1. PROJECT MANAGEMENT BY PERT-CPM
FEW MANAGEMENT APPLICATIONS OF NETWORK MODELS:
1) Construction of buildings, bridges , factories, highways, stadiums, irrigation project
2) Budget and auditing procedures.
3) Missile development programs.
4) Installation of a complex new equipment such as computers or large machinery.
5) Advertising programming and for development and launching of new products.
6) Planning of political campaigns.
7) Strategic and tactical military planning.
8) Research and developments of new products.
9) Finding the best traffic flow patterns in a large city.
10)Maintenance and overhauling complicated equipment in the chemical, power plants,
steel and petroleum industries.
11)Long-range planning and developing staffing plans.
12)Organization of big conferences , public works etc;
13)Shifting of manufacturing plant from one site to another site.
14)Preparation of bids and proposals for projects of large size.
15)Launching space programs.
BASIC STEPS IN PERT/CPM TECHNIQUES
Project scheduling by PERT/CPM consist of 4 main steps:
1. Planning: The planning phase is started by splitting the total project into small projects.
These smaller projects, in turn, are divided into activities and are analyzed by the
department or a section. The relationship of each activity with respect to another activity
are defined and established , and the corresponding responsibilities and the authority are
also stated. Thus, the possibility of overlooking any task necessary for the completion of the
project is reduced substantially.
2. Scheduling: the ultimate objective of scheduling phase is to prepare a time chat showing
the start and finish times for each activity as well as its relationship to other activities of the
project. More over, the schedule must pinpoint the critical path(in view of time) activities
which require special attention if the project is to be completed in time. For non- critical
activities the schedule must show the amount of slack of float times (defined later) which
can be used advantageously when such activities are delayed or when limited resources are
to be utilized effectively. In this phase, it is possible to resource requirements such as time,
man power , money machine etc
3. Allocation of resources: allocation of resources is performed to achieve the desired
objective. A resource is a physical variable such as labour, finance, equipment, and space
which will impose a limitation on time for the project. When resources are limited and
2. conflicting, demand are made for the same type of resources a systematic method for the
allocation of resources become essential. resource allocation usually incurs a compromise, and
the choice of this compromise depends on the judgment of managers.
4. Controlling: the final phase in the project management is controlling. Critical path methods
facilitate the application of the principle of management by project. By having progress
reports from time to time and updating the network continuosly, a better financial as well
as technical control over the project is exercised. Arrow diagrams and time charts are used
for making. Periodic progress reports. If necessary, new course of action is determined for
the remaining portion of project.
The OR techniques used for planning, scheduling and controlling the large and complex projects are
often referred to as ‘network analysis’ or ‘network planning’, and scheduling techniques. A network is a
graphical diagram consisting of a certain configuration of arrows and nodes for showing the logical
sequence of various tasks( or activities) to be performed to achieve project objectives. Network analysis is
the quite useful for designing, planning, coordinating, controlling and decision- making so that the project
could be economically completed in the minimum possible time with the limited available resources two
most popular form of this technique now used in many scheduling situations are the critical path
methods(CPM) and program evaluation and review technique.(PERT)
CPM: It differentiates between planning and scheduling. Planning refers to the determination of activities
that must be accomplished and the order in which such activities should be performed to achieve the
objectives of the project. Scheduling refers to the introduction of the time into the plan there by creating a
time table for the various activities to be performed. CPM uses two time and two costs estimates for each
activity (one time- cost estimate for the normal situation and the estimate for the crash situation). CPM
operates on the assumption that time taken by each activity in the project is already known precisely.
PERT: In PERT we usually assume that the time to perform each activity is uncertain and as such three
time estimates (the optimistic, the pessimistic, and the most likely) are used. Indeed in actual
implementation, the distinctions between PERT and CPM have become blurred as firms have integrated
the best features of both systems into their own efforts to manage projects effectively.
