Q.A project has activities A, B, C, D, E with the following predecessors and durations: A (—, 3 days), B (—, 4 days), C (A, 2 days), D (B, 6 days), E (C and D, 2 days). Find the critical path and the minimum project completion time.

The critical-path condition for activity $(i,j)$ is $E_j-E_i=L_j-L_i=t_{ij}$.

In network analysis (CPM), for each event we compute the earliest time $E$ and latest time $L$. An activity $(i,j)$ with duration $t_{ij}$ lies on the critical path when it has no float, which requires all three of these:

• the events are critical: $E_i=L_i$ and $E_j=L_j$ (zero slack), and …

A critical activity has no float, satisfying $E_j-E_i=L_j-L_i=t_{ij}$.

In CPM, an activity $(i,j)$ with duration $t_{ij}$ lies on the critical path when it has zero total float. The necessary conditions are:

$E_i = L_i,\qquad E_j = L_j$

and, for the activity itself,

$E_j - E_i = L_j - L_i = t_{ij}$

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The longest path of connected activities through a project network is called the critical path; its length equals the minimum project completion time.

In network analysis (CPM/PERT), a project is drawn as a network of activities. Several chains of activities lead from the start node to the end node. The longest such chain (the path with the greatest total duration) is the critical path, …

Activity $(i,j)$ is critical when $E_j - E_i = L_j - L_i = t_{ij}$, meaning it has zero float.

For an activity $(i,j)$ with duration $t_{ij}$, let $E_i,E_j$ be the earliest event times and $L_i,L_j$ the latest event times of its tail and head nodes.

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