Network Analysis - Pert and Cpm

PERT stands for programme evaluation and review technique.

1. Three time estimates are made.
2. It is based on probabilistic approach.
3. Time is directly proportional to time hence minimum cost will be achieved corresponding to minimum completion time.
4. Suitable for new type of projects.
5. Each activity follows β distribution.

(i) Expected completion time of an Activity: (t_E)

Where, t₀ = Optimistic time

t_p = Pessimistic time

t_m = Most likely time

(ii) Standard deviation of an Activity (𝛔) 

(iii) Variance of an activity: (𝛔²) 

(iv) Central limit theorem:

(a) The mean time of the project as a whole is  along the critical path.

Probability of completion of project in time t_E is 50%.

(b) The standard deviation of the project as a whole is  along the critical path.

Critical Path: The time wise longest path is called critical path. In this path any type of delay in any event will cause delay to the project. These are shown by double line or dark lines in a network.

An event is critical if its slack is zero.

Event Time

(i) Earliest expected event occurring time (T_E)

 When there is only one path.

Where,  = Expected completion time of an activity I – j

 …. when there are more than one path.

Where  = Earliest expected time of event i.

 = Earliest expected time of event j.

(ii) Latest allowable occurrence time (T_L):

 When there is only one path.

 When there are more than one path.

(iii) Slack (s): This is the time by which an event may be delayed without affecting the completion time of the project.

• Probability Factor (z)

Where, T_S = Given scheduled completion time of the project

T_E = Expected completion time of the project.

σ = Standard deviation

CPM

Definitions

CPM stands for critical path method. This is based on deterministic approach in which only one time estimate is made for activity completion.

1. A network diagram in CPM is activity oriented.
2. Cost is the most important criteria. Minimum is found corresponding to optimum time.
3. There is only single time estimate for each activity.
4. The probability of completion of activity in this estimated duration is 100%.
5. It is based on deterministic approach.
6. Suitable for repetitive type of work.
7. Normal distribution is followed.

Activity times

(i) Earliest start time 

(ii) Earliest start time

EFT = EST + Activity time

(iii) Latest finish time

LFT = T_L of head event

(iv) Latest finish time

LST = LFT - t_ij

Float

Float denotes the range within which activity time or its finish time may fluctuate without effecting the completion of the project.

(i) Total Float (F_T):

F_T = LST – EST or F_T = LFT – EFT

(ii) Free Total (F_F):

 Or 

Where S_j = Head event slack

(iii) Independent Float (F_ID):

Where S_i = Tail event slack

F_T = 0 – for critical path F_T > 0 –for subcritical path

F_T < 0 – for Supercritical path

(iv) Interfering float (F_IN)

It is the another name of head event slack.

CPM Systems

Mainly two systems are used in CPM analysis:

1. A-O-A System (Activity on arrow system)

An activity is graphically represented by an arrow.

The tail end and head end of arrow represent start and finish of an activity respectively.

2. A-O-N System (Activity on node system or precedence diagram). Activity is represented by circle or node. Events have no places. Arrows are used only to show the dependency relationship between activity nodes.

When two or more activities start parallel then an activity called DEBUT (D₀) is provided at the beginning. Likewise, a finish activity (F₀) is provided at the end when more than one activities finish parallel. Activity D & F have zero duration.