A Study on Project Planning Using the Deterministic and Probabilistic Models by Network Scheduling Techniques

Rama.S et al. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 7, Issue 3, ( Part -4) March 2017, pp.32-38
www.ijera.com DOI: 10.9790/9622- 0703043238 32 | P a g e
A Study on Project Planning Using the Deterministic and
Probabilistic Models by Network Scheduling Techniques
Rama.S1
, Sathya A2
, Shasikala A3
, and Cilfa Irene4
ABSTRACT:
Project planning is the important task in many areas like construction, resource allocation and many. A
sequence of activities has to be performed to complete one task. Each activity has its unique processing time and
all together to identify the critical activities which affect the completion of the project. In this paper the
probabilistic and deterministic models to determine the project completion time and also the critical activities
are considered. A case study on building construction project has been performed to demonstrate the application
of the above said models. The two project scheduling namely PERT and CPM are used to determine
numerically the different types of floating times of each activity and hence determined the critical path which
plays an important role in the project completion time. Also a linear programing model has been developed to
reduce the project completion time which optimize the resource allocation. To apply these techniques
numerically the primary data from a housing project company in a metropolitan city has been taken, the network
diagram of the activities involved in the building construction project has been drawn and the results are
tabulated.
Keywords: Project scheduling, Probabilistic model, Deterministic model, CPM, PERT, Linear programming.
I. INTRODUCTION
A project consists of various activities
which are mutually relative. There is a certain order
to implement each activity before completing it.
Some of the activities are independent to other
activities and can be started right away while some
other activities are dependent on completion of
preceding activity or done simultaneously.
Planning and scheduling are important aspects in
project management. Network scheduling is an
effective technique widely used in planning and
scheduling of projects especially in construction
projects. Nowadays housing construction project
are common. Building a house within reasonable
budget and in short time can be difficult without
any planning. Good scheduling can effectively help
in construction project management. Network
scheduling will help construction companies
(engineers) to finish their task in time and within
reasonable budget. In network scheduling, PERT
(Project Evaluation and Review Technique) and
CPM (Critical Path Method) are valuable
techniques used in planning, scheduling and
controlling of construction projects. These
techniques are used to calculate the earliest start,
earliest finish, latest start, latest finish, and
different types of float for each activity, hence
thereafter to calculate the critical activities and
critical paths. These techniques are used to provide
analytical means for scheduling the activities by
defining the project activities, their precedence
relation-ships and their specific time requirements.
The main advantage of Critical Path Analysis is to
find the minimum duration of time required to
complete the project (Adebowale et al. 2011). With
the help of PERT and CPM it is possible to follow
the actual progress of the project and evaluate the
proposed alternatives with respect to cost and time.
PERT method allows the calculation of the average
time needed for completion of a project,
identification of critical activities and estimations
of probabilities of achieving the planned deadlines.
(Gurau et al. 2012).
Dhirendra Rao (1978) explained the
application of project scheduling in two main
services of knowledge information center namely
abstracting services and referencing services and
concluded that balancing and completion of project
on time would be optimized using these techniques.
Gerald (1986) developed a mathematical model
which optimizes the monitoring behavior of
transcutaneous pCo2 which analyze the changes
happening in the premature infants having
respiratory diseases. Koteswara Rao et. al. (2008)
developed a prototype tool to select an optimal
policy obtained from fuzzy PERT network whereas
the main constraints are considered viz., cost,
manpower and unlisted points to crash the
time.Fahimifard et al. (2009) applied project
scheduling techniques in Agricultural Research
Center of University of Zabol and reduced the cost
and time estimation.Adebowaleet al. (2011) has
estimated the cost and time duration to complete
the project of a civil engineering company-
ALMEGA, Nig. Ltd. Adegoke (2011) used time
scheduling technique to solve parking space issue
RESEARCH ARTICLE OPEN ACCESS