NETWORK DIAGRAM REPRESENTATION:
In project scheduling, the first step is to sketch an arrow diagram which shows interdependencies and the precedence relationship among activities ( as defined below) of the project. In a
network representation of a project, certain basic definitions are used.
A) It has beginning and an end event
B) It is performed for the completion of a project
C) It requires some form of resources like material, personal etc; during time of its performance.
3. D) The constraint is the technological precedence, in the sense, that it cannot be started unless
the previous (preceding) activity or activities are completed.
An activity is represented as a ‘directed branch’ (called as ‘Arrow’) , in the project network.
Predecessor activity: activities that must be completed immediately prior to the start of another
activity are called predecessor activities.
a. Successor activities: activities that cannot be started until one of more of other activities
are completed, but immediately succeed them are called successor activities.
b. Concurrent activity: activities which can be accomplished concurrently are known as
concurrent activities. It may be noted that an activity can be a predecessor or a successor to
an event or it may be concurrent with one or more of the other activities
c. Dummy activity: a dummy activity is one which does not consume any resources and has
duration of time dij=0. Dummy activities are added to maintain the ‘Network logic ‘
A project network is an acyclic and directed network. As a project has a definite beginning
and an end event, the network also has a definite beginning and end nodes which are
connected by one or more number of activities.
Let us consider some of the conditions in the project network which call for the use of
(a) No two parallel activities have the same tail and head events;
4. (This type of dummy is called as’ identity dummy’)
(B) All events except at the starting and completion of the whole project must have at least one
activity entering and one activity leaving the node.
i.e; there should be no ‘dangling’ of nodes in the network.
(1-4) & (4-3) are dummies
© The activities should follow the correctly the logic of preceding and succeeding of the activities.
Let us consider the case : there are 4 activities A,B,C,D. activities A and B have no predecessors.
Activity C can start after completion of A. Activity D can start only after completion of A and B. the
logic can be followed by means of dummy activity.
2-3 Dummy activity
5. 2. Event: an event represents a point in time signifying the completion of some activities and the
beginning of new ones. This is usually represented by a circle ‘O’ in a network which is also called
called as node or connector. The events can be further classified into the following 3 categories.
1. Merge event: when more than one activity comes and joins an event is known as merge event.
2. Burst event: when more than one activity leaves an event, such event is known as burst event.
3. Merge and burst event: an activity may be a merge and burst at the same time as with respect to
some activities it can be a merge event and with respect to some other activities it may be burst
merge and burst event
Time estimates and critical path in network analysis:
Once the network of a project is constructed, the time analysis of the network becomes
essential for planning various activities of the project. An activity-time is a forecast for the
time an activity is expected to take from starting point to its completion(under normal
The main objective of the time analysis is to prepare a planning schedule of the project.
The planning schedule should include the following factors.
(1) Total completion time for the project.
(2) Earliest time when each activity can start.
(3) Latest time when each activity can be started without delaying the total project.
(4) Float for each activity, i..e., the amount of time by which the completion of an activity
can be delay.
(5) Identification of critical activities and critical path.
Basic scheduling computations:
We shall use the following notations for basic scheduling computations.
6. (ij) = activity (I,j) with tail event I and head event j.
TE = earliest occurrence time of event i
TL or LJ =latest allowable occurrence time of event j
DIJ =estimated completion time of activity (I,j)
(ES)IJ= earliest starting time of activity (I,j)
(Ef)ij= earliest finishing time of activity(I,j)
(Ls) ij= latest starting time of activity(I,j)
(Lf)ij= latest finishing time of activity(I,j)
The basic scheduling computations can be put under the following three groups
Forward pass computations (for earliest event time):
Before starting computations, the occurrence time of initial network event is fixed. Then, the
forward pass computation yields the earliest start and earliest finish time for each activity (I,j) ,
and indirectly the earliest expected occurrence time for each event. This is mainly done in three
Step: 1 the computations begin from the ‘start’ node and move towards the ‘end’ node. For
easiness in forward pass computations start by assuming the earliest occurrence time of zero
for the initial project event.
