# Numericals on PERT

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(PERT)

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1.Â Â Â Â Â Â Â Â  A project has the following activities and other characteristics :

 Activity A B C D E F G H I Preceding Activity – – A A C D B E,F G TIME Estimates (in weeks) O 4 1 6 2 5 3 3 1 4 M 7 5 12 5 11 6 9 4 19 p 16 15 30 8 17 15 27 7 28

Draw a network and estimate the earliest and the latest event times for all nodes and hence derive critical path and estimate the expected duration of the project and corresponding variance.

(a)Â Â Â Â  What is the probability that the project duration will exceed 60 days?

(b)Â Â Â Â  What is the chance of completing the project between 45 and 54 days?

(c)Â Â Â Â  What is the probability of completing of completing the project within 30 days?

2.Â Â Â Â Â Â Â Â  A small project consists of seven activities. Optimistic, most likely and pessimistic time estimates are given for each activity.

 Activity Preceding Activity Time (Days) Optimistic Most Likely Pessimistic A – 2 5 8 B – 2 5 14 C A 4 6 14 D A 5 7 15 E B, C 2 3 10 F D 3 3 3 G E 1 2 3

(1)Â Â Â Â  Draw the PERT network and find expected completion time of project.

(2)Â Â Â Â  What is the probability that the project will be completed in?

(3)Â Â Â Â  If the project manager wants an assurance of 95% that the project is completed on time, how many days before the scheduled date he should start the project.Â Â Â Â Â Â

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3.Â Â Â Â Â Â Â Â  As the Project Manager of a construction company, you are involved in drawing PERT network for a new project. The relevant 3 point time estimates for all activities of the project are given in the following table :

 Event Time Estimates in Weeks Optimistic Time Most Likely Time Pessimistic Time 1 â€“ 2 4 6 8 2 â€“ 5 8 10 12 2 â€“ 3 6 10 14 3 â€“ 4 0 0 0 2 â€“ 4 4 8 12 4 â€“ 5 2 4 6 5 â€“ 6 2 6 10

(i)Â Â Â Â Â  Construct the PERT network for the project and determine the critical path and the expected project completion time.

(ii)Â Â Â Â  What duration will have 95% confidence for project completion?

(iii)Â Â Â  If there is huge penalty for exceeding the project completion deadline of 27 weeks, what is the probability of not being penalized?

4.Â Â Â Â Â Â Â Â  A project manager has made following 3 point time estimates for various activities of a project.

 Event Estimates Duration (weeks) Optimistic Most Likely Pessimistic 1 â€“ 2 2 2 14 1 â€“ 3 2 8 14 1 â€“ 4 4 4 16 2 â€“ 5 2 2 2 3 â€“ 5 4 10 28 4 â€“ 6 4 10 16 5 â€“ 6 6 12 30 6 â€“ 7 2 4 6

(i)Â Â Â Â Â  Draw a PERT network and find out the expected project completion time.

(ii)Â Â Â Â  What project completion will have 90% confidence of completion?

(iii)Â Â Â  If there is a huge penalty for exceeding the project completion deadline of 44 weeks, what is the probability of being panelized?Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (April 2010)

5.Â Â Â Â Â Â Â Â  Details of nine activities of a project are as follows :

 Activity Time in days Optimistic Most Likely Pessimistic 1 â€“ 2 2 5 14 1 â€“ 6 2 5 8 2 â€“ 3 5 11 29 2 â€“ 4 1 4 7 3 â€“ 5 5 11 17 4 â€“ 5 2 5 14 6 â€“ 7 3 9 27 5 â€“ 8 2 2 8 7 â€“ 8 7 13 31
• Draw the network diagram and determine various paths and their durations.
• Calculate the probability of completion of the project in 38 days.
• Calculate the project duration if the probability of project completion is to be 94.5.Â Â  Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (April 2008)

6.Â Â Â Â Â Â Â Â  Nagaria and Associates are preparing for laying the foundation of state computer centre to be inaugurated by the Chief Minister by the end of August 2004. Following are the abbreviated activities and their predecessor activities with their three time estimates of completion time.

 Activities A B C D E F G H I J K Predecessor Activities – – A B C C C, D F, G E I H Optimistic Time Estimate 2 8 7 6 9 10 11 6 4 3 1 Presumptive Time Estimate 4 8 11 6 11 18 11 14 6 5 1 Most Likely Time Estimate 3 8 9 6 10 14 11 10 5 4 1

(a)Â Â Â Â  Draw the PERT network diagram.

(b)Â Â Â Â  Compute the slack for each activity and determine the critical path.

(c)Â Â Â Â  As per the contract a penalty of ` 5000/- is to be charged for any delay beyond 37 weeks. What is the probability that Nagaria & Associates will have to pay a maximum penalty of ` 15000/-?Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (April 2004)

(Note : Area under standard normal variate z = 0 to z = 1.4795 is 0.4306)

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