faced by various organization. Elambrouk (2011)
used project scheduling technique to identify the
critical activities and Linear Programming
technique to crash the activities so as to optimize
the cost and project completion time. Rotarescu
(2011) applied the project scheduling management
technique in human resource organization and
produced a mathematical model to optimize the
performance of the workers. Gurau et.al (2012)
explained the uses of PERT and CPM to expand
the furniture manufacturing company’s favorable
circumstances for sale on the internet and in
examination of critical path. This paper concludes
PERT/CPM of Win QSB helps us to choose the
appropriate economic approach and a new view of
facilitating decision for an enterprise. Rautela et
al.(2012) applied project scheduling technique in
large scale industries like shoe manufacturing
industry and completion time has been calculated
without delaying any activities.
Elmabrouk (2012) used linear programming and
CPM to develop the model which helps a project
manager to determine the crash cost and crash time
in construction. Paramveer Singh et al. (2013) used
the project scheduling techniques in designing and
replicating of the formation of human resources
and proved that the project can be complete before
the normal scheduled time which also would
increase confidence and fulfillment among the
workers. Rashmi Agarwal et al. (2013) explained
planning and control techniques of operations
research. Vikash Agarwal et al. (2013) had made a
comparative study on time-cost trade off problems
(TCTP) using CPM and PERT. Peng Wang (2013)
derived Ant colony optimization algorithm (ACO)
and Genetic Algorithm (GA) using PERT network
diagram. Tamrakar (2013) implemented the critical
path method in a project based company for
analyzing the completion of an arbitrary project.
Chatwal (2014) defined a simulated project
scheduling technique for wide range of simplex
problem which have large area of application. Aditi
et al. (2014) modeled the construction project
management by optimizing the time and cost of the
project using critical path method. Shailla (2014)
adapted a comparative study in hardware and
software product research and development using
CPM. Rajguru et al. (2016) proved mathematically
that cost and time are the main aspects to be
considered in the planning of every project by
adopting the project scheduling techniques.
The main objective of this paper to
minimize the schedule time and the cost for
completion of the project with adequate resources.
In this paper, the project scheduling techniques
CPM and PERT are used efficiently to optimize the
cost and the time required to construct a house in a
metropolitan city. CPM is used to produce a
graphical representation of the project and hence
find the critical activities which have to be focused
more on completion of project in time. These
critical activities form the critical path. PERT is
used to calculate the project completion time, by
considering the probabilistic model of the
construction project. To reduce the duration of any
project, a method called project crashing is used. It
is done by reducing the duration of each critical
activity.
II. CONSTRUCTION OF NETWORK
DIAGRAM
The sequences of activities involved in the
projects are listed out to draw the network diagram.
To draw the network diagram of the project the
precedence relationship of the activities are to be
determined. Network diagram is the graphical
representation of activities that have to be
attempted and completed to execute the project.
Arrows and circles are the two symbols used to
draw the network. Arrows are used to represent the
activities. Each activity is preceded and succeeded
by an event, represented by numbered circles and
are referred as tail event and head event
respectively. Preceding activity is an activity
which must be finished before starting the next
activity. Succeeding activity is an activity which
must begin only after the completion of preceding
activity. Any two events are joined only by one
activity. If any two activities have same tail event
or head event then a dummy activity would be
introduced by dotted arrow line. Clearly it is an
imaginary activity and is not a part of the project
activities. In the network diagram the events are
numbered from left to right by using Fulkerson’s
rule.
For numerical study a house construction project in
a metropolitan city has been considered and the
corresponding data produced in Table 1
Table.1 Data for construction of a house in metropolitan city
Activity code Name of the
Activity
Normal cost Immediate
predecessor
Estimated
duration
A Site clearing 3000 - 1
B Excavation 10000 A 2
C Pcc-Plain cement
concrete for bed1
5740 B 1
D Column footing and 102600 C 5

column raising up to
plinth level
E Foundation 60000 D 3
F Basement 39900 E 2
G Plinth Beam-RCC 65900 F 3
H Column raising up
to lintel level
45000 G 6
I Super structure
(wall)
90450 H 15
J Electrical pipes and
box fitting
150000 I 3
K Roof-RCC 131600 J 7
L Lintel and sunshade 20000 K 4
M Door and window
frame fixing
50000 L 3
N Plumbing and
sanitary pipes
50000 M 3
O Plastering 108200 N 5
P Hand rails fixing 80000 N,O 6
Q Flooring 66000 I,J,P 10
R Dadoing 28000 Q 3
S Door and window
shutter fixing
88778 N,M 3
T Painting 95534 S 10
U Interior fixtures 85000 Q 10
V Electrification with
all fitting
50000 N 5
W Cladding 30000 O 5
X Landscaping 20000 U 5
The precedence relationship for the activity is
A < B; B < C; C< D; D < E; E < F; F < G; G < H;
H < I; I < J; J < K; K < L; L< M; M < N; N < O;
O < P; I, J, P < Q; Q < R; N, M <S; S < T; Q<U;
N<V;O< W; U < X (1)
Using the precedence relationship given in
equation (1) the network diagram of the project has
been prepared and given in Figure 1.
Figure 1.The network diagram of the construction project in a metropolitan city.
III. ESTIMATION OF ACTIVITY
TIMES
3.1 Deterministic model:
CPM is a deterministic model whose
results have some certainty. By using this
deterministic model the different time estimates
and floats for each activity have been determined
by using to computation techniques namely
forward pass computation and backward pass
computation.
Using forward pass computation
ES(1)=EC(1)=0
ES(j)= )(
*),(
iECMax
ijPi
EC(j)=ES(j) + ti*j , for i=2:n (2)
Using backward pass computation
LC(n)=EC(n)
LC(i)= )(
*),(
iLCMin
jiSj
LS(i)=LC(i)-tij* , for i=n-1:-1:1 (3)
The total float is
T F = LS − ES = LC − EC (4)
The total time completion of the project
Tp=LC(n) (5)

