chapter 12: project management1 solutions to chapter 12 exercises

Chapter 12: Project Management 1

Solutions to Chapter 12 Exercises


Exercise 1•There are four paths:

A-C-I (7 days)

A-D-G-J (13 days)

B-E-G-J (11 days)

B-F-H-J (10 days)

•Thus, the critical path is A-D-G-J and these are the activities that should be monitored most closely because if they slip, the entire project will be delayed.


Exercise 1 concluded

• The time to complete the four paths is now:• A-C-I (5.5 days)

• A-D-G-J (11.5 days)

• B-E-G-J (12.5 days)

• B-F-H-J (11.5 days)

• The critical path is now B-E-G-J and the project completion is expected to be 12.5 days.

• Activities G and J remain as critical activities while A and D are no longer critical.


Exercise 2

Activity TES TEF TLS TLF Slacka 0 4 0 4 0b 0 3 2 5 2c 4 5 10 11 6d 4 9 4 9 0e 3 7 5 9 2f 3 4 6 7 3g 9 12 9 12 0h 4 9 7 12 3i 5 7 11 13 6j 12 13 12 13 0


Exercise 2: Path Slacks

Path A-C-I: Slack = 6

Path A-D-G-J: Slack = 0

Path B-E-G-J: Slack = 2

Path B-F-H-J: Slack = 10


Exercise 3: Network Diagram

11

22

33

44

66

55

77


Exercise 3: Activity Expected Times, Variances and Standard Deviations

1234567891011121314151617

A B C D E F GStandard

Activity Optimistic Most Likely Pessimistic te Variance Deviation1-2 5 11 11 10 1 11-3 10 10 10 10 0 01-4 2 5 8 5 1 12-6 1 7 13 7 4 23-6 4 4 10 5 1 13-7 4 7 10 7 1 13-5 2 2 2 2 0 04-5 0 6 6 5 1 15-7 2 8 14 8 4 26-7 1 4 7 4 1 1

Formulas:Cell E3 =(B3+(C3*4)+D3)/6 [copy to cells E4:E12]Cell F3 =((D3-B3)/6)^2 [copy to cells F4:F12]Cell G3 =SQRT(F3) [copy to cells G4:G12]


Exercise 3: Path Expected Times, Variances, and Standard Deviations

Path Path Path ExpectedExpected Path Standard Time + 2.33

Path Time Variance Deviation Standard Deviations1-2-6-7 21 6 2.45 26.711-3-6-7 19 2 1.41 22.291-3-7 17 1 1 19.33

1-3-5-7 20 4 2 24.661-4-5-7 18 6 2.45 23.71


Exercise 3: Probability Project Finishes within 17 Weeks

All paths must be considered since they all have a reasonable chance of taking longer than 17 weeks. Thus, there is a very small chance (i.e., .005%) that the project can be completed in 17 weeks or less.

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10111213141516

A B C D E FSpecified Time 17

Path PathExpected Path Standard

Path Time Variance Deviation z Probability1-2-6-7 21 6 2.45 -1.63 0.0511-3-6-7 19 2 1.41 -1.42 0.0781-3-7 17 1 1 0.00 0.500

1-3-5-7 20 4 2 -1.50 0.0671-4-5-7 18 6 2.45 -0.41 0.342

Product of Probabilites 0.005%

Formulas:Cell E6 =($B$1-B6)/D6 [copy to cells E7:E10]Cell F6 =NORMSDIST(E6) [copy to cells F7:F10]Cell F11 =PRODUCT(F6:F10)


Exercise 3: Probability Project Finishes within 24 Weeks

Only paths 1-2-6-7 and 1-3-5-7 need be considered.

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10111213141516



Path Time Variance Deviation z Probability1-2-6-7 21 6 2.45 1.22 0.8901-3-6-7 19 2 1.41 3.55 1.0001-3-7 17 1 1 7.00 1.000

1-3-5-7 20 4 2 2.00 0.9771-4-5-7 18 6 2.45 2.45 0.993



However, all paths can be easily included by simply changing cell B1 from 17 to 24. With all paths considered, there is a over an 86% chance of completing the project in 24 weeks.


Exercise 3: Should the Project be Pursued?

• Using the spreadsheet it can be easily determined that the probability of finishing within 18 weeks is .176% and the probability of the project taking 22 weeks or longer is 48.321%.

