https://file.aiquickdraw.com/mathbot/file/1728615992421aypaceeq.jpg

This problem involves project scheduling using the given priority list and two processors. The tasks are organized in a directed acyclic graph (DAG), where each task is labeled with its duration. We are to assign these tasks to two processors while considering the dependencies between the tasks and the priority list: \( T_2, T_7, T_9, T_1, T_3, T_5, T_6, T_4, T_8, T_{10} \).

### Steps to Solve:

1. **Identify Dependencies**: 
   - Task \( T_5 \) depends on \( T_1 \) and \( T_6 \),
   - Task \( T_9 \) depends on \( T_5 \),
   - Task \( T_6 \) depends on \( T_2 \),
   - Task \( T_8 \) depends on \( T_6 \),
   - Task \( T_{10} \) depends on \( T_8 \) and \( T_9 \).

2. **Assign Tasks**: We will use the given priority list to assign tasks to two processors while respecting dependencies.

### Priority List:
- \( T_2, T_7, T_9, T_1, T_3, T_5, T_6, T_4, T_8, T_{10} \).

#### Task Assignments Based on Dependencies:

1. **Processor 1** starts with \( T_2 \) at time 0 (duration 12).
2. **Processor 2** starts with \( T_7 \) at time 0 (duration 2).
3. After \( T_7 \) finishes, **Processor 2** takes \( T_9 \) at time 2 (duration 7).
4. **Processor 1** finishes \( T_2 \) at time 12, then takes \( T_1 \) (duration 8).
5. **Processor 2** finishes \( T_9 \) at time 9, but \( T_5 \) depends on \( T_1 \) finishing, so it waits.
6. After \( T_1 \) finishes at time 20, **Processor 1** can take \( T_5 \) (duration 10).
7. \( T_6 \) can only start after \( T_2 \), so **Processor 2** takes \( T_6 \) at time 12 (duration 9).
8. After \( T_6 \) finishes at time 21, **Processor 2** takes \( T_8 \) (duration 5).
9. \( T_{10} \) can only start after \( T_9 \) and \( T_8 \), so **Processor 2** takes \( T_{10} \) at time 26 (duration 11).

### Final Schedule:
- **Task 8** is done by **Processor 2** starting at time **21**.
- **Task 9** is done by **Processor 2** starting at time **2**.

Would you like more details about any of the steps? Here are five related questions to expand this further:

1. How do you handle more processors in a scheduling problem?
2. How would the schedule change if we added more dependencies between tasks?
3. Can you explain how critical paths are used in project scheduling?
4. How would this change if the priority list were different?
5. What happens if a task duration changes in the middle of the schedule?

**Tip**: When scheduling tasks with dependencies, always ensure the prerequisite tasks are completed before scheduling dependent ones!

Using the priority list T2, T7, T9, T1, T3, T5, T6, T4, T8, T10, schedule the project with two processors.

Explore how to schedule a project with two processors using a given priority list of tasks with dependencies. This example involves task dependencies, scheduling principles, and task assignments based on priority lists, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation