Question

3. Consider a project having the following activities, time, and cost:

                                                  Normal       Normal       Crash         Crash      Maximum

                        Immediate          Time         Cost         Time          Cost         Time

      Activity      Predecessors      (weeks)          ($)          (weeks)          ($)       Reduced

            a               none                  4            3,000             2            5,000            2

            b               a                        5            5,000             3            8,000            2

            c               a                        4            7,000             4            7,000            0

            d               b                        4            6,000             2            8,000            2

            e               c,d                     8            4,000             6            8,000            2

            f               c                        3            4,000             2            9,000            1

            g               e,f                      4            2,000             2            7,000            2

Assume partial crashing (not all maximum crashing time has to be used) is available.

1. Draw the network diagram. How long does project take using normal time?
2. Which activities should be crashed to reduce the completion time of the project by two weeks? How much is the additional cost? Show all work.

Answer 1

We find the crash cost per day for each activity as shown below:

Based on the predecessor relationship and durations given, we prepare a project network diagram as shown below:

The above network diagram in the form of formulas is shown below for better understanding and reference:

The above project diagram is prepared as per the legend shown in the top-left corner.

EST = Early start time

LST = Late start time

EFT = Early finish time

LFT = Late finish time

SLK = Slack = LST - EST or Slack = LFT - EFT

The EST of an activity = EFT of previous activity

EFT of an activity = EST + Duration

Similarly, LFT & LST are calculated during the backward pass.

There are multiple paths in the diagram:

a-b-d-e-g

a-c-e-g

a-c-f-g

Out of the above paths, path a-b-d-e-g is the longest path and has 0 slack. Hence, this is the critical path.

As seen from above, without crashing the earliest the project can be completed is 25 days

To reduce project duration, we have to crash activities on the critical path first.

As seen from the above table, we start with crashing activities with minimum crash cost per day to have a minimum cost impact.

Step-1: We crash activity "a" as it has the lowest crash cost. We crash A by 2 days which is the maximum possible. Additional cost = No. of days crashed * crash cost per day = 2*1000 = $2000

Hence, as seen from above, we have to crash activity "a" by 2 weeks with an additional cost of $2000

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