• The expected value of the project is thus:

(.176% $10,000) + (48.321% -$5,000) = -$2398.40

• Given the negative expected value, the firm should not bid.


Exercise 4

a 3

b 5

c 3

d 1

0

f 4

g 2h 3

e 3

i 1

The path with the longest expected time is a-b-dummy-f-h-i = 16.


Exercise 5123456789

1011121314151617181920212223


Activity Optimistic Most Likely Pessimistic te Variance DeviationA 6 7 14 8.00 1.78 1.33B 8 10 12 10.00 0.44 0.67C 2 3 4 3.00 0.11 0.33D 6 7 8 7.00 0.11 0.33E 5 5.5 9 6.00 0.44 0.67F 5 7 9 7.00 0.44 0.67G 4 6 8 6.00 0.44 0.67H 2.5 3 3.5 3.00 0.03 0.17



Path Time Variance DeviationA-D-G 21 2.33 1.53

A-C-E-G 23 2.78 1.67B-E-G 22 1.33 1.15B-F-H 20 0.92 0.96


Exercise 5a continued

123456789

10111213141516



Path Time Variance Deviation z ProbabilityA-D-G 21 2.33 1.53 0.00 0.500

A-C-E-G 23 2.78 1.67 -1.20 0.115B-E-G 22 1.33 1.15 -0.87 0.193B-F-H 20 0.92 0.96 1.04 0.852




Exercise 5b continued

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10111213141516




A-C-E-G 23 2.78 1.67 -0.60 0.274B-E-G 22 1.33 1.15 0.00 0.500B-F-H 20 0.92 0.96 2.09 0.982




Exercise 5c concluded

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10111213141516




A-C-E-G 23 2.78 1.67 1.20 0.885B-E-G 22 1.33 1.15 2.60 0.995B-F-H 20 0.92 0.96 5.22 1.000




Exercise 6

# 1

# 2

# 3

# 4# 5

# 6

# 7

# 8

# 9


Exercise 6 continued123456789

10111213141516


Activity Optimistic Most Likely Pessimistic te Variance Deviation1 8 10 13 10.17 0.69 0.832 5 6 8 6.17 0.25 0.503 13 15 21 15.67 1.78 1.334 10 12 14 12.00 0.44 0.675 11 20 30 20.17 10.03 3.176 4 5 8 5.33 0.44 0.677 2 3 4 3.00 0.11 0.338 4 6 10 6.33 1.00 1.009 2 3 4 3.00 0.11 0.33




Path 2-3-4-5-7-8-9 has the longest expected time. Thus, the project would be expected to take 66.33.


Path Time Variance Deviation1-4-5-6-9 50.67 11.72 3.42

1-4-5-7-8-9 54.67 12.39 3.522-3-4-5-6-9 62.33 13.06 3.61

2-3-4-5-7-8-9 66.33 13.72 3.70


Exercise 7

55 70

Actual

Plan

40

35

Time, Days

Pro

gres

s, %

Plan

55 70

1720

30

24

Time, Days

$ (000)



• Spending variance = EC - AC = 17,000 - 30,000 = -$13,000. This indicates a cost overrun of $13,000.

• Schedule variance = EC - SC = 17,000 - 24,000 = -$7,000 indicating the project is behind.

• Time variance = TE - TA = 55 -70 = -15 days indicating a 15 day delay.


Exercise 8

Plan

ValueActual

1 TE 2 3

Time, Months

$ (000)

84

81

78


Exercise 8 concluded• Spending variance = EC - AC = 81,000 - 78,000 =

$3,000.• Schedule variance = EC - SC = 81,000 - 84,000 = -

$3,000.• TE can be estimated from chart by determining

how much time should have elapsed according to the plan to achieve the current value completed. In this case TE is approximately 1.5 months.

• Thus, the spending variance = TE - TA = 1.5 - 2 = -0.5 month.


Exercise 9

• Spending variance = EC - AC = 39,000 - 34,000 = $5,000.

• Schedule variance = EC - SC = 39,000 - 42,000 = -$3,000.


