Calculating Risk of Cost Using Monte Carlo Simulation

with Fuzzy Parameters in Civil Engineering

Michał BętkowskiAndrzej Pownuk

Silesian University of Technology, Poland

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Risk of cost overruns

We can define risk as possibility of occurrence of loss.

There is always the difference between the planned costs and real costs.

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Direct costs (DC)

Indirect costs (IC)

Direct costs are expenses that are directly linked to the project

For example: materials, labour, equipment etc.

Other costs.

For example: management costs, cost of insurance etc.

Calculating of cost

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Estimating of direct cost (DC)

The project can be decomposed into elementsiDC

i

iDCDC

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Direct cost (DC)

DC = Cost 1 + Cost 2+ Cost 3

Cost 1 Cost 2 Cost 3

Materials Labour Equipment

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Methods of calculating of directional cost

Deterministic

Probabilistic

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Deterministic methods of calculating costs

- appearance of task is deterministic

- cost of each task is deterministic

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Calculating Risk in deterministic methods

Risk in deterministic methods is taken into account as additional constant component of cost.

(It is possible to express the risk in percent)

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Typical problems with deterministic methods of

calculating of costs

Unknown characteristics of costs (labour, whether),

- Alternative tasks, - Additional tasks.

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Probabilistic methods

Alternative tasks

Additional task

Changeable costs of tasks

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Alternative tasks

hamburger

Cola Beer

Begin

End

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Additional task

hamburger

Cola Beer

chips

End

Begin

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Changeable costs of tasks

Old car is cheaper than the new one

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Probabilistic definition of risk

0,0, 1 TTTT ccPccPR

- real cost (random variable)

- fixed cost

Tc

0,Tc

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Probabilistic definition of risk

Probability density function Tc cf

T

Cumulative distribution function

0,0, TTTc ccPcT

0,0, 1 TT ccR Risk of cost

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Beta Pert distribution

6

4 pmo ccccE

11

, 1

xxxf

4

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Beta Pert distribution

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Beta Pert distribution

op

om

cc

cc

4 4

po

p

po cc

cxf

ccxf ,,

1)(

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Alternative tasks

Cost 1

Cost 2 Cost 3

1p 2p

121 ppTotal cost = Cost 1 + Cost 2

orTotal cost = Cost 1 + Cost 3

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Existing software

- Pert Master, - Risk, - MS Project

Etc.

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Advantages of probabilistic methods

- Express realistic character of the realization of the process.

- Using probabilistic methods it is possible describe random parameters (unpredictable weather, material cost, inaccurate materials estimates)

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Limitation of pure probabilistic

methods

- unique character of many civil engineering project

- different conditions of the realization (weather, geological conditions, geographical region etc.)

Because of that we do not know reliable statistical data

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Main problem

It is very difficult to obtain exact values of probabilistic characteristics of the structure

For example: m, σ etc.

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Basic assumption According to many experiments parameters of the system can be

characterized by typical probability distribution of cost (if we know the data):

- Normal distribution- Beta-Pert distribution- Lognormal distribution etc.

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However we do not know the parameters

x

xf1

xf 2 xfi

Probability density function of costs

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What we know? We know deterministic values of costs

from the catalogue

We have expert knowledge about particular cost (i.e. what happened usually)

Sometimes we have some experimental data

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Information from experts Lower bound Upper bound Most probable

cost

oc

ocpc

mc

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If we have many experts then we can get more information

Lower bound

Upper bound

Most probable cost

pp cc ,

mm cc ,

oo cc ,

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Fuzzy numbers (clouds)

We can also ask experts about confidence intervals

for different probability levels(alpha-cuts, degree of membership)

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Confidence intervals

},,ˆ:{1 ,, ooo cccP hhh

,oc

,oc

1

oc

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Calculation of fuzzy numbersusing the data

dPXmidxc ˆ

PXmidxc ˆ

xxXP ,ˆ:1

xx

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Advantages of fuzzy sets description (clouds)

