and {µ; σ}

1 h

0 0

α ^α-cut

Possibility

distribution [a; c; b]

Probability distribution {µ; σ}

Alternative interpretations

h = 2/(b-a+d-c)

μ = h((b ³ -d ³ )/(b-d)-(c ³ -a ³ )/(c-a))/6 σ ² = (3(r+2s+t) ⁴ +6(r ² +t ² )(r+2s+t) ² -(r ² -t ² ) ² )/(12(r+2s+t)) ²

r s t

1

h = 2/(b-a) μ = (a+b+c)/3

σ ² = (a ² +b ² +c ² -ab-ac-bc)/18

h

α ^α-cut

Possibility

distribution [a; c; b]

Triangular representation [a; c; b] and {µ; σ}

Probability distribution {µ; σ}

Alternative

interpretations

{μ; σ} = {μ

; σ

} # {μ

; σ

} [a; b] = [a

; b

] # [a

; b

] [a; c; b] = [a

; c

; b

] # [a

; c

; b

]

Addition μ = μ

+ μ

; σ

= σ

+ σ

a = a

+ a

; b = b

+ b

a = a

+ a

; c = c

+ c

; b = b

+ b

Subtraction μ = μ

- μ

; σ

= σ

+ σ

a = a

- b

; b = b

- a

a = a

- b

; c = c

- c

; b = b

- a

Multiplication μ = μ

·μ

; σ

≅ σ

·μ

+ σ

·μ

a = min(a

a

, a

b

, b

a

, b

b

);

b = max(a

a

, a

b

, b

a

, b

b

)

a = min(a

a

, a

b

, b

a

, b

b

);

c = c

c

;

b = max(a

a

, a

b

, b

a

, b

b

)

Division

μ = μ

/μ

;

σ

≅ σ

/μ

+ σ

·μ

/μ

, if μ

≠ 0

a = min(a

/b

, a

/a

, b

/b

, b

/a

,);

b = max(a

/b

, a

/a

, b

/b

, b

/a

), if 0 ∉ [a

; b

]

a = min(a

/b

, a

/a

, b

/b

, b

/a

,);

c = c

/c

;

b = max(a

/b

, a

/a

, b

/b

, b

/a

), if 0 ∉ [a

; b

]

Table 1. Formulas for basic calculations with alternative representations of uncertain variables.

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(X ₁ , X ₂ ,…, X _n ).

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.

The actual economic problem is modelled by a function Y of n independent  uncertain variables Y = Y(X ₁ , X ₂ ,…, X _n ).

Linear approximation

Y is approximated by means of a Taylor series

Y ≅ Y(μ ₁ ,…, μ _n ) + ∂Y/∂X ₁ ∙(X ₁ ‐μ ₁ ) + ∂Y/∂X ₂ ∙(X ₂ ‐μ ₂ ) + … + ∂Y/∂X _n ∙(X _n ‐μ _n ),

where ∂Y/∂X _i is the partial derivative of Y with respect to X _i calculated at (μ ₁ ,…,  μ _n ).

The expected value is given by E(Y) = μ = Y(μ ₁ ,…, μ _n ).

The variance is approximated by

VAR(Y) = σ ² ≅ (∂Y/∂X ) ² ∙σ ² +…+ (∂Y/∂X ) ² ∙σ ² .

Monte Carlo

simulation

0 5 10 15 20 25 30

0,00 0,05 0,10 0,15 0,20 0,25

Y = X(1-X)

pdf

Global optimisation X = [0; 1]

Y = [0; 0,25]

(Normalised as pdf)

Monte Carlo simulation X = RiskUniform(0; 1)

Y = {0,167; 0,075}

Y = X(1-X), X = [0; 1]

0,00 0,01 0,02 0,03 0,04 0,05 0,06

60 70 80 90 100 110 120 130 140 150 160

pdf

Independent variables Monte Carlo simulation

N{110; 7,3}

Fuzzy variables Fuzzy arithmetic

[70; 150]

(normalized as pdf)

Sum of 10 identical trapezoidal cost elements [7; 9; 11; 15]

0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08

70 80 90 100 110 120 130 140 150

pdf

Independent variables Monte Carlo simulation

N{106; 5,4}

Fuzzy variables Fuzzy arithmetic [70; 90; 110; 150]

(normalized as pdf)

0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08

70 80 90 100 110 120 130 140 150

pdf Fuzzy variables

Fuzzy arithmetic [70; 100; 150]

(normalized as pdf) Independent variables

Monte Carlo simulation

N{107; 5,2}

Sum of 10 identical triangular cost elements [7; 10; 15]

Sum of 10 identical triangular cost elements [7; 10; 15]

Sum of 10 identical triangular cost elements [7; 10; 15]

0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08

80 90 100 110 120 130 140 150 160

Y = sum of X

, i = 1,…,10

pdf

Monte Carlo simulation X = RiskTriangular(8; 10; 16)

(100% correlated variables) Y = {113,3; 17,00}

Fuzzy arithmetic X = [8; 10; 16]

Y = [80; 100; 160]

(Normalised as pdf)

Monte Carlo simulation X = RiskTriangular(8; 10; 16)

(Uncorrelated variables) Y = {113,3; 5,35}

cdf

n

Y _n = ∑ X _i , X _i = [90/n;100/n;140/n]

i =1

n = 1

n = 5

n = 10

n = 15

• 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

3. Some principles of New Budgeting

• ”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

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

• 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

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

For the i’th activity A _i , we get the modified estimated cost C _i ^τ at time τ C _i ^τ =

= (p _i + ∆p _i1 + ∆p _i2 + … + ∆p _ij + … + ∆p _im ) · (q _i + ∆q _i1 + ∆q _i2 + … + ∆q _ij + … + ∆q _im )