Exercise 10a1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253

A B C D E F G H I JActivity Activity Activity Activity Activity Activity Path 1 Path 2 Path 3 Project

A B C D E F (A-C-F) (B-D-F) (B-E) Finish Time31.74 26.63 22.08 30.16 42.55 37.92 91.74 94.72 69.18 94.7230.57 23.56 24.88 26.19 48.86 32.67 88.12 82.42 72.42 88.1232.39 24.15 18.61 23.46 25.97 35.06 86.07 82.67 50.12 86.0733.63 22.74 25.68 22.19 31.69 33.66 92.97 78.59 54.43 92.9733.54 24.12 25.44 23.69 38.36 36.76 95.74 84.57 62.48 95.7434.18 17.68 19.09 26.78 32.29 29.92 83.19 74.37 49.96 83.1929.48 24.19 22.35 26.72 36.38 36.12 87.95 87.03 60.57 87.9531.82 20.48 21.27 23.22 40.52 31.10 84.19 74.81 61.00 84.1933.41 25.09 21.82 14.95 33.93 31.71 86.95 71.75 59.01 86.9530.80 26.16 19.69 17.69 35.33 33.53 84.02 77.37 61.49 84.0231.27 28.68 25.52 26.88 33.76 33.88 90.67 89.44 62.44 90.6730.07 22.68 24.16 32.87 32.59 30.60 84.83 86.15 55.27 86.1529.88 22.28 24.16 27.82 40.50 40.36 94.40 90.46 62.78 94.4030.93 25.05 24.01 27.75 46.97 23.91 78.85 76.71 72.03 78.8531.17 23.88 23.58 23.07 48.14 36.78 91.53 83.73 72.02 91.5329.56 23.95 20.84 25.02 32.60 32.78 83.18 81.75 56.55 83.1831.42 29.49 19.06 20.17 30.52 40.86 91.34 90.52 60.00 91.3431.62 24.72 22.33 30.88 41.09 27.52 81.47 83.11 65.80 83.1132.26 24.15 23.74 21.84 34.14 33.73 89.73 79.72 58.29 89.7331.66 30.27 20.64 24.08 35.54 33.53 85.84 87.89 65.81 87.8931.71 22.64 17.65 25.07 32.56 41.25 90.60 88.96 55.21 90.6031.66 28.54 20.31 13.84 31.63 37.55 89.51 79.93 60.17 89.5133.71 24.55 18.78 24.92 22.78 36.90 89.39 86.36 47.34 89.3932.00 24.40 26.39 25.76 40.33 38.55 96.94 88.71 64.73 96.9431.88 29.59 20.08 29.10 36.93 34.24 86.20 92.94 66.52 92.9431.48 25.72 25.59 25.78 27.38 33.83 90.90 85.33 53.10 90.9034.47 25.95 22.79 31.15 33.61 40.33 97.59 97.43 59.56 97.5933.14 23.87 27.24 17.60 21.37 40.63 101.02 82.10 45.25 101.0234.95 26.72 20.24 25.69 23.76 31.95 87.15 84.36 50.47 87.1531.31 24.39 22.42 28.50 31.28 32.51 86.24 85.39 55.66 86.2434.09 22.32 20.29 21.12 32.69 28.93 83.31 72.37 55.01 83.3130.17 21.20 26.01 23.78 31.51 32.84 89.02 77.82 52.71 89.0232.75 25.23 21.75 25.85 24.26 33.50 88.00 84.59 49.49 88.0033.18 21.56 21.95 17.10 36.17 35.12 90.26 73.78 57.73 90.2634.40 30.61 22.24 18.72 30.16 29.11 85.75 78.44 60.77 85.7532.00 24.21 19.67 37.33 28.22 40.12 91.79 101.66 52.43 101.6631.47 19.68 23.33 21.92 28.16 33.39 88.19 74.98 47.84 88.1932.91 23.56 27.31 22.08 42.91 25.76 85.98 71.39 66.47 85.9831.64 27.57 23.64 17.89 21.53 38.41 93.69 83.87 49.09 93.6933.01 23.41 19.38 28.45 28.79 31.92 84.31 83.77 52.20 84.3130.37 22.57 19.36 29.80 43.93 28.36 78.08 80.73 66.51 80.7331.08 20.36 23.00 21.78 33.03 36.10 90.18 78.24 53.40 90.1830.27 25.58 23.16 29.07 40.26 34.39 87.82 89.04 65.84 89.0431.66 24.09 20.97 23.61 32.56 45.16 97.79 92.86 56.65 97.7932.06 26.37 24.14 27.89 40.92 35.00 91.19 89.26 67.29 91.1932.13 22.62 20.15 21.86 42.77 37.79 90.07 82.27 65.39 90.0731.71 23.53 22.47 29.62 29.14 34.77 88.95 87.92 52.67 88.9534.73 25.49 23.51 23.34 41.17 42.19 100.43 91.01 66.65 100.4330.01 23.57 23.51 29.64 25.61 28.49 82.01 81.70 49.19 82.0131.22 25.26 21.79 29.28 45.08 37.49 90.50 92.04 70.33 92.0429.01 16.93 24.44 24.72 39.28 32.26 85.70 73.91 56.22 85.70


