Critical Path in Network Analysis

Basic Scheduling Computations

The notations used are

(i, j) is Activity with tail event i and head event j

E_i is Earliest occurrence time of event i

L_j is Latest allowable occurrence time of event j

D_ij is Estimated completion time of activity (i, j)

(Es)_ij is Earliest starting time of activity (i, j)

(Ef)_ij is Earliest finishing time of activity (i, j)

(Ls)_ij is Latest starting time of activity (i, j)

(Lf)_ij is Latest finishing time of activity (i, j)

The process is as follows

1. Determination of Earliest time (E_j): Forward Pass computation 

• Step 1

The calculation commences from the start node and goes towards the end node. For easiness, the forward pass computation begins by assuming the first occurrence time of zero for the first project event.

• Step 2

                                i.            Earliest starting time of activity (i, j) is also the earliest event time of the tail end event, that is (Es)_ij = E_i

                              ii.            Earliest finish time of activity (i, j) is also the earliest starting time + the activity time, that is    (Ef)_ij = (Es)_ij + D_ij or (Ef)_ij = E_i + D_ij

                            iii.            Earliest event time for event j is the maximum of the earliest finish times of each and every activities end in to that event, that is E_j = max [(Ef)_ij for all immediate predecessor of (i, j)] or E_j =max [E_i + D_ij]

1. Backward Pass computation (for latest allowable time)

• Step 1

For end event suppose E = L. Remember that all E's have been calculated by forward pass computations.

• Step 2

Latest finish time for activity (i, j) is equivalent to the latest event time of event j, that is (Lf)_ij = L_j

• Step 3

Latest starting time of activity (i, j) is equal to the latest completion time of (i, j) - the activity time or (Ls)_ij =(Lf)_ij - D_ij  or (Ls)_ij = L_j - D_ij  

• Step 4

Latest event time for event 'i' is the least of the latest start time of all activities originating from that event, that is L_i = min [(Ls)_ij for all immediate successor of (i, j)]  = min [(Lf)_ij - D_ij]  = min [L_j - D_ij]

1. Determination of floats and slack times

There are three types of floats

• Total float - The amount of time with which the completion of an activity could be postponed beyond the earliest probable completion time without affecting the on the whole project duration time.

Mathematically

(Tf)_ij = (Latest start - Earliest start) for activity ( i - j)

(Tf)_ij = (Ls)_ij - (Es)_ij  or (Tf)_ij = (L_j - D_ij) - E_i

• Free float - It is the time with which the completion of an activity can be postponed beyond the earliest finish time with no affect in the earliest start of a succeeding activity.

Mathematically

(Ff)_ij = (Earliest time for event j - Earliest time for event i) - Activity time for ( i,  j)

(Ff)_ij = (E_j - E_i) - D_ij

• Independent float - The amount of time through which the start of an activity can be postponed without effecting the earliest start time of any directly following activities, supposing that the preceding activity has completed at its latest finish time.

Mathematically

(If)_ij =  (E_j - L_i) - D_ij 

The negative independent float is always considered as zero.

• Event slack - The difference between the latest event and earliest event times is known as event slack.

Mathematically

Head event slack = L_j - E_j, Tail event slack = L_i - E_i

1. Finding out the critical path

• Critical event - The events with zero slack times are known as critical events. If E_i = L_i  then the event i is said to be critical

• Critical activity - The activities or actions with zero total float are called as critical activities. In other words an activity is referred to be critical when there is a delay in its start will cause an additional delay in the completion date of the whole project.

• Critical path - The series of critical activities in a network is known as critical path. The critical path is the longest path in any network from the starting to ending event and states the minimum time needed for completion of the project.