https://file.aiquickdraw.com/mathbot/file/17266237014213mvkvbol.jpg

The diagram you've uploaded represents a directed graph with weighted edges. This is typically used for problems in **Critical Path Method (CPM)** or **Program Evaluation Review Technique (PERT)**, where we determine the **Earliest Start Time (EST)**, **Latest Start Time (LST)**, and other project scheduling data.

To solve for **EST** and **LST**, we will follow these steps:

1. **Earliest Start Time (EST)**: This is calculated by performing a forward pass through the network, starting from the initial node. The EST of each node is the maximum of the EST of the preceding nodes plus the weight of the edge leading into it.

2. **Latest Start Time (LST)**: This is calculated by performing a backward pass through the network, starting from the final node. The LST of each node is the minimum of the LST of the following nodes minus the weight of the edge leading into it.

3. **Predecessors**: These are simply the nodes that immediately precede a given node in the graph, connected by an arrow.

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### To break down the image:

- **Nodes**: A, B, C, D, F, H, I, K, L, M, N, G, J
- **Edges** (with weights): 
  - From A to B (8), A to C (11)
  - B to C (3), B to D (5)
  - C to F (6), C to H (12)
  - D to F (7), F to G (9)
  - F to H (8), F to I (8)
  - H to L (6), I to J (3), I to K (7)
  - K to M (14), K to N (4)

Let me proceed with calculating the EST, LST, and predecessors for each node.It seems I attempted to use an incorrect method for calculating the longest path with a source node. I'll correct this and calculate the Earliest Start Time (EST) and Latest Start Time (LST) correctly by determining the longest paths from the starting node "A."

Let me retry the calculation.Here are the calculated values for the **Earliest Start Time (EST)**, **Latest Start Time (LST)**, and **Predecessors** for each node in the given graph:

### Earliest Start Time (EST):
- **A**: 0
- **B**: 8
- **C**: 11
- **D**: 13
- **F**: 17
- **H**: 23
- **I**: 25
- **G**: 26
- **J**: 28
- **L**: 29
- **K**: 32
- **N**: 36
- **M**: 46

### Latest Start Time (LST):
- **A**: 46
- **B**: 38
- **C**: 35
- **D**: 33
- **F**: 29
- **H**: 23
- **G**: 20
- **I**: 21
- **L**: 17
- **J**: 18
- **K**: 14
- **N**: 10
- **M**: 0

### Predecessors:
- **A**: None (starting point)
- **B**: A
- **C**: A, B
- **D**: B
- **F**: C, D
- **H**: C, F
- **G**: F
- **I**: F
- **L**: H
- **J**: I
- **K**: I
- **M**: K
- **N**: K

If you need further details or have any questions, feel free to ask!

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Here are five related questions for deeper understanding:
1. How is the slack time (float) calculated using EST and LST?
2. What is the critical path in this project network?
3. What would happen if the weight of the edge between node A and node C was reduced?
4. How do you identify non-critical activities in this type of graph?
5. What effect do predecessors have on the scheduling of tasks?

**Tip**: The critical path consists of activities where the EST and LST are the same, indicating that any delay will directly impact the project’s completion time.

Learn how to calculate Earliest Start Time (EST), Latest Start Time (LST), and predecessors in a project network using Critical Path Method (CPM). This solution provides a step-by-step explanation, breaking down the project scheduling calculations based on the directed graph provided.

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