AP08_1.vp


1 Introduction

A simulation of construction activities on the basis of pro-
duction rate makes it possible to monitor the reliability of the
expected time schedule and the total costs. The input param-
eters are production rate, scope of the work, the time sched-
ule, bonding conditions, maximum and minimum deviations
from the scope of the work, and the production rate.

The simulated model can be used at many levels of project
management. Clients are able to make decisions about imple-
menting their intentions; competitors can assess a bid price;
building contractors can make a detailed calculation of the
costs and the time schedule of construction activities, and they
can optimize the construction process.

The simulation model stems form the research of Haas,
Hajek [4] and then Beran, Dlask [1] and [2], which has been
carried out at the Faculty of Civil Engineering, Czech Techni-
cal University in Prague during the last ten years.

2 Definition of the problem
Within the framework of initiating a simulation of a build-

ing project, it is necessary to define the problem as such. The
application obtains input data by means of the Module of input
data, which defines the particular construction activities, and
the volume Q of these particular construction activities is ex-
pressed in physical or financial units, its production rate v and
bonding conditions � Dconnection activities linking particular
activities. TABproject characterizes the calculation as a meta
problem called Dynamic progress chart (flow-sheet). N generally
characterizes the sequential networks Ni [2]. The set expres-
sion is as follows:
TAB

D

project

connection activitie

�

� �N f risk riski D Q v( , ,� �� �s ) , , ,i � 1 �
(1)

where i are partial processes and D is the set of activity dura-
tions, while risk influence is a conditioned externality (see Ta-

30 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 48  No. 1/2008

Dynamic Simulations in Cost and Time
Estimation of the Construction Process
V. Beran, E. Hromada

This paper describes a model which is able to simulate the costs and the duration of construction for a building project. The model predicts the
set of expected costs and the duration of the project depending on input parameters such as production rate, scope of the work, the time
schedule, bonding conditions, maximum and minimum deviations from the scope of the work, and the production rate. Clients are able to
make proper decisions concerning the time and cost schedules of their investments.

Keywords: simulation, time scheduling, cost scheduling, reliability and risk.

Key:
… Noncritical activity
… Critical activity
… Total reserve
… Waiting for activity
… Waiting for activity – on critical way

Table 1: Dynamic progress chart of a building project



ble 1). The notation is supplement by conditionality of the
breach of the assumed input parameters – the scope of the
work and the production rate.

A practical solution of the calculation according to the dy-
namic progress chart (1) is based on inputting the work vol-
ume, the production rate and a time schedule of the particu-
lar activities. The time duration in the dynamic progress chart
is calculated as the quotient of quantities Q and v, or more
precisely D Q vi i i� . The input data included in the Module of
input data in the Connection activities sheet defines the bonding
conditions among the particular production activities. The
Deviations of project parameters 1 and 2 sheet contain input data
about the minimum / maximum deviations of the scope of the
work and the production rate of the particular activities,
based on the expected parameters of the building process [6].

3 Solution and an example
On the basis of the Excel VBA application, the algorithm

enable us to calculate an instant dynamic progress chart of the
building project, including a time schedule of resources in
terms of expression (1), the dynamic progress chart is differ-
ences calculated on the basis of a common progress chart.
The calculation is based on the production rates and the indi-
vidual activities, which are described in columns Start and End
(Tab. 1), which present the links between the individual ac-
tivities. It practically represents the relations between the
declared function f risk risk( , , )Q v �Dconnection activities from
expression (1) and the composition of the task as a consecu-
tive process on the basis of the time duration of the individual
processes N [D] [1, 2, 5].

The dynamic progress chart creates a comprehensive me-
thodically uniform model. The outputs of the model include
information about the deadlines for the start and the end of
the production activities, and information about cost sched-
ules. The application creates a graphic visualization of the
demand for resources in time (see Fig. 1).

The question of continuity of the project realization is in-
terconnected with the cost-cutting management measures.

The varying construction rate causes changes in construction
costs. The flow of the construction costs is a significant indica-
tor of the economy of the capital employment.

The calculation and software application described here
can be used for evaluating of bid proposals for investment
projects. The approach carries out a two-dimensional simula-
tion. The project described in propositions time and costs will
be marked as a predefined project. The discrete probabilistic
variables (T; C) obtain the values (ti; cj). We mark it as follows:
P( ; )T t C c pi i ij� � � .

The SW application simulates the presumed development
of the examined construction phase, the whole construction
project or just a set of construction activities. We can identify
the effects of changes and we can view management changes
in the scope of particular jobs (construction activities) and the
probability (reliability) of meeting the proposed (contracted)
completion deadline tfin and the proposed contracting cost
limit cfin. In general we are searching for acceptable

F t c T t C c
ct

( ; ) ( ; )fin fin fin fin� � �		 P
for the selected project activities Ak

(2)

or their activity sets Ak, Al, …, Ax, functionality-designed into a
network.