= C _i + p _i · (∆q _i1 + ∆q _i2 + … + ∆q _ij + … + ∆q _im ) + (∆p _i1 + ∆p _i2 + … + ∆p _ij + … + ∆p _im ) · q _i + ∆p _i1 · (∆q _i1 + ∆q _i2 + … + ∆q _ij + … + ∆q _im )

+ ∆p _i2 · (∆q _i1 + ∆q _i2 + … + ∆q _ij + … + ∆q _im ) + …

+ ∆p _ij · (∆q _i1 + ∆q _i2 + … + ∆q _ij + … + ∆q _im ) + …

+ ∆p _im · (∆q _i1 + ∆q _i2 + … + ∆q _ij + … + ∆q _im )

Convenient set‐up for calculations

Anchor  Budget

Event Impact  Matrix

Table 1. Convenient set-up for calculations, n = 5, m = 3.

Risk  Budget

Event Impact Matrix at t = τ Anchor Budget

at t = 0 E

E

E

Inter-

action Sum

Risk Budget at t = τ Acti-

vity

p q Cost ∆p ∆q ∆Cost ∆p ∆q ∆Cost ∆p ∆q ∆Cost ∆Cost ∆Cost p q Cost A

p

q

C

∆p

∆q

∆C

∆p

∆q

∆C

∆p

∆q

∆C

∆c

∆C

p

q

C

A

p

q

C

∆p

∆q

∆C

∆p

∆q

∆C

∆p

∆q

∆C

∆c

∆C

p

q

C

A

p

q

C

∆p

∆q

∆C

∆p

∆q

∆C

∆p

∆q

∆C

∆c

∆C

p

q

C

A

p

q

C

∆p

∆q

∆C

∆p

∆q

∆C

∆p

∆q

∆C

∆c

∆C

p

q

C

A

p

q

C

∆p

∆q

∆C

∆p

∆q

∆C

∆p

∆q

∆C

∆c

∆C

p

q

C

Sum C ∆C

∆C

∆C

∆c

∆C

C

Anchor Budget at t = 0

E

E

E

Inter-

action Sum

Risk Budget at t = τ Acti-

vity

p q Cost ∆p ∆q ∆Cost ∆p ∆q ∆Cost ∆p ∆q ∆Cost ∆Cost ∆Cost p q Cost A

100 1,000 100,000 20 100 32,000 25 2,500 500 35,000 120 1,125 135,000 A

50 10,000 500,000 –200 –10,000 –10,000 50 9,800 490,000 A

200 500 100,000 30 15,000 15,000 230 500 115,000 A

1,000 150 150,000 10 10,000 10,000 1,000 160 160,000

A

150 300 45,000 45,000 150 300 45,000

Sum 850,000 47,000 12,500 35,000 500 95,000 945,000

Table 2. Numerical test example without event and impact uncertainty.

A

E

: ∆Cost = (p+∆p)·(q+∆q) - Cost = p·∆q + ∆p·q + ∆p·∆q = 100·100 + 20·1,000 + 20·100 = 32,000 A

E

: ∆Cost = (p+∆p)·(q+∆q) - Cost = p·∆q + ∆p·q + ∆p·∆q = 100·25 + 0·1,000 + 0·25 = 2,500

A

Interaction: ∆Cost = 20 · 25 = 500

4.  Introducing uncertainty in the cost model

• 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

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]

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

5.  Numerical examples

E₁ E₂ E₃ Interaction Sum Acti-

vity

∆p ∆q ∆Cost ∆p ∆q ∆Cost ∆p ∆q ∆Cost ∆Cost ∆Cost A₁ [18; 25] [95; 125] [29,210; 40,625] [21; 31] [2,100; 3,100] [378; 775] [31,688; 44,500]

A₂ [-210; -175] [-10,500; -8,750] [-10,500, -8,750]

A₃ [25; 45] [12,500; 22,500] [12,500; 22,500]

A₄ [7; 16] [7,000; 16,000] [7,000; 16,000]

A₅ [145; 165] [280; 350] [40,600; 57,750] [40,600; 57,750]

Sum [41,710; 63,125] [9,100; 19,100] [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).

Risk Budget at t = τ Acti-

vity

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).

Acti- vity

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).

Event Impact Matrix at t = τ

E₁ E₂ E₃ Interaction Sum

Acti- vity

∆p ∆q ∆Cost ∆p ∆q ∆Cost ∆p ∆q ∆Cost ∆Cost ∆Cost A1 {21.0; 1.47} {106.7; 6.56} {33,907; 1,802} {25.7; 2.05} {2,567; 205} {539; 57.9} {37,012; 1,851}

A₂ {195; 7.36} {-9,750; 368} {-9,750; 368}

A₃ {33.3; 4.25} {16,667; 2,125} {16,667; 2,125}

A₄ {11.0; 1.87} {11,000; 1,871} {11,000; 1,871}

A₅ {153.3; 4.25} {310; 14.7} {47,534; 2,619} {47,534; 2,619}

Sum {50,573; 2,807} {13,567; 1,884} {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).

vity

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).

E

E

E

Probabilities of

combinations Cost C

Cost C

pr

=0.6 pr

=0.3 pr

=0.2 pdf cdf [a; c; b] {µ; σ}

no no no 0.224 0.224 850,000 850,000 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 C

for representations by triple estimates and probabilities.

1,000,000 900,000

800,000 1.0

0.0

E

E

E

E

, E

E

, E

E

, E

E

, E

, E

cdf

Uncorrelated input parameters

100% correlated input parameters

Triangular probability Monte Carlo simulation of Risk Budget.

(Anchor Budget of Table 2, Event Impact Matrix of Table 6).

Max: 982,000 Uncorrelated

input parameters

Min: 931,288

Most poss.:

945,000

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

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