It is a group or combination of activities that must be executed within a certain fixed order before the entire task is completed. Activities are inter-related in a logical sequence, in the sense that some activities can only be started when all the activities earlier to it have been completed.


Event denotes the point of time or accomplishment occurring at a movement. Is used to denote their initial and the final point of an activity which neither consume any time nor resources for its completion.


It is a recognizable part of a project which consumes time and resources for its completion, as it may involve the physical or mental work.


Network Diagram:

Network Diagram

It is a graphical representation used to represent the logical sequence in which different activities are interrelated to one another while completing a project.

Rules for network construction:

  • An activity can only be started when all the activities earlier to it are completed
  • No two or more activities may have the same head and the tail event

Dummy Activity:


An activity which only shows the dependency, logical or relationship of one activity over the other, but does not consume any time or resources for its completion is called dummy activity.

It is represented by dotted line arrow.

The length and direction of the arrow is indicative, along with the time flow from left to right on the network diagram.

Dummy activity should only be performed, when it is necessary but there is no restriction on the number of dummy activity used.

There should be no lopping and dangling in the network diagram


When the initial activity, other than the final activity does not have any successor activity then the situation is called as dangling. Such activity should be connected directly to the last event of the network diagram.

Types of network Diagrams:

Event on node:

The event on node is also known as the activity on arrow.

Activity on node:

Activity on node diagram does not require dummy activity, and is considered to be simple and easy process irrespective of these advantages event on the node diagram, are used in PERT and CPM.

Difference between the PERT and CPM

             S.NO              PERT           CPM
1 Project evaluation and review techniques Critical Path Method
2 It is event oriented It is activity oriented
3 It is associated with probabilistic activities Associated with deterministic activities
4 It is based upon three time estimate to complete an activity Based upon a single time to complete an activity
5 It is used where time is required to complete various activities is not certain Used for repetitive job where one has prior experience of handling similar projects
6 It usually does not consider cost analysis It gives importance to cost analysis and crashing is done to minimize the cost of CPM project
7 It is used mainly for research and development project Used mainly for construction


It is based upon the three times estimate, those are:

Optimistic time:

It is used to denote the minimum time required to complete an activity when everything goes according to the plan.

Pessimistic time:

It denotes the maximum time required to complete an activity when everything goes against the plan.

Most Likely time:

It is the time required to complete an activity when executed under normal working conditions. It is denoted by t_m or m.

The fundamental assumption is PERT. It is three times estimate from the end point of the distribution curve and activity is assumed to follow β distribution. It is also assumed that the probability of company activity in time a (or) b is equal and the probability of completing activity in time m is four times of either a (or) b.

The average or expect time to complete an activity is given by

\mu \:\: (or) \:\: t_E\(\frac{a+4m+b}{6}\)=\(\frac {t_0+4t_m+t_p}{6}\)



Higher the variance or standard deviation then we can see higher will be the uncertainty.

Critical Path:

It is the minimum time consuming path from the 1st event to the last event in a network diagram. The time taken along the critical path is known as the expected project completion time, and the activities along the critical path are termed as critical activities. These are termed as critical, for the reason that if any activity on this path is delayed by a certain amount of time, then the entire project must be delayed by the same amount of time.

Possibility of completing project with in the scheduled time:

The possibility of completing project with in the scheduled time  T_s is given by



T_E  = Expected project completion time

\sigma = standard deviation along the critical path


The procedure for finding out the critical path is similar for both the PERT and CPM. It consists of 2 phases

Forward pass computation:

In this process we compute the time at which event is expected to be completed at the earliest.

Forward pass computation

E_j=maximum \:\:of \:\: all \:\:the \:\:values \:\: [E_i+t_E^{ij}]

E_i= Earliest expected time for the event i

E_j= Earliest expected time for the event j

t_E^{ij} = Expected time for activity ij

Backward Pass Computation:

In this process we compute the time by which an event must be completed at the latest.

Backward Pass Computation

L_j=maximum \:\:of \:\: all \:\:the \:\:values \:\: [L_j-t_E^{ij}]

L_i= Latest allowable time for the event i

L_j= Latest allowable time for the event j

t_E^{ij} = Expected time for Entity ij

For any activity to be critical the 3 conditions are to be followed and it must be satisfied.

Backward Pass Computation-2

  • Head event slack = 0, \(L_j-E_j=0\)
  • Tail event slack = 0, \(L_i-E_j=0\)
  • L_j-L_i=E_j-E_i=t_E^{ij}

In case of PERT if there is more than one critical path in order to find probability, always select the path having highest standard deviation.

Slack and float


Slack is also known as event slack. The slack is defined as for a particular event of the difference between E_i  the  and L_j . Slack denotes the amount of time by which a particular event can be delayed without delaying the expected project completion time.

Slack on node  = L_i-E_i

Slack on node  = L_j-E_j


Total float:

The amount of time by which an activity can be delayed without delaying the expected project completion time is known as the total float. So, the extra time available for an activity without delaying the project schedule is known as Total float.

  • If the total float value is positive then the resources are surplus and can be allocated for other activities
  • If the total float valve is zero then the resources are just sufficient to complete the activity on time.
  • If the total float valve is negative then the resources are not sufficient to complete the activity on time.

Total cost = LFT – EFT = LST – EST

Free cost:

It is the amount of total float value which can be used without affecting the float of succeeding activity. So the extra time available by which an activity can be delayed so that the succeeding activity can be stored at their earliest start time Is known as free cost.


Independent float:

It is the amount of float time which can be used without affecting either the head or the tail event.


Independent float

Crashing or Time Cost:

It is an extension of critical path method that considers a compromise between the time and cost required to complete a project. The total cost of any project consists of both the direct and the indirect cost that is to be involved in its completion.


Direct cost:

It is the cost directly involves in the execution of an activity and it includes direct material, labor, equipment, cost of the machine and many more other cost.Direct cost:

Crash time:

It is the minimum activity duration to which an activity can be compressed by increasing the resources, and therefore by increasing the direct cost. The slope line in the diagram represents the amount increased in the direct cost per unit time for crashing an activity. The slope is termed as cost time slope.

Cost time slope = \frac {\Delta C}{\Delta T}= \tan \theta = \frac {C_c-C_N}{T_N-T_C}

Indirect cost-1

Indirect cost-2

The objective of crashing a network is to determine the optimum project duration corresponding to minimum cost of project, as the steps involved are:

  • In the critical path select the critical activity having minimum cost slope
  • Reduce the duration of this activity by one time unit
  • Revise the network diagram by adjusting the time and cost of crashed activity. Again find critical path , project duration and total cost of the project
  • If the optimum project duration is obtained then stop the process, else repeat from the first step.

Click Here To Know More About: Forecasting and Types of demand variations