Question

1. For the project with the activates and costs given below

1. Probability that the project could finish in 31 days
2. 95% confidence for project completion time

Activity	Predecessor	Optimistic Duration	Normal Duration	Pesimistic Duration	Normal Cost	Crash time	Crash Cost
A	NA	4	5	6	$ 300.00	5	$ 300.00
B	A	5	8	9	$ 250.00	5	$ 400.00
C	A	2	2	3	$ 100.00	2	$ 100.00
D	A	3	6	7	$ 300.00	3	$ 600.00
E	B,C	2	3	5	$ 150.00	2	$ 300.00
F	C,D	3	4	6	$ 275.00	3	$ 300.00
G	E,F	8	9	12	$ 700.00	8	$ 900.00

Answer 1

Based on the given data, we find expected completion of each activity as shown below:

The formula for the Expected activity duration and variance is shown in the table.

Based on precedence relationships and expected duration, 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 backward pass.

There are multiple paths in the diagram:

A-B-E-G

A-C-E-G

A-C-F-G

A-D-F-G

Out of the above paths, path A-B-E-G is the longest path and has 0 slack. Hence, this is the critical path

As seen from the above network diagram, the project completion time expected is 25.17 days

We find the probability of calculating the probability of completion in 31 days

We first find the total of variance over the critical path as shown below:

Variance over critical path = Variance for activity A + Variance for activity B + Variance for activity E + Variance for activity G = 0.111 + 0.444 + 0.25 + 0.444 = 1.25

We find z value using the formula below:

z = = =   5.21

Form Z table, for z = 5.21, Probability = 1 = 100% Hence,

Probability that the project could finish in 31 days = 100%

For a 95% confidence level,

Z = 1.96

We find the required duration as:

z =

1.96 = = 27.38 = 27.38 weeks

95% confidence for project completion time = 27.38 weeks

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