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Supplementary Problems
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  1. A project consists of 10 activities, lettered A through J, as shown in the accompanying precedence diagram, with the activities on the arrows. For each activity, the deterministic time estimate in weeks is shown in the following table.
     

    Activity

    		 

    Time

    		 

    Activity

    		 

    Time

     

    A

    B

    C

    D

    E

    		 

    2

    4

    3

    5

    6

    		 

    F

    G

    H

    I

    J

    		 

    3

    4

    4

    3

    1

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    1. List all of the paths through the network.
    2. What is the duration of each of the paths?
    3. What is the critical path?
    4. What is the second-most critical path?
    5. What is the slack time for each activity?
    6. Calculate the ES, EF, LS and LF times for each activity.

  2. A project consists of 11 activities, lettered A through K, below. For each activity, the preceding activity is given, and a deterministic estimate of the length of time required to complete it in weeks.
     

    Activity

    Time

    Preceding Activity

    Activity

    Time

    Preceding Activity

     

    A

    1 week

    -

    G

    4 weeks

    E

     

    B

    3

    A

    H

    6

    F

     

    C

    2

    A

    I

    2

    G

     

    D

    4

    C

    J

    1

    H,I

     

    E

    2

    -

    K

    1

    B,D,J

     

    F

    3

    E

    		   

    1. Draw the precedence network for this project, with the activities on the arrows.
    2. List all of the paths through the network.
    3. What is the duration of each of the paths?
    4. What is the critical path?
    5. What is the slack time for each activity?
    6. The materials required to accomplish activity F have been delayed for two weeks by a strike at the supplier's plant. What effect will this have on the length of time required to complete the project?
    7. An equipment breakdown has delayed activity B for one week. What effect will this have on the length of time required to complete the project?

  3. A project consists of 8 activities, lettered A through H. below. For each activity, the preceding activity is given, and a probabilistic estimate of the time required to complete it. Times are in days.
      

    Preceding

    Optimistic

    Most Likely

    Pessimistic

     

    Activity

    Activity

    Time

    Time

    Time

     

    A

    --

    2 days

    4 days

    6 days

     

    B

    A

    3

    6

    9

     

    C

    A

    2

    5

    11

     

    D

    --

    2

    10

    12

     

    E

    C,

    4

    8

    15

     

    F

    B,E

    2

    4

    12

     

    G

    D

    3

    4

    11

     

    H

    F,G

    1

    1

    1


    1. Determine the expected time for each activity.
    2. Determine the variance for each activity
    3. Draw the PERT network for this project, with the activities on the arrows.
    4. List all of the paths through the network.
    5. What is the duration of each path?
    6. What is the variance of each path?
    7. What is the critical path?
    8. What is the second-most critical path?
    9. Activity D was delayed three days by an earthquake in the area. What effect does this have on the length of time required to complete the project?

  4. Refer to Problem 3 and the original critical path before the earthquake.
    1. What is the mean of the probability distribution for the completion time?
    2. What is the standard deviation of the probability distribution for the completion time?
    3. What is the probability of finishing the project within 24 days?
    4. What is the probability of finishing the project within 22 days?
    5. What is the probability that the project will take longer than 28 days?

  5. After the earthquake, in Problem 3-i., management faced the problem of how to make up the lost time, and how much it would cost. Estimates of crash times and costs for each activity are given below.
     

    Activity

    Crash Time

    Cost per Day

     

    A

    Not possible

    -

     

    B

    3 days

    $200 per day

     

    C

    not possible

    -

     

    D

    not possible*

    -

     

    E

    5 days

    $300 per day

     

    F

    2 days

    $500 per day

     

    G

    not possible

    -

     

    H

    not possible

    -

    *(because of the earthquake)
    1. Which activity should be crashed?
    2. How many days should it be crashed?
    3. How much will it cost?

  6. Construct the PERT network for Problem 3 with the activities on the nodes.

  7. (One step beyond) Use the time estimates and the network for Problem 2 for solving this problem. Today is Monday, May 1, and work on the project is just beginning. The project is scheduled to be finished on Friday, August 11. Use this information to calculate the ES, EF, LS, and LF.
    1. Calculate the slack for each activity.
    2. What do you notice about the slack along the critical path?





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