Question

MS3053 MANAGEMENT SCIENCE & OPERATIONS TECHNOLOGY 6.1 Practice

The Farmer’s American Bank of Leesburg is planning to install a new computerized accounts system. Bank management has determined the activities required to complete the project, the precedence relationships of the activities, and activity time estimates, as shown in the following table:

			Time Estimates (weeks)
Activity	Activity Description	Activity Predecessor	a	b	c
A	Position recruiting	---	5	8	17
B	System development	---	3	12	15
C	System training	A	4	7	10
D	Equipment training	A	5	8	23
E	Manual system test	B, C	1	1	1
F	Preliminary system changeover	B, C	1	4	13
G	Computer-personnel interface	D, E	3	6	9
H	Equipment modification	D, E	1	2.5	7
I	Equipment testing	H	1	1	1
J	System debugging and installation	F, G	2	2	2
K	Equipment changeover	G, I	5	8	11

1. Using the PERT approach, use the three estimates of duration to calculate a mean duration and standard deviation for all activities.

2. Construct/draw a network diagram for this problem using the mean durations, calculate LS(Foll.), ES(Prec.), and the total float for each task, and hence identify the critical path. What is the standard deviation of the critical path?

3. What is the 94% confidence interval for the length of the critical path?

4. What is the probability of completing the project within 40 weeks?

5. Answer the project manager’s question:

   “I want to tell the client a project length which I am 89.44% sure that we can meet - what figure should I give them?”

6. What is the probability of completing the project between 35 and 39 weeks?

Answer 1

a) Based on the given data, we find mean duration Variance for all activities as shown below:

b) Based on the above, we find critical activities by preparing 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. However, path A-D-G-K is the longest and has 0 slack. hence, it is the critical path.

As seen from the above network diagram, the project completion time expected is 33 weeks

c) We find the probability of calculating the probability of completion in 40 weeks

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 D + Variance for activity G + Variance for activity K = 4 + 9 + 1 + 1 = 15

Hence, Standard deviation over critical path = = = 3.873

We calculate Z value as shown below:

Z = = = 1.807

For z = 1.807, probability from Z table = 0.9649 = 96.49%

Hence, probability for completion within 40 weeks = 96.49%

d) For 94% confidence interval, z = 1.56

Hence, we calculate required duration as:

Z =

1.56 =

Required duration = 39.04 = 39 weeks

e) For 89.44% confidence interval, z = 1.25

Hence, we calculate required duration as:

Z =

1.25 =

Required duration = 39.04 = 37.84 weeks

f) Probability of completion between 35 and 39 weeks = Probability of completion within 39 weeks - Probability of completion within 35 weeks

As we found from part d, probability for completing within 39 week = 94%

For 35 weeks, we calculate probability as:

We calculate Z value as shown below:

Z = = = 0.5164

For z = 0.5164, probability from Z table = 0.695 = 69.50%

Hence, probability for completion within 35 weeks = 69.50%

Hence, Probability of completion between 35 and 39 weeks = 94 - 69.5 = 24.5%

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