(1) Earliest starting time of activity (I,j) is the earliest event time of the tail and event
(2) Earliest finish time of activity (I,J) is the earliest starting time + the activity time
i.e.,(EF)=(ES)ij+ DIJ (EF)ij= EI+Dij.
(3) Earliest event time for event j is maximum of the earliest finish times for all activities
and ending into that event. That is,
Ej=max [(Ef) ij for all immediate predecessor of (I,j)] or Ej=max[Ei+Dij]
The computed ‘E’ values are put over the respective circles representing each event.
Backward pass computations (for latest allowable time)
7. for latest event time (L) indicates the time by which all activities entering into that
event must completed without delaying the completion of the project. These can be
computed by reversing the method calculation used for earliest event tomes. This is
done in to the following steps
step1: for ending event assume E=L. Remember that all E’s have been computed by
forward pass computations.
Step2: latest finish time activity (I,j) is equal to the latest event time of event
Step3: latest starting time of activity (I,j)= the latest completion time of(I,j) activity time
or (Ls)ij=(Lf)ij-Dij or (LS)ij=Lj-Dij
Step4: latest event time for event I is the minimum of the latest start time of all
activities originating for that event.i..e,
Li=min [Ls)ij for all immediate successors of (I,j)]=min[(Lf)ij-Dij]= min [Lj-Dij]
The compound ‘L’ values are put over the respective circles representing each event.
Float or slack values (slack is sued for events only, floats is applied for
float or slack refers to the amount of time by which is particular activity or event can be
delayed without effecting the times schedule of the network , there are in general four
types of floats:
(1) Total float
(2) Free float
(3) Interference float
(4) Independent float
Total float: it is the amount of tome by which the completion of an activity can be
delayed beyond the earliest expected completion time without affecting the overall project
duration time. Difference between latest finish time and earliest finish time of the activity=
IF-ef=total float. This is also equal to latest start time earliest start time of the activity.
8. Thus total float=IF-ef=Is-es
Free float: It is the amount of time by which the completion of an activity can be delayed
beyond the earliest finish time without affecting the earliest start of a succeeding activity. This
is based on the assumption that all activities start as early as possible. Free float=earliest
start time of successor-earliest finish time of the activity=total float-head event float.
Head event float=(Tl)-(TE)J
After use of the float in earlier activities whatever float is left as balance can be used freely for that
activity and hence the name.
Note: negative float. If the slack of events on the critical path is negative, the implication is
that the project cannot be completed by the target date, without replanning. For instance if the
length of the critical path is 28 days and the target project completion time is 25 days, then
there will be negative slack 3 days for events on the critical path.
Interfacing float: it can be used for the activity or it is possible to use it up (partly or fully) on
the subsequent activities. Interfacing float is that part of the total float which causes a
reduction in the float of successor activities. It is the difference between the latest finish time of
activity in question and the earliest starting time of the following activity or zero whichever is
larger. It represents the portion of the float of an activity which cannot be consumed without
adversely affecting (or interfacing) the float of the subsequent activities and hence the name.
(Interfacing float=total float-free float)
Independent float+ duration
Free float+ duration
Total float+ duration
9. Note: independent float< free float< total float (see fig.12-6 for illustration)
Refer to fig.12.9. We take into consideration activity 6-7 with duration of 11
(1) We have TE(tail)+total float+ duration=TL(head)
: 5+total float+11=18
: Total float=18-5-11=2
(2)TE(tail)+free float duration=TE(head)
: 5+free float+11=16 or free float=16-11-5=0
(3)TL(tail)+independent float duration =TE(head)
Or 7+independent float+11=16
Or independent float =16-7-11=2
(2) Interface float=TL =TE=18-16=2(also equal to total float-free float)
Determination of critical path:
Before defining critical path, let us first discuss about the meaning of critical event and critical
Critical event: since the slack of an event is the difference between the latest and earliest
event time i.e., slack (i)=Li=Ei, the events with zero slack times are called critical events.
