PERT And CPM

PERT and CPM techniques are used for planning and scheduling large projects.

PERT :  Program Evaluation and Review Technique.
CPM: Critical Path Method.

Terminology:

Network: It consist of series of activities arranged in a logical sequence and shows inter relationship between them.

Activity: It is time consuming effort which is required to perform part of work. It is represented by an arrow ($\to$).

Event: It is the start or completion point of an activity. It is represented by a circle.

Predecessor and Successor Activities: Activities that must be completed before another activity can be started called its predecessor activities. Successor activity occurs following predecessor activities.

Dummy Activity: An activity that consumes no time but shows precedence or relationship among activities. It is represented by dotted line in network.

Each event has two important times associated with it:

Earliest Time: Time when an event can occur when all predecessor activities finished at the earliest possible time.

Latest Time: It is the latest time when each activity can start without delaying the total project.

Forward Pass Computation/ Earliest Start Time

Earliest Start Time (EST): It is the earliest start time of an activity when all predecessor activities are started at their earliest time. 

Earliest Finish Time (EFT): if activity time is 't' then EFT = EST + t

Backward Pass Computation / Latest Time

Latest Finish Time (LFT): The latest finish time for an activity without delaying the project.

Latest Start Time (LST)  = LFT - t

Example:

Example 

Forward Pass:

Starts from initial event to move to end event.

$EST_1$ = 0

$EST_2$ = $EST_1$ + t = 0 + 12 = 12

$EST_3$ = $EST_2$ + 8 = 20

$EST_4$ = $EST_2$ + 10 = 22

$EST_5$ = max{ ($EST_3$+14) , ($EST_4$ + 8) } = max{34, 30} = 34

$EST_6$ = $EST_5$ + 4  = 38

Backward Pass:

Starts from end event to come to first event.

$LST_6$ = 38

$LST_5$ = $LST_6$ - 4 = 34

$LST_4$ = $LST_5$ - 8 = 26

$LST_3$ = $LST_5$ - 14 = 20

$LST_2$ = min{ ($LST_3$ - 8), ($LST_4$ - 10)} = min{12, 16} = 12

$LST_1$ = $LST_2$ - 12 = 0

Slack: Difference between the latest time and earliest time of an event is called slack. An activity with zero slack is known as critical activity.

Critical Path : Critical path is the on the network of project activities which takes longest time from start to finish.
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Total Float: It is the time which completion of an activity can be delayed beyond earliest expected completion time without affecting total project time. 

Total float = Latest start time - Earliest start time

Free Float: It is the time which the completion an activity can be delayed without delaying the earliest start of any succeeding activity.

Free Float = Total Float - Head Event Slack

Independent Float: It is the time which the start of an activity can be delayed without affecting the earliest start time of any immediately following activities assuming that the preceding activity has finished its LFT.

Independent Float = Free Float - Tail Event Slack

PERT :  Program Evaluation and Review Technique

It is used when activity times are not known with certainty. 

The fundamental assumption in PERT is that the three estimates of time ($t_m, t_o, t_p) follow β - distribution curve.

Optimistic Time ($t_o$): It is an estimate of minimum possible time to complete the activity under ideal condition. 

Most Likely time ($t_m$): This lies between optimistic and pessimistic time estimates.

Pessimistic Time ($t_p$): It is the longest time taking into consideration odds.

 Expected time (mean) :

\[t_e = \frac{t_o + 4t_m + t_p}{6}\]

Standard deviation        \( σ = \frac{t_p - t_o}{6}\)

Variance                        \( σ^2 = (\frac{t_p - t_o}{6})^2\)

• Expected time of project ($T_{e}$) is the sum of the expected time of all activities lies on critical path.
• Expected time of activity is assumed to be β - distribution and  expected time for project is normally distributed.
• Variance of the expected time of project ($σ_{cp}^2$) is the sum of the variance of the expected time of all activities along the critical path.
• Probability that the project will be completed  in a given time (T)

Normal distribution $ Z = \frac{T - T_e}{ σ_{cp}}$

Probability P = φ(Z).

For a normal deviate of Z = +1, the corresponding probability is 84.1% , for Z = –1 corresponding P = 15.9 % and for Z = 0 corresponding P = 50%.

Difference Between CPM and PERT:

CPM	PERT
It is activity oriented	It is event oriented
Activity time is deterministic	Activity time is probabilistic
One time estimate	Three time estimates