calculating risk of cost using monte carlo simulation with fuzzy parameters in civil engineering...
TRANSCRIPT
![Page 1: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/1.jpg)
Calculating Risk of Cost Using Monte Carlo Simulation
with Fuzzy Parameters in Civil Engineering
Michał BętkowskiAndrzej Pownuk
Silesian University of Technology, Poland
![Page 2: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/2.jpg)
2/56
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.
![Page 3: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/3.jpg)
3/56
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
![Page 4: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/4.jpg)
4/56
Estimating of direct cost (DC)
The project can be decomposed into elementsiDC
i
iDCDC
![Page 5: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/5.jpg)
5/56
Direct cost (DC)
DC = Cost 1 + Cost 2+ Cost 3
Cost 1 Cost 2 Cost 3
Materials Labour Equipment
![Page 6: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/6.jpg)
6/56
Methods of calculating of directional cost
Deterministic
Probabilistic
![Page 7: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/7.jpg)
7/56
Deterministic methods of calculating costs
- appearance of task is deterministic
- cost of each task is deterministic
![Page 8: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/8.jpg)
8/56
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)
![Page 9: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/9.jpg)
9/56
Typical problems with deterministic methods of
calculating of costs
Unknown characteristics of costs (labour, whether),
- Alternative tasks, - Additional tasks.
![Page 10: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/10.jpg)
10/56
Probabilistic methods
Alternative tasks
Additional task
Changeable costs of tasks
![Page 11: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/11.jpg)
11/56
Alternative tasks
hamburger
Cola Beer
Begin
End
![Page 12: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/12.jpg)
12/56
Additional task
hamburger
Cola Beer
chips
End
Begin
![Page 13: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/13.jpg)
13/56
Changeable costs of tasks
Old car is cheaper than the new one
![Page 14: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/14.jpg)
14/56
Probabilistic definition of risk
0,0, 1 TTTT ccPccPR
- real cost (random variable)
- fixed cost
Tc
0,Tc
![Page 15: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/15.jpg)
15/56
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
![Page 16: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/16.jpg)
16/56
Beta Pert distribution
6
4 pmo ccccE
11
, 1
xxxf
4
![Page 17: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/17.jpg)
17/56
Beta Pert distribution
![Page 18: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/18.jpg)
18/56
Beta Pert distribution
op
om
cc
cc
4 4
po
p
po cc
cxf
ccxf ,,
1)(
![Page 19: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/19.jpg)
19/56
Alternative tasks
Cost 1
Cost 2 Cost 3
1p 2p
121 ppTotal cost = Cost 1 + Cost 2
orTotal cost = Cost 1 + Cost 3
![Page 20: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/20.jpg)
20/56
Existing software
- Pert Master, - Risk, - MS Project
Etc.
![Page 21: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/21.jpg)
21/56
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)
![Page 22: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/22.jpg)
22/56
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
![Page 23: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/23.jpg)
23/56
Main problem
It is very difficult to obtain exact values of probabilistic characteristics of the structure
For example: m, σ etc.
![Page 24: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/24.jpg)
24/56
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.
![Page 25: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/25.jpg)
25/56
However we do not know the parameters
x
xf1
xf 2 xfi
Probability density function of costs
![Page 26: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/26.jpg)
26/56
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
![Page 27: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/27.jpg)
27/56
Information from experts Lower bound Upper bound Most probable
cost
oc
ocpc
mc
![Page 28: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/28.jpg)
28/56
If we have many experts then we can get more information
Lower bound
Upper bound
Most probable cost
pp cc ,
mm cc ,
oo cc ,
![Page 29: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/29.jpg)
29/56
Fuzzy numbers (clouds)
We can also ask experts about confidence intervals
for different probability levels(alpha-cuts, degree of membership)
![Page 30: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/30.jpg)
30/56
Confidence intervals
},,ˆ:{1 ,, ooo cccP hhh
,oc
,oc
1
oc
![Page 31: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/31.jpg)
31/56
Calculation of fuzzy numbersusing the data
dPXmidxc ˆ
PXmidxc ˆ
xxXP ,ˆ:1
xx
![Page 32: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/32.jpg)
32/56
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)
![Page 33: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/33.jpg)
33/56
Dependency problem
It is not a good idea to convert interval probability density function to interval cumulative distribution function(overestimation problem)
![Page 34: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/34.jpg)
34/56
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
221
P-box method consider all possible probability distribution i.e. some of them do not corresponds to any parameters a, b
![Page 35: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/35.jpg)
35/56
Dependency problem
X
Envelop does not correspond to any combination of parameters
![Page 36: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/36.jpg)
36/56
Probability density function with fuzzy parameters
111),,(
xxxf
omp ccc ,, omp ccc ,,
ompompomp ccccccxfcccxf ,,,,,,),,,(
![Page 37: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/37.jpg)
37/56
Application of extension principle
0,Tc RRR ,ˆ
,0 ,0ˆ ˆ 1 , :
TT c TR c c h h h
RRRRsupR ˆ,:
![Page 38: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/38.jpg)
38/56
- 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
![Page 39: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/39.jpg)
39/56
Modified extension principle(clouds)
RcRRP o hhh ,,ˆ:1
RRRsupR ,:
RRR ,ˆ
![Page 40: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/40.jpg)
40/56
Discretization of α-cuts
iiiihhh ˆ...ˆˆˆ h
miiimi jmjjjjj hhh ,,,2,,1,,...,,, ,...,,
2121 h
imi jjj hh ˆ,...,,, 21
![Page 41: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/41.jpg)
41/56
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
![Page 42: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/42.jpg)
42/56
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
![Page 43: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/43.jpg)
43/56
Graph description of the system
![Page 44: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/44.jpg)
44/56
Numerical results
![Page 45: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/45.jpg)
45/56
Numerical results
![Page 46: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/46.jpg)
46/56
Numerical results
![Page 47: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/47.jpg)
47/56
Numerical results
![Page 48: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/48.jpg)
48/56
Computer implementation of the algorithm
Algorithm was implemented in C++ language.
GSL library was also applied.
![Page 49: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/49.jpg)
49/56
Numerical data for node 0
![Page 50: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/50.jpg)
50/56
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
![Page 51: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/51.jpg)
51/56
Types of nodes - Normal distribution
(with uncertain parameters) - Beta Pert distribution
(with uncertain parameters) - constant value
- intervals (not implemented) - fuzzy numbers (not implemented)
![Page 52: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/52.jpg)
52/56
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)
![Page 53: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/53.jpg)
53/56
Results:
Classic extension principle:- fuzzy probability
New extension principle:- fuzzy probability- fuzzy number (confidence intervals, clouds)
![Page 54: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/54.jpg)
54/56
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.
![Page 55: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/55.jpg)
55/56
Additional information necessary for computation
ResultsXmin 820, Xmax 1120, NumberOfSimulations 10000, NumberOfClasses 20,NumberOfGrids 3, DistributionType 2End
TcTc
Tc
Tc cfT
![Page 56: Calculating Risk of Cost Using Monte Carlo Simulation with Fuzzy Parameters in Civil Engineering Michał Bętkowski Andrzej Pownuk Silesian University of](https://reader035.vdocument.in/reader035/viewer/2022062722/56649f395503460f94c55d76/html5/thumbnails/56.jpg)
56/56
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.