Chapter 8

Appendix B – Chapter 8

Project Management: A Managerial Approach, sixth edition

Problems

NOTE: Many of the AON graphics in this solutions set depict the start day of the successor activity to be the same day as the completion of the predecessor. This is consistent with the presentation in the text. It is not consistent with the result that would be obtained using Microsoft® Project, where the start day of the successor is always the next working day after the completion of the predecessor.

Problem 2:

Problem 4:

a) The critical path is B-E-G.

b) 23 work periods.

Problem 6:

PDM Diagram 6a

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PDM Diagram 6b

Problem 8:

Please see note about network depiction preceding Problem 1

a) The critical path activities are A, C, E, and G.

b) The project’s duration is 22 days.

c) Yes, activity B can be delayed one day without delaying the completion of the project.

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Problem 10:

a) The critical path is AC CB BE EF.

b) The only event with slack is “D” at 3 days.

c) If “D” were the final event in the network, then the critical path would be AC CB BD.

d) The following spreadsheet excerpt illustrates the calculation of the probability of completion in 14 days:

Task a m b Expected Variance Std Dev.

AB 3 6 9 6.0 1.00 1.00

AC 1 4 7 4.0 1.00 1.00

CB 0 3 6 3.0 1.00 1.00

CD 3 3 3 3.0 0.00 0.00

CE 2 2 8 3.0 1.00 1.00

BD 0 0 6 1.0 1.00 1.00

BE 2 5 8 5.0 1.00 1.00

DF 4 4 10 5.0 1.00 1.00

DE 1 1 1 1.0 0.00 0.00

EF 1 4 7 4.0 1.00 1.00

Desired Duration

Expected Project

Sum of Variances Z Probability

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DurationCritical

Path

14 16.0 4.00 -1 15.9%

e) If CD slips to six days the critical path is unchanged but slack on D is reduced. If CD slips to seven days then there are two critical paths: AC CB BE EF and AC CD DF. If CD slips to eight days then the critical path shifts to AC CD DF and the project duration extends to 17 days.

Problem 12:

Figure 8.12a shows the PDM network for the data from Table A of Problem 8-12 assuming that the data were applied as shown in Figure 8.12b.


1) The critical path is 2,3,4,5,7,8,9.

2) The slack for activity 1 is 11.7 days. The slack for activity 6 is 4 days.

2) The following table shows the calculation of the expected completion time:

Activity a m b Expected

1 8 10 13 10.2

2 5 6 8 6.2

3 13 15 21 15.7

4 10 12 14 12.0

5 11 20 30 20.2

6 4 5 8 5.3

7 2 3 4 3.0

8 4 6 10 6.3

9 2 3 4 3.0

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Expected Project

Duration

66.4

Problem 14:

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Figure 8.14a shows the original network diagram for problem 14.


1) The critical path activities are A, D, G, and J. Activities B and E should be closely monitored as a near critical path.

Figure 8.14b shows the impact of the project’s performance to date.

2) The project will be completed in 12.5 days instead of the 13 days originally expected. The near critical path (B, E, G, J) is now critical. Activities A, D, F, and H are now near critical activities.

Problem 16:

Using critical path analysis with the data provided gives the following table:

Activity Expected Std Dev. Variance

a 2.0 2.00 4.00

b 3.0 1.00 1.00

c 4.0 0.00 0.00

d 2.0 3.00 9.00

e 1.0 1.00 1.00

f 6.0 2.00 4.00

g 4.0 2.00 4.00

h 2.0 0.00 0.00

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Desired Duration

Expected Project

Duration

Sum of Variances

Critical Path Z Probability

12 13.0 9.00 -0.33 36.9%

13 13.0 9.00 0.00 50.0%

16 13.0 9.00 1.00 84.1%

17.3 13.0 9.00 1.43 92.5%

For this problem the variance has to be calculated from the standard deviation, and the durations provided are assumed to be the expected durations. As can be seen there is about an 84% chance of completing the project within the drop dead time. If a little more than a week is added to the duration, the chance of completing the project on time rises to 92.5%.