Using the data given in table.1 the different time
estimates and floats have been calculated and are
tabulated in table 2. The symbols used are
explained in Appendix A.
Table 2. Time estimates and total float using CPM method
Activity
code
Time Earliest start Earliest
Finish
Latest Start Latest
Finish
Total Float
A 1 0 1 0 1 0
B 2 1 3 1 3 0
C 1 3 4 3 4 0
D 5 4 9 4 9 0
E 3 9 12 9 12 0
F 2 12 14 12 14 0
G 3 14 17 14 17 0
H 6 17 23 17 23 0
I 15 23 38 23 38 0
J 3 38 41 38 41 0
K 7 41 48 41 48 0
L 4 48 52 48 52 0
M 3 52 55 52 55 0
N 3 55 58 55 58 0
O 5 58 63 58 63 0
P 6 63 69 63 69 0
Q 10 69 79 69 79 0
R 3 79 82 91 94 12
S 3 58 61 81 84 23
T 10 61 71 84 94 23
U 10 79 89 79 89 0
V 5 58 63 89 94 31
W 5 63 68 89 94 26
X 5 89 94 89 94 0
3.2 Probabilistic model:
The three time duration of each activity namely
optimistic, most likely and pessimistic time are
considered. The expected time and variance for
each activity are given
te= (t0+4tm+tp)/6 (6)
Variance=((tp-t0)/6)2
(7)
Sum of variance of all critical activities will give
variance of total time of the project. Table 3 gives
the probabilistic data for the construction project
and the estimated variance and average time.
Table.3 Estimated time for probabilistic model by PERT
Activity
code
Immediate
predecessor
t0 tm tp te Variance
A - 1 1 1 1 0
B A 1 2 3 2 1/9
C B 1 0.5 3 1 1/9
D C 3 5 7 5 4/9
E D 2 3 4 3 1/9
F E 1 2 3 2 1/9
G F 2 3 4 3 1/9
H G 1 6 11 6 25/9
I H 6 16 20 15 49/9
J I 1 3 5 3 4/9
K J 4 7 10 7 1
L K 2 4 6 4 4/9
M L 1 3 5 3 4/9
N M 2 3 4 3 1/9
O N 3 5 7 5 4/9
P N,O 4 5.5 10 6 1

Q I,J,P 5 10 15 10 25/9
R Q 1 3 5 3 4/9
S N,M 1 3 5 3 4/9
T S 8 10 12 3 4/9
U Q 7 10 13 10 1
V N 2 5 8 5 1
W O 3 5 7 5 4/9
X U 4 5 6 5 1/9
IV. DETERMINATION OF CRITICAL
ACTIVITIES AND CRITICAL
PATH
Any activity whose total float is zero has
been considered as critical activity. The path
connecting the critical activities forms the critical
path. These critical activities delay would affect
the project on time completion. The activities
which are having positive value as total float would
not affect the project complete if the activities are
delayed. For the construction project from Table 2
the critical activities and critical path are
determined
Critical Activities: A B C D E F G H I J K L M N
O P Q U and X
Critical Path:-1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-
11 11-12 12-13 13-1414-15 15-16 16-18 18-19 19-
20 20-21.
The total estimation time of the project has been
calculated as 94 days. The probability to delay the
project for 4 more days has been estimated as 83%.
V. TIME-COST TRADE-OFF
Even though CPM/PERT minimizes the
project duration and the cost, sometimes the project
managers are compelled to complete the project
before the deadline. This has been done by
increasing the labor and material cost. Crashing is a
method used to decrease the project duration by
reducing the duration of critical activities from it’s
the normal duration. Cost Slope for each activity
has been calculated as
Cost slope = (Crash cost – Normal Cost) /(Normal
time – Crash time) (8)
Crash time is the minimum duration in which the
particular activity should be finished. Table 4 gives
the crash slope for each activity considered in the
project.
Table 4 Crash slope for each activity
Activity
code
Normal
Time
Normal
Cost
Crash
Time
Crash
cost
∆c ∆t r=∆c/∆t
A 1 3000 ½ 4050 1050 ½ 2100
B 2 10000 1 15000 5000 1 5000
C 1 5740 ½ 8610 2870 ½ 5740
D 5 102600 3 138510 35910 2 17955
E 3 60000 2 72000 12000 1 12000
F 2 39900 1 53865 13965 1 13965
G 3 65900 2 82375 16475 1 16475
H 6 45000 5 51750 6750 1 6750
I 15 90450 13 113060 22610 2 11305
J 3 150000 2 202500 52500 1 52500
K 7 131600 5 177660 46060 2 23030
L 4 20000 2 31000 11000 2 5500
M 3 50000 2 62500 12500 1 12500
N 3 50000 2 65000 15000 1 15000
O 5 108200 3 146070 37870 2 18935
P 6 80000 5 100000 20000 1 20000
Q 10 66000 8 89100 23100 2 1550
R 3 28000 2 35000 7000 1 7000
S 3 88778 2 119848 31070 1 31070
T 10 95534 9 109864 14330 1 14330
U 10 85000 8 106250 21250 2 10625
V 5 50000 3 70000 20000 2 10000
W 5 30000 4 40500 10500 1 10500
X 5 20000 3 25000 5000 2 2500
To minimize this crash cost a linear
programming model has been formulated. Let Z as
the overall cost for crashing the activities and Xi,
(i=A, B, C…X) are the decision variables, denote

