Answer :
To determine the project's Early Start (ES), Early Finish (EF), Late Start (LS), Late Finish (LF), slack for each activity, the total project completion time, and the critical path, follow these steps:
### Activity Data
Based on the given activity data:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Activity} & \text{Time (weeks)} & \text{Predecessors} \\ \hline A & 6 & - \\ B & 6 & - \\ C & 3 & A \\ D & 3 & A \\ E & 4 & B \\ F & 7 & B \\ G & 9 & C, E \\ H & 8 & D, F \\ \hline \end{array} \][/tex]
### Early Start (ES) and Early Finish (EF)
The ES and EF for each activity are calculated as follows:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Activity} & \text{ES} & \text{EF} \\ \hline A & 0 & 6 \\ B & 0 & 6 \\ C & 6 & 9 \\ D & 6 & 9 \\ E & 6 & 10 \\ F & 6 & 13 \\ G & 10 & 19 \\ H & 13 & 21 \\ \hline \end{array} \][/tex]
### Late Start (LS) and Late Finish (LF)
The LS and LF for each activity are calculated as follows:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Activity} & \text{LS} & \text{LF} \\ \hline A & 0 & 6 \\ B & 0 & 6 \\ C & 6 & 9 \\ D & 6 & 9 \\ E & 6 & 10 \\ F & 6 & 13 \\ G & 10 & 19 \\ H & 13 & 21 \\ \hline \end{array} \][/tex]
### Slack Calculation
The slack (or float) for each activity is calculated as:
[tex]\[ \text{Slack} = \text{LS} - \text{ES} = \text{LF} - \text{EF} \][/tex]
Using the given LS and ES values:
[tex]\[ \begin{array}{|c|c|} \hline \text{Activity} & \text{Slack} \\ \hline A & 0 \\ B & 0 \\ C & 0 \\ D & 0 \\ E & 0 \\ F & 0 \\ G & 0 \\ H & 0 \\ \hline \end{array} \][/tex]
### Total Project Completion Time
The total project completion time is the EF of the last activity in the critical path, which is 21 weeks.
### Critical Path
An activity is part of the critical path if it has zero slack (Slack = 0). Therefore, the activities in the critical path are:
[tex]\[ [A, B, C, D, E, F, G, H] \][/tex]
### Summary
- Slack for each activity:
[tex]\[ \begin{array}{|c|c|} \hline \text{Activity} & \text{Slack} \\ \hline A & 0 \\ B & 0 \\ C & 0 \\ D & 0 \\ E & 0 \\ F & 0 \\ G & 0 \\ H & 0 \\ \hline \end{array} \][/tex]
- Total project completion time: 21 weeks
- Critical path: [tex]\[ [A, B, C, D, E, F, G, H] \][/tex]
### Activity Data
Based on the given activity data:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Activity} & \text{Time (weeks)} & \text{Predecessors} \\ \hline A & 6 & - \\ B & 6 & - \\ C & 3 & A \\ D & 3 & A \\ E & 4 & B \\ F & 7 & B \\ G & 9 & C, E \\ H & 8 & D, F \\ \hline \end{array} \][/tex]
### Early Start (ES) and Early Finish (EF)
The ES and EF for each activity are calculated as follows:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Activity} & \text{ES} & \text{EF} \\ \hline A & 0 & 6 \\ B & 0 & 6 \\ C & 6 & 9 \\ D & 6 & 9 \\ E & 6 & 10 \\ F & 6 & 13 \\ G & 10 & 19 \\ H & 13 & 21 \\ \hline \end{array} \][/tex]
### Late Start (LS) and Late Finish (LF)
The LS and LF for each activity are calculated as follows:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Activity} & \text{LS} & \text{LF} \\ \hline A & 0 & 6 \\ B & 0 & 6 \\ C & 6 & 9 \\ D & 6 & 9 \\ E & 6 & 10 \\ F & 6 & 13 \\ G & 10 & 19 \\ H & 13 & 21 \\ \hline \end{array} \][/tex]
### Slack Calculation
The slack (or float) for each activity is calculated as:
[tex]\[ \text{Slack} = \text{LS} - \text{ES} = \text{LF} - \text{EF} \][/tex]
Using the given LS and ES values:
[tex]\[ \begin{array}{|c|c|} \hline \text{Activity} & \text{Slack} \\ \hline A & 0 \\ B & 0 \\ C & 0 \\ D & 0 \\ E & 0 \\ F & 0 \\ G & 0 \\ H & 0 \\ \hline \end{array} \][/tex]
### Total Project Completion Time
The total project completion time is the EF of the last activity in the critical path, which is 21 weeks.
### Critical Path
An activity is part of the critical path if it has zero slack (Slack = 0). Therefore, the activities in the critical path are:
[tex]\[ [A, B, C, D, E, F, G, H] \][/tex]
### Summary
- Slack for each activity:
[tex]\[ \begin{array}{|c|c|} \hline \text{Activity} & \text{Slack} \\ \hline A & 0 \\ B & 0 \\ C & 0 \\ D & 0 \\ E & 0 \\ F & 0 \\ G & 0 \\ H & 0 \\ \hline \end{array} \][/tex]
- Total project completion time: 21 weeks
- Critical path: [tex]\[ [A, B, C, D, E, F, G, H] \][/tex]
