Questions: Consider the following project. Activity Predecessor(s) Duration (days) A - 4 B - 3 C A, B 5 D B 4 E C 4 F D, E 7 G E 6 H F, G 4 I D 9 J G, I 6 Develop the project schedule using the critical path method, indicating the early start, early finish, late start, late finish, and slack for each activity. What is the project duration, and which activities are critical? As a hint, note that both activities A and B have no predecessor activities, and both activities H and J have no successor activities. You do not need to draw the project network diagram.

Solution Steps

To solve this problem using the Critical Path Method (CPM), we need to follow these steps:

1. Identify all paths through the project network: List all possible paths from start to finish, considering the dependencies.
2. Calculate the duration of each path: Sum the durations of activities in each path.
3. Determine the critical path: The critical path is the longest path through the network, which determines the project duration.
4. Calculate early start (ES) and early finish (EF) for each activity: Start from the beginning of the project and move forward, calculating the earliest times each activity can start and finish.
5. Calculate late start (LS) and late finish (LF) for each activity: Start from the end of the project and move backward, calculating the latest times each activity can start and finish without delaying the project.
6. Calculate slack for each activity: Slack is the difference between the late start and early start (or late finish and early finish). Activities with zero slack are critical.

The project duration is determined by the longest path through the network of activities. From the calculations, the project duration is given by:

\[ \text{Project Duration} = 25 \text{ days} \]

Critical activities are those with zero slack, meaning any delay in these activities will directly impact the project duration. The critical activities identified are:

\[ \text{Critical Activities} = \{A, C, E, G, J\} \]

The values for each activity are as follows:

• For Activity \(A\): \[ ES_A = 0, \quad EF_A = 4, \quad LS_A = 0, \quad LF_A = 4, \quad \text{Slack}_A = 0 \]

• For Activity \(B\): \[ ES_B = 0, \quad EF_B = 3, \quad LS_B = 1, \quad LF_B = 4, \quad \text{Slack}_B = 1 \]

• For Activity \(C\): \[ ES_C = 4, \quad EF_C = 9, \quad LS_C = 4, \quad LF_C = 9, \quad \text{Slack}_C = 0 \]

• For Activity \(D\): \[ ES_D = 3, \quad EF_D = 7, \quad LS_D = 6, \quad LF_D = 10, \quad \text{Slack}_D = 3 \]

• For Activity \(E\): \[ ES_E = 9, \quad EF_E = 13, \quad LS_E = 9, \quad LF_E = 13, \quad \text{Slack}_E = 0 \]

• For Activity \(F\): \[ ES_F = 13, \quad EF_F = 20, \quad LS_F = 14, \quad LF_F = 21, \quad \text{Slack}_F = 1 \]

• For Activity \(G\): \[ ES_G = 13, \quad EF_G = 19, \quad LS_G = 13, \quad LF_G = 19, \quad \text{Slack}_G = 0 \]

• For Activity \(H\): \[ ES_H = 20, \quad EF_H = 24, \quad LS_H = 21, \quad LF_H = 25, \quad \text{Slack}_H = 1 \]

• For Activity \(I\): \[ ES_I = 7, \quad EF_I = 16, \quad LS_I = 10, \quad LF_I = 19, \quad \text{Slack}_I = 3 \]

• For Activity \(J\): \[ ES_J = 19, \quad EF_J = 25, \quad LS_J = 19, \quad LF_J = 25, \quad \text{Slack}_J = 0 \]

Final Answer