The results of particular simulations

P P( ; ) ( ( ; ))T C sim T C� TABproject (2a)

for the example described below are T � �45; 89� and
C � �2411;3131�. Simulation data is continuously recorded on
the basis of (1), [2]. The simulation is based on the time sched-
ule given in Table 1. 50 000 simulations can be ranged into
30×30 categories. When a simulation is finished, the re-
corded data serves as a basis for a statistical analysis of the
construction processes. The data file serves for the final analy-
sis and the inter alia is a basis for modified 3D visualization, as
shown in Fig. 2. The calculation of the expected or fixed
probability starts, ends and reserves the results, as shown in
Fig. 3 and Fig. 4.

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Acta Polytechnica Vol. 48  No. 1/2008

0

20

40

60

80

100

120

140

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Time duration (working days)

T
h

o
u

s
a

n
d

E
U

R

0

500

1 000

1 500

2 000

2 500

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Time duration (working days)

T
h

o
u

s
a
n

d
E

U
R

Fig. 1: Required cash flow of capital needed in the course of the construction period and the cumulative need for capital



Table 2 contains structural data of the comprehensive sim-
ulation example. The particular items are calculated as a
construction bid proposal which is further described by a sim-
ulation study. This shows how practically competitive and
realistic the supposed completion date and costs are.

P P

P

( ; ) ( ( ; ))

(

( , ) ( , )T C sim T C

s

45 89 2411 3131 � �TABproject

� � �im N f risk riski D Q v� �( , , ) ,Dconnection activities
for su �bprojects or subactivities i � 1, ).�

(2b)

The construction project [3] is proposed with the time
schedule and the scope of the work as given in the 3D bar
chart. The ellipse in Table 2 shows the shift of probability
in time and costs. Using this approach, he results of the simu-
lations can be specified more precisely. The occurrence
frequencies of the particular scenarios of building project bids
are comparable. The highest simulation frequency values in
the 3D bar graph indicate the highest probabilities of poten-
tial success scenarios for the construction project. In the given
case, the building project will be completed with satisfactory
commercial probability, within the range of 57 to 59 days, and
the construction cost will be in the range of 2 731 to 2 751
thousand EUR.

Within the framework a building project simulation, the
calculation frequently reveals a unique regular solution. An
example of a 3D probability bar chart with a unique regular
solution is shown in Fig. 2.

In the case of complicated bonding conditions and other
additional interdependences between the particular activities,
the solution of the simulation may not be unique.

Fig. 5 and Fig. 5a present a building project in which the
input parameters contain a specific interdependence within
the first activity (ground works). Let us compare these results
with Fig. 2. In the event that the first activity takes more than
25 days, the building ground machinery must without delay
be removed to another major activity (another building pro-
ject). This situation causes slippage of dates within the range

32 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 48  No. 1/2008

Fig. 2: Example of a 3D probability bar chart as an expression on the basis of (1)

Fig. 3: Probability 3D bar chart for a construction project with
fixed cost scope C

Fig. 4: Probability 3D bar chart for a construction project with
fixed time duration T



of 21 days. After that period, the ground works can be
resumed.

This specific condition leads to a heterogeneous solution
of the simulation. It is difficult to find the solution of this
building project using standard statistical methods. It is con-
venient to take advantage of visualization techniques and par-
ticular simulation calculations.

Important information regarding a proposal for a future
project time schedule is specified by tests of the potential
scenarios of the project development with current fix-
ing of certain parameters of the building organizational
model. Important information can be obtained about critical
parameters of the planned project, for example by fixing the

deviations of the scope of the work for particular activities, see
Q in expression (1) [7].

It is common practice to present the probability of the to-
tal construction time of a building project without the cost
viewpoint (Fig. 4). A better-expressed project cost is presented
as a respected fixed value that will be stable and independent
from the duration of the project. Addressing this notion, the
proposed approach that simulates the interrelated time and
cost values shown in Fig. 3, is more understandable and more
comprehensive than the information shown in Fig. 4, where

P( ) ( ) ( )a T b F t F t
T

b a
 
 � �	 (3)

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Acta Polytechnica Vol. 48  No. 1/2008

Table 2: The example of a 3D bar chart illustrating the result of 50 000 simulations

2
4
4
7

2
4
8
9

2
5

3
1

2
5
7
3

2
6
1
5

2
6
5
7

2
6
9
9

2
7
4
1

2
7

8
3

2
8
2
5

2
8
6
7

2
9

0
9

2
9
5
1

2
9
9
3

3
6

4
2

4
8

5
4

6
0

6
6

7
2

7
8

8
4

9
0

9
6

1
0

2

1
0
8

1
1
4

0 000.

0 001.

0 002.

0 003.

0 004.

0 005.

0 006.