Exercise 10a (concluded)54555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106

A B C D E F G H I J33.84 24.39 22.82 22.38 30.70 34.80 91.46 81.57 55.09 91.4630.56 25.37 25.88 30.13 30.90 40.78 97.23 96.28 56.26 97.2331.32 31.16 21.56 28.61 26.64 31.11 83.99 90.88 57.80 90.8833.01 19.87 19.25 27.95 37.54 39.25 91.51 87.07 57.41 91.5132.66 19.77 21.28 25.74 34.28 28.73 82.67 74.24 54.05 82.6733.15 17.46 25.10 23.48 39.57 33.31 91.56 74.24 57.03 91.5632.81 28.23 22.75 16.31 26.50 34.54 90.10 79.08 54.73 90.1030.45 31.96 24.26 36.33 24.67 31.12 85.83 99.40 56.63 99.4030.76 23.39 20.73 33.46 34.40 41.09 92.58 97.94 57.79 97.9432.93 21.50 19.29 31.90 51.24 36.21 88.44 89.61 72.74 89.6132.49 26.30 21.55 30.11 30.49 37.61 91.65 94.02 56.79 94.0230.97 26.64 18.60 37.18 21.54 39.69 89.26 103.51 48.18 103.5131.81 21.96 26.53 25.83 25.75 39.64 97.99 87.43 47.71 97.9932.26 27.11 21.01 24.61 33.07 28.30 81.57 80.02 60.18 81.5732.77 26.65 25.65 30.91 40.72 38.32 96.74 95.88 67.37 96.7432.27 23.96 18.42 26.15 39.00 41.17 91.86 91.28 62.96 91.8631.01 19.80 22.48 27.48 31.20 31.64 85.12 78.92 51.00 85.1234.36 27.74 20.86 17.46 32.91 31.80 87.03 77.00 60.65 87.0332.68 26.97 24.46 25.33 36.79 32.88 90.03 85.18 63.75 90.0332.19 23.51 27.25 28.90 40.35 37.06 96.50 89.47 63.87 96.5033.10 22.46 20.11 22.59 31.75 41.14 94.34 86.18 54.21 94.3433.13 27.27 23.15 20.02 39.43 35.52 91.81 82.82 66.71 91.8131.34 25.68 27.06 34.25 37.48 25.96 84.36 85.89 63.16 85.8930.99 26.32 21.15 30.72 28.80 33.39 85.53 90.43 55.12 90.4333.43 31.07 24.21 30.37 36.69 35.46 93.10 96.90 67.76 96.9030.66 25.45 20.67 30.29 33.16 38.60 89.92 94.33 58.61 94.3330.23 21.17 21.88 23.45 28.66 35.64 87.75 80.25 49.82 87.7532.95 26.09 19.81 30.30 33.65 29.42 82.18 85.81 59.74 85.8132.87 20.79 22.50 20.41 26.10 36.35 91.71 77.55 46.89 91.7134.75 26.35 19.76 26.12 19.13 39.31 93.82 91.78 45.49 93.8233.83 24.11 21.58 22.74 35.97 37.71 93.12 84.56 60.09 93.1233.66 25.53 23.44 29.48 39.06 32.51 89.62 87.51 64.59 89.6232.24 25.71 23.70 27.42 44.41 28.87 84.81 82.00 70.12 84.8132.10 25.01 20.68 32.09 28.78 40.28 93.06 97.37 53.79 97.3732.64 23.76 22.22 28.55 29.19 37.30 92.17 89.60 52.95 92.1732.07 20.42 21.26 23.83 34.53 33.22 86.55 77.47 54.95 86.5530.83 31.17 19.08 30.65 32.66 31.33 81.23 93.15 63.84 93.1529.97 29.55 22.05 28.79 28.27 38.04 90.05 96.37 57.82 96.3733.09 27.85 25.54 29.40 36.10 33.76 92.40 91.02 63.95 92.4032.63 20.02 22.41 37.54 33.62 37.46 92.50 95.02 53.64 95.0232.84 21.14 21.97 21.59 34.08 35.42 90.23 78.14 55.21 90.2332.36 26.25 24.93 16.63 43.11 37.69 94.97 80.57 69.36 94.9730.87 24.83 21.42 28.33 43.27 31.48 83.77 84.64 68.10 84.6433.59 24.57 18.85 21.36 37.76 31.52 83.95 77.45 62.33 83.9531.73 25.10 24.57 22.93 28.40 33.05 89.35 81.08 53.50 89.3531.09 18.77 22.64 42.31 31.15 34.51 88.24 95.58 49.92 95.5831.11 23.80 20.09 27.48 23.14 31.23 82.44 82.52 46.94 82.5231.59 21.69 21.88 26.99 41.61 28.94 82.41 77.62 63.31 82.4131.56 21.51 19.63 24.54 32.15 37.14 88.32 83.19 53.65 88.32

