[Geneneral Math 3-4] Chapter 14E: Using crashing to reduce the completion time on of a project

Hôm nay Gia Sư Toán Cao Cấp và Xác Suất Thống Kê sẽ giới thiệu đến các bạn dạng bài tập Crashing trong môn General Math 3-4

1. Altering completion times

- The minimum time it takes to complete a project depends upon the times it takes to complete the individual activites of the project, and upon the predecessors each of the activities have.

- Critical path analysis can be completed to find the overall minimum completion time.

- Sometime, The managers of a project might arrange for one or more activities within the project to be completed in a shorter time than originally planned. Changing the conditions of an acitivites within a project, and recalculating the minimum completion time for the project, is called crashing.

2. A simple crashing example.

- A simple activity network is shown in the diagram on the below. The forwards and backwards scanning processes have been completed and the critical path is shown in red on the diagram.

- The minimum time for completion is currently 13 hours. In order to reduce this overall time, The manager of the project should try to complete one, or more, of the activities in a shorter time than normal. Reducing the time taken to complete activity A,B or C would not achieve this goal however. These activites are not on the critical path and so they already slack time. Reducing their completion time will not shorten the overall time taken to complete the project.

- Activity D and E, on the other hand, lie on the critical path. Reducing the duration of these activities will reduce the overall time for the project. If activities D was reduced in time to 4 hours instead, the project will be completed in 11, not 13, hours.

3. Crashing with Cost

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Example 1:

Solution:

Example 2:

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Example 3: 

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Example 4:

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Example 5:

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Exercise 1: The activity network for a project is shown in the diagram below. The duration for each activity is in hours.

a. List all four paths from the Start to the Finish of the project, With their respective completion times.

b. Identify the critical path and the minimum completion time for the project.

c. If  Activity E is reduced by 3 hours, identify the new minimum completion time for this project.

Solution:

1a. A-D

B-E-F

B-E-G-I

C-H-I

1b. B-E-G-I: 21  hours

1c. 18

Exercise 2: The directed network below shows the sequence of 8 activites that are needed to complete a project. The time, in days, that is takes to complete each activity is also shown. 

a. Write down the critical path for this project.

b. What is the minimum completion time for the project ? Activity B can be reduced by a maximum of 3 days at a cost of  $100 per day.

c. What is the new minimum completion time for the project ?

d. What is the minimum cost that will achieve the greatest reduction in time taken to complete the project ?

Solution:

2a. A-B-F-G

2b. 21 days.

2c. 20 days

2d. $100

Exercise 3: The activity network for a project is shown in the diagram on the below. The duration for each activity is in hours.

a. Identify the critical path for this project.

b. What is the maximum number of hours that the completion time for activity E can be reduced by without changing the minimum completion time of the project ?

c. What is the maximum number of hours that the completion time for activity H can be reduced without changing the minimum completion time of the project ?

d. Every activity can be reduced in duration by a maximum of 2 hours. If every activity was reduced by the maxmimum amount possible, what is the new minimum completion time for the project ? 

Solution:

3a. B-E-H-J

3b. 2 hours

3c. 6 hours

3d. 14 hours

Exercise 4: The activity network for a project is shown in the diagram below. The duration for each activity is in hours.

a. How many activities could be delayed by 4 hours without altering the minimum completion for the project ?

b. If the project is to be crashed by reducing the completion time of one activity only, what is the minimum time, in hours, that the project can be completed in ?

c. Activity G can be reduced in time at a cost of $200 per hour. Activity J can be reduced in time  at a cost of $150 per hour, What is the cost of reducing the completion time of this project as much as possible ?

Solution:

4a. 4

4b. 17 hours

4c. $1200