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Problem 18:

Problem 20:

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Figure 8.20 shows the network diagram for problem 20.


a) The critical path is A, D, E, G, I, J.

b) The slack on process confirmation (F) is 20 days.

c) The slack on test pension plan (C) is 61 days.

d) The slack on verify debt restriction compliance (H) is 20 days.

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Problem 22:

Figure 8.22 shows the network diagram for problem 22.


b) The critical path is B, F, H.

c) Week 9.

d) If activity E requires one extra week, the time will be absorbed in free float and will not affect any other activity. If activity E requires two extra weeks, then a second critical path will be created for activities B, E, G. If activity E requires three weeks, negative float will be created and the project cannot complete in nine weeks. The new completion time will rise to 10 weeks.

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Problem 24:

Figure 8.24a shows the network diagram for problem 24a.

b) The critical path is B, E, G, H.

Figure 8.24c shows the network diagram solution to problem 24c.

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d) Given a float value of 6 weeks, activity F seems to be the best candidate to supply resources needed to crash the project. Since the float is almost 50% of the activity’s duration, using its resources to work other activities is unlikely to convert activity F into a near-critical activity. Since activity D is both critical and concurrent to activity F, the resources should be transferred there.

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Problem 26:

Figure 8.26a shows the network, critical path and slack times.

Tabulating the calculations for expected durations and probability looks like this:


1-2 6 8 10 8 0.44 0.67

1-3 5 6 7 6 0.11 0.33

1-4 6 6 6 6 0.00 0.00

2-6 0 0 0 0 0.00 0.00

2-7 10 11 12 11 0.11 0.33

3-6 12 14 16 14 0.44 0.67

4-5 5 8 11 8 1.00 1.00

4-9 7 9 11 9 0.44 0.67

5-6 8 10 12 10 0.44 0.67

5-9 0 0 0 0 0.00 0.00

6-7 14 15 16 15 0.11 0.33

6-8 10 12 14 12 0.44 0.67

7-10 9 12 15 12 1.00 1.00

8-10 0 4 14 5 5.44 2.33

9-11 5 5 5 5 0.00 0.00

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10-11 7 8 9 8 0.11 0.33

Desired Duration

Expected Project Duration

Sum of Variances Critical Path Z Probability

61.69 59.0 2.67 1.65 95.0%

The next longest path is 1-3, 3-6, 6-7, 7-10, 10-11 at 55 days. It will only be a concern if under some circumstances; its duration exceeds the actual critical path of 59 days. Using the same technique for calculating the probability of exceeding a particular duration gives the following table for this path:

Probability for path 1-3-6-7-10-11

Desired Duration

Expected Path

DurationSum of Path

Variances Z Probability

59 55.0 1.78 3.00 99.9%

Clearly the chance of exceeding 59 days is quite small. The same technique can be applied to the next longest path 1-4, 4-5, 5-6, 6-8, 8-10, 10-11 which while relatively short has high variance:

Probability for path 1-4-5-6-8-10-11

Desired Duration

Expected Path

DurationSum of Path

Variances Z Probability

59 49.0 7.44 3.67 100.0%

Again it is clear that it is unlikely that this path will cause problems with the overall project duration.

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Problem 28:

Figure 28a shows the PDM network diagram for problem 28.

The following table tabulates the variances and probability for this project:


1 6 10 14 10 1.78 1.33

2 0 1 2 1 0.11 0.33

3 16 20 30 21 5.44 2.33

4 3 5 7 5 0.44 0.67

5 2 3 4 3 0.11 0.33

6 7 10 13 10 1.00 1.00

7 1 2 3 2 0.11 0.33

8 0 2 4 2 0.44 0.67

9 2 2 2 2 0.00 0.00

10 2 3 4 3 0.11 0.33

11 0 1 2 1 0.11 0.33

12 1 2 3 2 0.11 0.33

Desired Duration

Expected Project

Duration

Sum of Variances

Critical Path Z Probability

44 41.0 8.22 1.05 85.2%

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Problem 30:

The setup for problem 30 is similar to that for problem 29. First the spreadsheet in Excel is prepared with the calculations for the paths:

Then, similar to problem 29, triangle distributions are established to calculate the durations for all activities except 9 (no variation in the estimate).

The resulting forecast for the duration of the project and corresponding statistics are:

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Note that the probability of completing the project in 44 days has dropped to about 70%.

