Chapter 12: Project Manag ement 1 Solutions to Chapter 12 Exercises
Chapter 12: Project Management 1
Solutions to Chapter 12 Exercises
Chapter 12: Project Management 2
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.
Chapter 12: Project Management 3
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.
Chapter 12: Project Management 4
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
Chapter 12: Project Management 5
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
Chapter 12: Project Management 6
Exercise 3: Network Diagram
11
22
33
44
66
55
77
Chapter 12: Project Management 7
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]
Chapter 12: Project Management 8
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
Chapter 12: Project Management 9
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.
123456789
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$1-B6)/D6 [copy to cells E7:E10]Cell F6 =NORMSDIST(E6) [copy to cells F7:F10]Cell F11 =PRODUCT(F6:F10)
Chapter 12: Project Management 10
Exercise 3: Probability Project Finishes within 24 Weeks
Only paths 1-2-6-7 and 1-3-5-7 need be considered.
123456789
10111213141516
A B C D E FSpecified Time 24
Path PathExpected Path Standard
Path Time Variance Deviation z Probability1-2-6-7 21 6 2.45 1.22 0.8901-3-6-7 19 2 1.41 3.55 1.0001-3-7 17 1 1 7.00 1.000
1-3-5-7 20 4 2 2.00 0.9771-4-5-7 18 6 2.45 2.45 0.993
Product of Probabilites 86.298%
Formulas:Cell E6 =($B$1-B6)/D6 [copy to cells E7:E10]Cell F6 =NORMSDIST(E6) [copy to cells F7:F10]Cell F11 =PRODUCT(F6:F10)
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.
Chapter 12: Project Management 11
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) = -$2398.40
• Given the negative expected value, the firm should not bid.
Chapter 12: Project Management 12
Exercise 4
a 3
b 5
c 3
d 1
0
f 4
g 2h 3
e 3
i 1
The path with the longest expected time is a-b-dummy-f-h-i = 16.
Chapter 12: Project Management 13
Exercise 5123456789
1011121314151617181920212223
A B C D E F GStandard
Activity Optimistic Most Likely Pessimistic te Variance DeviationA 6 7 14 8.00 1.78 1.33B 8 10 12 10.00 0.44 0.67C 2 3 4 3.00 0.11 0.33D 6 7 8 7.00 0.11 0.33E 5 5.5 9 6.00 0.44 0.67F 5 7 9 7.00 0.44 0.67G 4 6 8 6.00 0.44 0.67H 2.5 3 3.5 3.00 0.03 0.17
Formulas:Cell E3 =(B3+(C3*4)+D3)/6 [copy to cells E4:E10]Cell F3 =((D3-B3)/6)^2 [copy to cells F4:F10]Cell G3 =SQRT(F3) [copy to cells G4:G10]
Path PathExpected Path Standard
Path Time Variance DeviationA-D-G 21 2.33 1.53
A-C-E-G 23 2.78 1.67B-E-G 22 1.33 1.15B-F-H 20 0.92 0.96
Chapter 12: Project Management 14
Exercise 5a continued
123456789
10111213141516
A B C D E FSpecified Time 21
Path PathExpected Path Standard
Path Time Variance Deviation z ProbabilityA-D-G 21 2.33 1.53 0.00 0.500
A-C-E-G 23 2.78 1.67 -1.20 0.115B-E-G 22 1.33 1.15 -0.87 0.193B-F-H 20 0.92 0.96 1.04 0.852
Product of Probabilites 0.947%
Formulas:Cell E6 =($B$1-B6)/D6 [copy to cells E7:E19]Cell F6 =NORMSDIST(E6) [copy to cells F7:F9]Cell F11 =PRODUCT(F6:F9)
Chapter 12: Project Management 15
Exercise 5b continued
123456789
10111213141516
A B C D E FSpecified Time 22
Path PathExpected Path Standard
Path Time Variance Deviation z ProbabilityA-D-G 21 2.33 1.53 0.65 0.744
A-C-E-G 23 2.78 1.67 -0.60 0.274B-E-G 22 1.33 1.15 0.00 0.500B-F-H 20 0.92 0.96 2.09 0.982
Product of Probabilites 10.010%
Formulas:Cell E6 =($B$1-B6)/D6 [copy to cells E7:E19]Cell F6 =NORMSDIST(E6) [copy to cells F7:F9]Cell F11 =PRODUCT(F6:F9)
Chapter 12: Project Management 16
Exercise 5c concluded
123456789
10111213141516
A B C D E FSpecified Time 25
Path PathExpected Path Standard
Path Time Variance Deviation z ProbabilityA-D-G 21 2.33 1.53 2.62 0.996
A-C-E-G 23 2.78 1.67 1.20 0.885B-E-G 22 1.33 1.15 2.60 0.995B-F-H 20 0.92 0.96 5.22 1.000
Product of Probabilites 87.689%
Formulas:Cell E6 =($B$1-B6)/D6 [copy to cells E7:E19]Cell F6 =NORMSDIST(E6) [copy to cells F7:F9]Cell F11 =PRODUCT(F6:F9)
Chapter 12: Project Management 17
Exercise 6
# 1
# 2
# 3
# 4# 5
# 6
# 7
# 8
# 9
Chapter 12: Project Management 18
Exercise 6 continued123456789
10111213141516
A B C D E F GStandard
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
Formulas:Cell E3 =(B3+(C3*4)+D3)/6 [copy to cells E4:E10]Cell F3 =((D3-B3)/6)^2 [copy to cells F4:F10]Cell G3 =SQRT(F3) [copy to cells G4:G10]
Chapter 12: Project Management 19
Exercise 6 concluded
Path 2-3-4-5-7-8-9 has the longest expected time. Thus, the project would be expected to take 66.33.
Path PathExpected Path Standard
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
Chapter 12: Project Management 20
Exercise 7
55 70
Actual
Plan
40
35
Time, Days
Pro
gres
s, %
Plan
55 70
1720
30
24
Time, Days
$ (000)
Chapter 12: Project Management 21
Exercise 7 concluded
• 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.
Chapter 12: Project Management 22
Exercise 8
Plan
ValueActual
1 TE 2 3
Time, Months
$ (000)
84
81
78
Chapter 12: Project Management 23
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.
Chapter 12: Project Management 24
Exercise 9
• Spending variance = EC - AC = 39,000 - 34,000 = $5,000.
• Schedule variance = EC - SC = 39,000 - 42,000 = -$3,000.
Chapter 12: Project Management 25
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
Chapter 12: Project Management 26
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
Chapter 12: Project Management 27
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.
Chapter 12: Project Management 28
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
Chapter 12: Project Management 29
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%
Chapter 12: Project Management 30
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.