the activity duration. Yi-start time of ith
activity
(i=B,C,D,E,F,G,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,
X). The problem has been formulated as
Objective function:
Min z= 2100 XA + 5000 XB + 5740 XC + 17955 XD
+ 12000 XE + 13965 XF + 16475 XG + 6750 XH +
11305 XI + 52500 XJ + 23030 XK + 5500 XL +
12500 XM + 15000 XN + 18935 XO + 20000 XP +
11550 XQ + 7000 XR + 31070 XS + 14330 XT +
10625 XU + 10000 XV + 10500 XW + 2000 XX
(9)
Slope time constraints:
XA 0.5 , XB 1 , XC 0.5 , XD 2 , XE 1 ,
XF 1 , XG 1 , XH 1 , XI 2 , XJ 1 , XK 2 ,
XL 2 , XM 1 , XN 1 , XO 2 , XP 1 , XQ 2 ,
XR 1 , XS 1 ,XT 1 , XU 2 , XV 2 , XW 1 ,
XX 2 (10)
Start time constraints:
YB+XA≥1,YC-YB+XB≥2,YD-YC+XC≥1,
YE-YD+XD≥5,YF-YE+XE≥3, YG-YF+XF≥2,
YH-YG+XG≥6, YI-YH+XH≥6 , YJ-YI+XI≥15,
YK-YJ+XJ≥3, YL-YK+XK≥7, YM-YL+XL≥4,
YN-YM+XM≥3, YO-YN+XN≥3, YP-YO+XO≥5,
YP-YN+XN≥3 ,YQ-YI+XI≥15, YQ-YJ+XJ≥3 ,
YQ-YP+XP≥6 ,YR-YQ+XQ≥10, YS-YN+XN≥3,
YS-YM+XM≥3, YT-YS+XS≥3, YU-YQ+XQ≥10,
YV-XN+YN≥3,YW-YO+XO≥5,YX-U+XU≥10,
YFinish-YR+XR≥3,YFinish-YT+XT≥10,YFinish-V+XV≥5,
YFinish-YW+XW≥5 ,YFinish-YX+XX≥5 , YFinish≤9 (11)
Non-negative constraint for decision variable
XA, XB , XC , XD , XE , XF, XG, XH, XI, XJ,, XK, XL,
XM, XN, XO, XP, XQ, XR, XS, XT, XU, XV, XW,
XX 0 (12)
Non-negative constraint for start time variables
YA ,YB , YC , YD , YE , YF , YG , YH , YI , YK , YL ,
YM , YN , YO , YP , YQ , YR , YS, YT , YU , YV , YW ,
YX YFinish 0 (13)
Here the target of the project completion
time has been taken as 90days. By crashing of 4
days from the duration of the project the target
would be achieved. By solving the above LPP
model, the crash cost and a crash time for each
activity has been estimated and minimizes the total
crash cost of the project.
VI. CONCLUSION
By using deterministic and probabilistic
technique in construction project, the project
duration and cost are optimized. Initially the
project duration was 120 days. After applying these
techniques the project duration has been reduced to
94days. Using linear programming model the
project duration of 94 days has been crashed to 90
days. This paper concludes that CPM and PERT
techniques are useful for the optimizing time
schedule of any construction project. The
complexity of this model increases when more
activities are considered and their precedence
relationship is complicated. While preparing the
network diagram less number of dummy activities
increase the computation time and also the
probabilistic model is mostly preferable by all the
construction based companies as the activity
durations are mainly depend on the workers, the
raw materials arrival and other circumstances.
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Appendix A
n-last node
P(j)={immediate predecessors of node j}
S(i)={immediate successors of node i}
tij-duration time of the activity (i,j)
ES(j)-earliest start of the activity (k,j)
EC(j)-earliest completion of the activity (k,j)
LS(i)-Latest start of the activity (i,k)
LC(i)-Latest completion of the activity (i,k)
TF-Total float
IF-independent float
FF-free float
Tp-total project completion time
t0- optimistic time
tm-most likely time
tp - pessimistic time
te-expected time

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