Problem 32:

The “Pert Entry Form” in Microsoft® Project is used to enter the three durations. After they are in the “Calculate Pert” button is clicked to populate the Duration field with the expected durations. Note that MSP uses the non-standard terminology “Expected” in lieu of “Most Likely.”

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Using the calculated durations, the Gantt chart looks like this:

ID Task Name Duration Start Finish Predecessors1 Start 0 days Sun 12/11/05 Sun 12/11/05

2 a 7.5 days Mon 12/12/05 Wed 12/21/05 1

3 b 8 days Mon 12/12/05 Wed 12/21/05 1

4 c 6 days Wed 12/21/05 Thu 12/29/05 2

5 d 14.5 days Thu 12/22/05 Wed 1/11/06 2,3

6 e 7 days Wed 1/11/06 Fri 1/20/06 4,5

7 f 11.5 days Fri 1/20/06 Mon 2/6/06 3,6

8 g 8 days Tue 2/7/06 Thu 2/16/06 7

9 End 0 days Thu 2/16/06 Thu 2/16/06 8

12/11

2/16

4 11 18 25 1 8 15 22 29 5 12 19 26Dec '05 Jan '06 Feb '06

The figure shows the default Gantt chart view of the problem, with a project start day of Sunday December 11, 2005. Note that MSP moves the beginning of the first task to the first workday of Monday the 12th. This display shows the default calendar of 5 day 40 hr. weeks with no holidays. A “Start” and “End” milestone have been inserted to insure that all activities have at least one predecessor and successor.

The tracking Gantt view can be used to display the critical path:

ID Task Name Duration1 Start 0 days

2 a 7.5 days

3 b 8 days

4 c 6 days

5 d 14.5 days

6 e 7 days

7 f 11.5 days

8 g 8 days

9 End 0 days

12/11

0%

0%

0%

0%

0%

0%

0%

2/16

4 11 18 25 1 8 15 22 29 5 12 19 26Dec '05 Jan '06 Feb '06

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The network diagram can be displayed directly from MSP using the “Network Diagram” view. A portion of it with the default format settings looks like this:

aStart: 12/12/05 ID: 2

Finish: 12/21/05 Dur: 7.5 days

Res:

bStart: 12/12/05 ID: 3

Finish: 12/21/05 Dur: 8 days

Res:

Start

Milestone Date: Sun 12/11/05

ID: 1

The slack values are automatically calculated by MSP. They can be revealed in a number of different views:

ID Task Name Start Finish Late Start Late Finish Free Slack Total Slack1 Start Sun 12/11/05 Sun 12/11/05 Mon 12/12/05 Mon 12/12/05 0 days 0 days

2 a Mon 12/12/05 Wed 12/21/05 Mon 12/12/05 Wed 12/21/05 0 days 0.5 days

3 b Mon 12/12/05 Wed 12/21/05 Mon 12/12/05 Wed 12/21/05 0 days 0 days

4 c Wed 12/21/05 Thu 12/29/05 Tue 1/3/06 Wed 1/11/06 9 days 9 days

5 d Thu 12/22/05 Wed 1/11/06 Thu 12/22/05 Wed 1/11/06 0 days 0 days

6 e Wed 1/11/06 Fri 1/20/06 Wed 1/11/06 Fri 1/20/06 0 days 0 days

7 f Fri 1/20/06 Mon 2/6/06 Fri 1/20/06 Mon 2/6/06 0 days 0 days

8 g Tue 2/7/06 Thu 2/16/06 Tue 2/7/06 Thu 2/16/06 0 days 0 days

9 End Thu 2/16/06 Thu 2/16/06 Thu 2/16/06 Thu 2/16/06 0 days 0 days

9 days

27 4 11 18 25 1 8 15 22 29 5 12 19 26 5 12 19 26 2 9 16 23Nov '05 Dec '05 Jan '06 Feb '06 Mar '06 Apr '06

This view shows the View “Detail Gantt” combined with the “Schedule” Table. Note that the Gantt chart also displays the slack as a green line.

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Chapter 8

Documents

b problem

near critical path b

activity b

appendix b chapter

project management

activities b

critical path activities

expected project duration