In other words, the event (i) is said to be critical if Ei=Li
Critical activity: since, the difference between the latest start time and earliest start time of
an activity is usually called as total float, the activities with zero total float are known as critical
In other words, an activity is said to be critical if a delay in its start will cause a further delay in
the completion date of the entire project.
10. Obviously, a non-critical activity is such the time between its earliest start and its latest
completion dates (as allowed by the project) is longer than its actual duration. In this case, noncritical activity is said to have a slack or float time.
Critical path: the sequence of critical activities in a network is called the critical path. The
critical path is the longest path in the network from the starting event to ending event and
defines the minimum time required to complete the project.
By the term path we mean a sequence of activities such that it begins at the
starting event and end at the final event. The length of a path is the sum of the individual times
of activities lying on the path.
If the activities on the critical path are delayed by a day, the project would also be
delayed by a day unless the time of the future critical activities are reduced by a day by
different means. The critical path is denoted by double or darker lines to make distinction from
the other non- critical paths.
Main features of critical path. The critical path has two main features:
(1) If the project has to be shortened, then some of the activities on that path must also be
shortened. The application of additional resources on other activities will not give the
desired result unless that critical path is shortened first.
(2) The variation in actual performance from the expected activity duration time will be
completely reflected in one-to-one fashion in the anticipated completion of the whole
The critical path identifies all critical activities of the project. The method of
determine such a path is explained by the following numerical examples.
Idealization of the pert model:
PERT is based on the following idealization
1. Activity duration as probabilistic and impendent.
2. Each activity duration can be expressed as
(1) Optimistic time
(a) The time likely to happen when everything is going alrigth.
2) Pessimistic time
(b) The time likely to happen when everything goes wrong
3) Most likely to happen most of the time
(m) The time likely to happen most of the time
11. 4) The duration of time distribution of each activity has the mean( µ) and standard
deviation ( σ) as:
The mean (µ ) =1/6( a+4m+b)
The standard deviation (σ) = 1/6(b-a)
The variance (V) = 1/36(b-a) ^2
5) the probability distribution of project completion time (T) follows ‘normal distribution’, with its
mean=sum of the mean values of the distribution times of the activities along the critical path; and
its variance= sum of the variable of the distribution times of the activities along the critical path.
The beta distribution is expressed as:
P(t)=k(t-a)^α (b-t)^β, where
P(t)=probability density function
a= the optimistic time
b= the pessimistic time
α,β = exponents depending upon the model (m) mean (µ) of the distribution. It is to be noted that
P(t)= 0 when t=b, beta distribution is continuous and uni model. Beta distribution is not fully
described by the mean and standard deviation. The values of mean = (a+4m+b)/6
, and the standard deviation=b-a/6 are only approximations.
The most likely time (m) is the ‘mode’ of the beta distribution
Why bets distribution is assumed, because
1) It is continuous
2) It is uni model
3) There are two non negative intercepts in the abscissa with b>a. (hence it is east to define ‘a’ and
‘b’ as positive values)
4) The error due to its assumption is found to be minimum.
Probability of completion time:
The following procedure is used in PERT calculations, to determine the probability distribution of
completion time of the project.
Setp1: draw the project network diagram>( similar to CPM)
Step2: calculate the mean and variance of each activity.
Step3: perform the forward and backward passes, using the mean values of the activities (as
duration times, dij for each activity)-(similar to CPM)
13. Step4: determine the critical path as the one connecting the nodes having E=L, in sequence (similar
Step5: determine the sum of the mean duration of the activities in the critical path.(i.e.., the length
of the critical path=E value of the final node)
Step6: determine the sum of the variances of the activities along the critical path. this is the
variance (v) of the project completion time. The standard deviation () +
Step7: the probability distribution of project completion time, follows normal distribution, with
mean= and standard deviation =
Note; and refer to mean and variance for the activity time. And refer to the project completion times