Avg 89.08 85.31 58.51 90.33S.D. 4.72 7.30 6.93 5.16Max 101.02 103.51 72.74 103.51Min 78.08 71.39 45.25 78.85


Exercise 10b

•From the above spreadsheet, the average

project completion time is 90.33 days with

a standard deviation of 5.16 days.

•Across the 100 replications of the

simulation model, the project finished as

quickly as 78.85 days and took as long as

103.51 days.


Exercise 10cc .

T h e a b o v e h is to g ra m c le a r ly il lu s t ra te s the u nc e r ta in t y a s so c ia te d w ith the a c tu a la m o u n t o f t im e it w ill t a k e to co m p le te the p ro je c t . A c c o rd in g to th e h is to g ra m ,th e re is a s m a ll c h a n c e th e p ro je c t c a n be c o m p le te d in le s s tha n 8 0 d a ys a n d as lig h t ly la rg e r c h a nc e it w ill t a k e m o re th a n 1 0 0 d a ys . In th is c a se , if th e m a na g e rfo c u se d o n ly o n th e p a th w ith th e lo n g e s t e xp e c te d c o m p le t io n t im e ( i.e . , p a th A -C -F ) a nd e s t im a te d th e p ro je c t c o m p le t io n t im e o n th e ba s is o f th is p a th a s 8 8 .8d a y s, the re su lt s o f the s im u la t io n su g g e s t th a t the re w o u ld b e a 6 3 p e rc e n t c h a n c eo f e x c e e d in g th is e s t im a te . In fa c t , in 2 7 p e rc e n t o f th e s im u la t io n re p lic a t io n s ,p a th B -D -F tu rn e d o u t to b e the c r it ic a l p a th .

Distribution of Project Completion Times

0

5

10

15

20

25

30

35

40

80 85 90 95 100 105

Project Completion Time

Fre

qu

en

cy


Exercise 10d• Path A-C-F has an expected duration of 88.8 days with a standard deviation of

4.81 days. Based on this, the probability that it is completed in 80 days or less can be calculated as:

• From the z table in the Appendix A, the probability of the path being completed in 80 days or less is 3.51%. Alternatively, this can be found using Excel as follows: =normdist(80,88.8,4.81,true) = 3.37. Incidentally, the value found using Excel is actually more accurate since rounding the value of z was not required.

• Path B-D-F has an expected duration of 85.2 days and standard deviation of 7.31 days. Using Excel the probability it is completed in 80 days or less is found as: =normdist(80,85.2,7.31,true) = 23.84%.

• Path B-E has an expected duration of 59.0 days and standard deviation of 6.93 days. Using Excel the probability it is completed in 80 days or less is found as: =normdist(80,59,6.93,true) = 99.88%.

• The probability the entire project is completed in 80 days or less is calculated as (.0337)(.2384)(.9988) = 0.80%


Exercise 10e

• The results shown in the spreadsheet to part a are consistent with the probability calculated in part d.

• In only one of the 100 replications was the project completed in 80 days or less (i.e., rep number 14).

• One percent of the replications is quite close to the 0.8% calculated in part d.

chapter 12: project management1 solutions to chapter 12 exercises

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