In order to define the worst case (intervals) we do not need many information

Confidence intervals can be defined for set valued data (random sets)

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Dependency problem

It is not a good idea to convert interval probability density function to interval cumulative distribution function(overestimation problem)

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Dependency problem

ba,x ,1

ba,x ,0

1

ab

xf X

ba,x ,1

ba,x ,0

2

ab

xf X

bbaxabab

xba

baaxab

ax

bax

xf XX

2,,1

,2,2

2,2,0

2

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P-box method consider all possible probability distribution i.e. some of them do not corresponds to any parameters a, b

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Dependency problem

X

Envelop does not correspond to any combination of parameters

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Probability density function with fuzzy parameters

111),,(

xxxf

omp ccc ,, omp ccc ,,

ompompomp ccccccxfcccxf ,,,,,,),,,(

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Application of extension principle

0,Tc RRR ,ˆ

,0 ,0ˆ ˆ 1 , :

TT c TR c c h h h

RRRRsupR ˆ,:

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- Risk for particular cost

h,0,Tc - Cumulative distribution function

npnm

nopmo cccccc ,,,...,,, 111h

,0 ,0ˆ ˆ 1 , :

TT c TR c c h h h

- vector of uncertain parameters

RRR ,ˆ

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Modified extension principle(clouds)

RcRRP o hhh ,,ˆ:1

RRRsupR ,:

RRR ,ˆ

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Discretization of α-cuts

iiiihhh ˆ...ˆˆˆ h

miiimi jmjjjjj hhh ,,,2,,1,,...,,, ,...,,

2121 h

imi jjj hh ˆ,...,,, 21

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Monte Carlo simulations

1 2

,0

,0 , , ,..., 1 1 , : ,..., 1,...,

i

T i m

T

c T j j j m

R c

min c j j k

h

kjjcmax

cR

mjjjTc

T

miT

i

,...,1 ,...,:,1 1,...,,,0,

0,

21

h

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Advantages of Monte Carlo method

- it is possible to get full description of probability density function of the results

- the method is able to take into account any type of uncertainty and dependency

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Graph description of the system

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Numerical results

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Numerical results

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Numerical results

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Numerical results

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Computer implementation of the algorithm

Algorithm was implemented in C++ language.

GSL library was also applied.

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Numerical data for node 0

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Description of the node

NodeNumberOfNode 0, NumberOfChildren 2, Children 1 3, Probability 0.415,IntervalProbability 0.088, xMinMin 198.766, xMinMax 206.016, xMidMin 215.688, xMidMax 219.313, xMaxMin 231.391, xMaxMax 238.641, ProbabilityGrids 3NumberOfGrid 3End

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Types of nodes - Normal distribution

(with uncertain parameters) - Beta Pert distribution

(with uncertain parameters) - constant value

- intervals (not implemented) - fuzzy numbers (not implemented)

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Sum of fuzzy number and probability density function Using presented algorithm it is possible

to calculate sum of probability density function and fuzzy number (clouds).

In calculation one can apply:- min-max extension principle (classical solution - controversial)- new extension principle(recommended, has clear interpretation based on clouds)

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Results:

Classic extension principle:- fuzzy probability

New extension principle:- fuzzy probability- fuzzy number (confidence intervals, clouds)

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One more remark about dependency problem

Due to nonlinearity alpha cat method is not always good method of transformation of confidence intervals.

Because of that we have to check some additional conditions before using this method.

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Additional information necessary for computation

ResultsXmin 820, Xmax 1120, NumberOfSimulations 10000, NumberOfClasses 20,NumberOfGrids 3, DistributionType 2End

TcTc

Tc

Tc cfT

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Conclusions

Presented method allows estimating the direct cost risk of civil engineering projects in the case when there are no credible data.

In presented algorithm the costs can be deterministic, probabilistic, fuzzy number.

It is also possible to take into account the cost which is modeledby probability density function with fuzzy parameters.

The method shows the relation between the assumed maximaldirect costs, the risk of overrun and the uncertainty of the statisticaldata.