Note that since activity i-j comes out of Node i and goes in to Node j, by definition, LC_ij=LC_j and ES_ij=ES_i

Critical Activity:
One that has a total float equal to 0, i.e., activity i-j is critical if TF_ij=0. Note that TF_ij=0 $\Rightarrow$ (LC_ij - ES_ij) = D_ij, i.e., ES_ij + D_ij = LC_ij. Thus, a critical activity has no excess time available for its completion - it MUST start as soon as possible (at ES_ij) so that it is done no later than the time beyond which the project will be delayed (LC_ij).

Critical Path:
A path of critical actvities in the project network that link the start node (Node 1) and the end node (Node n). It is possible for a project to have more than one critical path!
 

CPM: Critical Path Method

1. Forward Path: Start at Node 1 with ES₁=0 and find ES₂, ES₃,…ES_n via ES_j = Max_i {ES_i + D_ij} $\forall$ defined i-j.
2. Backward Path: Start at Node n with LC_n=ES_n and find LC_n-1, LC_n-2,…LC₁ via LC_i=Min_j {LC_j-D_ij} $\forall$ defined i-j.
3. Compute LC_ij=LC_j and ES_ij=ES_i for each activity i-j in the project network
4. Compute TF_ij= {(LC_ij - ES_ij) - D_ij} for each activity i-j in the project network and identify the critical path.

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