A project has the following tasks:

Task Time Precursor(s)
\(T_1\) 3  
\(T_2\) 5  
\(T_3\) 5 \(T_1\)
\(T_4\) 4  
\(T_5\) 3 \(T_2\)
\(T_6\) 3 \(T_4\)
\(T_7\) 1 \(T_5\)
\(T_8\) 3 \(T_6\)
\(T_9\) 6 \(T_1\)
\(T_{10}\) 3 \(T_3\)
\(T_{11}\) 2 \(T_6\)
\(T_{12}\) 2 \(T_9, T_{10}\)
\(T_{13}\) 4 \(T_7, T_8\)
\(T_{14}\) 2 \(T_{11}, T_{12}\)
\(T_{15}\) 8 \(T_{11}, T_{12}\)
\(T_{16}\) 3 \(T_{13}, T_{14}\)

Complete the following (solutions):

  1. Draw an activity graph (e.g., PERT chart or task network) for the tasks.

    task network

  2. Compute the following for each task:

    earliest start time
    maximum finish time of all predecessors
    latest start time
    minimum latest start time of all successors - duration of task
    slack
    latest start time - earliest start time
    Task Early Late Slack
    \(T_{1}\) 0 0 0
    \(T_{2}\) 0 5 5
    \(T_{3}\) 3 3 0
    \(T_{4}\) 0 4 4
    \(T_{5}\) 5 10 5
    \(T_{6}\) 4 8 4
    \(T_{7}\) 8 13 5
    \(T_{8}\) 7 11 4
    \(T_{9}\) 3 5 2
    \(T_{10}\) 8 8 0
    \(T_{11}\) 7 11 4
    \(T_{12}\) 11 11 0
    \(T_{13}\) 10 14 4
    \(T_{14}\) 13 16 3
    \(T_{15}\) 13 13 0
    \(T_{16}\) 15 18 3

  3. Identify the critical path where the slack at every node is 0.

    \[T_1, T_3, T_{10}, T_{12}, T_{15}\]

  4. Draw a Gantt chart for the tasks.

    Gantt chart

    The dark blue bar indicate the duration of the task, and the light blue bar indicates the slack time (if any). Bars without slack are on the critical path.