Question

Consider the project with the listed activities. Normal durations and costs, as well as crash durations and costs are listed for each activity. Precedence relationships are implicitly given by the activity names (e.g., activity (1,2) is represented as an arc from node 1 to node 2).
                                Crash     Crash                                     Normal                 Normal

Activity                 Time      Cost                                       Time                      Cost

(1, 2)                      5              $41,000                                 8                              $29,000

(1, 3)                      2              $18,000                                 3                              $10,000

(2, 4)                      10           $50,000                                 12                           $46,000

(2, 5)                      4              $26,000                                 6                              $20,000

(3, 5)                      9              $18,000                                 11                           $15,000

(4, 6)                      4              $22,000                                 5                              $20,000

(5, 6)                      4              $13,000                                 4                              $13,000

(5, 7)                      8              $30,000                                 10                           $25,000

(6, 8)                      3              $15,000                                 5                              $10,000

(7, 8)                      4              $10,000                                 5                              $7,000

*Times are in weeks

(a) Assuming that all activities are completed according to their normal durations, how long will it take to complete the project?

(b) Suppose that you wanted to shorten the duration of the project by 3 weeks. Which activities would you shorten, and by how much? What additional cost would you incur by doing this?

Answer 1

The project network diagram for the given scenario is as follows:

The duration of the various paths are as follows:

Path A: 1 - 2 - 4 - 6 - 8 = 5 + 10 + 4 + 3 = 22 Weeks

Path B: 1 - 2 - 5 - 6 - 8 = 5 + 4 + 4 + 3 = 16 Weeks

Path C: 1 - 2 - 5 - 7 - 8 = 5 + 4 + 8 + 4 = 21 Weeks

Path D: 1 - 3 - 5 - 6 - 8 = 2 + 9 + 4 + 3 = 18 Weeks

Path E: 1 - 3 - 5 - 7 - 8 = 2 + 9 + 8 + 4 = 23 Weeks

The highest duration Path is 1 -- 3 -- 5 -- 7 -- 8. Therefore, it is the critical path.

(a) If all the activities are completed according to their normal durations, the project would get completed in 23 weeks.

(b) If the duration is to be shortened by 3 weeks:

The cost slope information for the activities are as follows:

In order to decrease the duration of the project, we need to first decrease the duration of the critical path.

The activity having least cost slope among the critical path activities is 3 -- 5. If we reduce 1 week by crashing Activity 3-5, we would incur additional cost of $1,500.

Then the project duration would become 22 Weeks.

With 22 weeks duration, we would have two critical paths and those are, Path A and Path E.

Now we need to crash the activities from each of the critical path.

Crashing activity 2-4 by 2 weeks would incur additional crashing cost of $4,000.

Crashing activity 3-5 by one week and activity 5-7 by one week would incur the additional crashing cost of $4,000 ($1,500 + $2,500)

Therefore, the activities which should be shortened are:

Activity 3-5 by 2 weeks;

Activity 5-7 by 1 week;

Activity 2-4 by 2 weeks.

Additional cost of crashing = $1,500 + $4,000 + $4,000 = $9,500