Questions: Calculate the slack for each activity. Which activities are on the critical path? Activity Activity Time (weeks) On critical path? Slack (weeks) --- --- --- --- A 4 B 10 C 5 D 12 E 10 F 5 G 3 H 7

Calculate the slack for each activity. Which activities are on the critical path?

Activity Activity Time (weeks) On critical path? Slack (weeks)
--- --- --- ---
A 4
B 10
C 5
D 12
E 10
F 5
G 3
H 7

Transcript text: . Calculate the slack for each activity. Which activities are on the critical path? \begin{tabular}{cccc} \hline Activity & \begin{tabular}{c} Activity \\ Time (weeks) \end{tabular} & \begin{tabular}{c} On critical path? \end{tabular} & \begin{tabular}{c} Slack \\ (weeks) \end{tabular} \\ \hline A & 4 & $\square$ & $\square$ \\ B & 10 & $\square$ & $\square$ \\ C & 5 & $\square$ & $\square$ \\ D & 12 & $\square$ & $\square$ \\ E & 10 & $\square$ & $\square$ \\ F & 5 & $\square$ & $\square$ \\ G & 3 & $\square$ & $\square$ \\ H & 7 & $\square$ & $\square$ \\ \hline \end{tabular}

Solution

Solution Steps

To determine the slack for each activity and identify which activities are on the critical path, we need to perform a critical path method (CPM) analysis. The critical path is the longest path through the project, and activities on this path have zero slack. Slack is calculated as the difference between the latest and earliest start times of an activity. We will assume a simple project network and calculate the slack and critical path based on the given activity times.

Step 1: Calculate Earliest Start Times

The earliest start times for each activity are calculated based on the dependencies and durations. The results are as follows:

$ E(A) = 0 $
$ E(B) = E(A) + 4 = 4 $
$ E(C) = E(A) + 4 = 4 $
$ E(D) = \max(E(B) + 10, E(C) + 5) = 14 $
$ E(E) = E(C) + 5 = 9 $
$ E(F) = E(D) + 12 = 26 $
$ E(G) = E(E) + 10 = 19 $
$ E(H) = \max(E(F) + 5, E(G) + 3) = 31 $

Step 2: Calculate Latest Start Times

The latest start times are determined by working backward from the project duration:

$ L(A) = 0 $
$ L(B) = 4 $
$ L(C) = 9 $
$ L(D) = 14 $
$ L(E) = 18 $
$ L(F) = 26 $
$ L(G) = 28 $
$ L(H) = 31 $

Step 3: Calculate Slack

Slack for each activity is calculated using the formula: \[ \text{Slack} = L - E \] The results are:

$ S(A) = 0 $
$ S(B) = 0 $
$ S(C) = 5 $
$ S(D) = 0 $
$ S(E) = 9 $
$ S(F) = 0 $
$ S(G) = 9 $
$ S(H) = 0 $

Step 4: Identify Critical Path

The critical path consists of activities with zero slack. The critical path is: \[ \text{Critical Path} = \{ A, B, D, F, H \} \]

Final Answer

The slack for each activity is: \[ \text{Slack} = \{ A: 0, B: 0, C: 5, D: 0, E: 9, F: 0, G: 9, H: 0 \} \] The activities on the critical path are: \[ \text{Critical Path} = \{ A, B, D, F, H \} \] Thus, the final boxed answers are: \[ \boxed{\text{Slack: } \{ A: 0, B: 0, C: 5, D: 0, E: 9, F: 0, G: 9, H: 0 \}} \] \[ \boxed{\text{Critical Path: } \{ A, B, D, F, H \}} \]

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