0 007.

0 008.

0.009

P
ro

b
a

b
il

it
y

Construction cost

(Thousand EUR)
Time duration

(Working days)

Fig. 5: Example of a 3D probability bar chart with heterogeneous
solutions

Fig. 5a: Example of a graph with interdependences between time
duration and construction cost (frequencies)



or than the calculation with a fixed scope of work

P P( ) ( ( ))45 89
 
 �	 T sim T
T

TABproject . (3a)

A similar situation arises if we fix the alternation of time
schedules for the project. The scope of the work given as C is
specified as

P( ) ( ) ( )x T y F c F c
C

y x
 
 � �	 (3b)

for the data simulated in Table 2 and demonstrated in Fig. 4.

P P( ) ( ( ))2411 3131
 
 �	 C sim C
C

TABproject , (3c)

in the calculation with a fixed time ratio.
Fig. 4 shows the changes in the project cost with fixed du-

ration of the observed project.
The expected time duration of the total construction pro-

ject is given by its mean value

� �E T C t P T t ti i
T

project const� � � �	. ( ) (4)

In the same way, we can quantify the expected scope of the
work on the total construction project by its mean value

� �E C T c P C c ci i
C

project const� � � �	. ( ) (5)

4 The search for reliable construction
cost and time duration
The simulation model is able to calculate the adequate

construction costs and the time duration of a project on the
basis of the input probability level. The reciprocal view aims to

find out the adequate level of probability for construction cost
and activity durations.

There are two ways to calculate an adequate level of proba-
bility. The first way consists in fixing one variable parameter
and investigating the changes in the remaining parameter.
The second way involves a simultaneous investigation of the
deviations of both parameters.

The approach used in this paper is based on expression
(1) and Table 3 (discrete probability density table), and it en-
ables us according to the data in Table 2 to calculate the level
of probability as a cumulative density function

F T C P T t C ci i
CT

( ; ) ( ; )� � �		 , (6)

where ti and ci run through the set of all possible values of T
and C and

P T t C ci i
CT

( ; )� � �		 1. (6a)

In the course of a closer investigation of the results of par-
ticular simulations, the dependence between the level of
probability and the construction cost and time duration was
found. The following figures show the bilateral interactions of
these project parameters.

5 Conclusion
The model described here enables us to predict the ex-

pected costs and the duration schedule of a project depend-
ing on input parameters such as the production rate, the
scope of the work, the time schedule, bonding conditions,
maximum and minimum deviations from the scope of the
work, and the production rate. The results present a useful
risk evaluation for projects or project activities.

34 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 48  No. 1/2008

Table 3: Example of calculating the level of Discrete Probability Distribution (DPD) for 50 000 simulations



Acknowledgement
This paper was developed with financial support from the

Ministry of Education, Youth and Sports of the Czech Repub-
lic, project No. 1M6840770001.

References
[1] Beran, V., Dlask, P.: Management udržitelného rozvoje. Pra-

ha: Academia, 2005.
[2] Beran, V., Dlask, P., Heralová, R., Berka, V.: Dynamický

harmonogram, rozvrhování výroby do času. 1. vyd. Praha:
Academia, 2002. 170 s. ISBN 80-200-1007-6.

[3] Dlask, P., Beran, V.: MDM 2004 – teoretická příručka.
1. vyd. Praha: Vydavatelství ČVUT, 2004. 90 s.
ISBN 80-01-03072-5.

[4] Haas, Š., Hájek, V.: Systémové plánování a řízení ve staveb-
nictví. Praha: SNTL, 1981.

[5] Heralová, R., Frková, J., Tománková, J.: Decision Making
and Bids in Construction Industry. Integrated Risk Manage-
ment. Berlin: Humbolt Universität Berlin, 2002.

[6] Hromada, E., Beran, V.: Reliable Construction Cost and
Time Estimation on the basis of the Dynamic Flow-Chart
(Software for Project Reliability Estimation and Risk
Evaluation). In: Advanced Engineering Design AED 2006
[CD-ROM]. Prague: CTU, 2006, vol. 1,
p. P3.14-1–P3.14-15. ISBN 80-86059-44-8.

[7] Hromada, E., Beran, V.: Management časových a finanč-
ních potřeb stavebních zakázek. In: Stavebnictví a interiér.
2005, roč. 2005, č. 12, s. 98–100. ISSN 1211-6017.

Doc. Ing. Václav Beran, DrSc.
phone: +4202 2435 3720
Fax: +420 224 355 439
e-mail: beran@fsv.cvut.cz

Ing. Eduard Hromada, Ph.D.
e-mail: eduard.hromada@fsv.cvut.cz

Czech Technical University in Prague
Faculty of Civil Engineering
Thákurova 7
166 29 Praha 6, Czech Republic

©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 35

Acta Polytechnica Vol. 48  No. 1/2008