**PLEASE CHECK NUMBERS AND PROVIDE AN EXPLAINATION WITH FORMULAS** Problem 5-6 T
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Question
**PLEASE CHECK NUMBERS AND PROVIDE AN EXPLAINATION WITH FORMULAS**
Problem 5-6
The following represents a project that should be scheduled using CPM:
b. What is the critical path?
c. What is the expected project completion time? (Round your answer to 3 decimal places.)
Project completion time= days
d. What is the probability of completing this project within 22 days? (Do not round intermediate calculations. Round your answer to 4 decimal places.)
Probability =
IMMEDIATE PREDECESSORS TIMES (DAYS) ACTIVITY a m b A — 1 4 7 B — 3 5 10 C A 2 5 8 D A 4 6 11 E B 1 2 3 F C,D 3 5 7 G D,E 1 2 6 H F,G 2 5 6Explanation / Answer
Following table calculates Expected time as well as Standard deviation of each task :
Time ( Days)
Activity
a
M
B
Expected time , e = ( a + 4.m +b)/6
Standard deviation, Sd = ( b - a)/6
A
1
4
7
4
1.00
B
3
5
10
5.5
1.17
C
2
5
8
5
1.00
D
4
6
11
6.5
1.17
E
1
2
3
2
0.33
F
3
5
7
5
0.67
G
1
2
6
2.5
0.83
H
2
5
6
4.67
0.67
The predecessor diagram as follows :
A
B
C
D
E
F
G
H
The possible paths and their corresponding expected durations as follows :
A-C-F-H = 4 + 5 + 5 + 4.67 = 18.67 days
A-D-F-H = 4 + 6.5 + 5 + 4.67 = 20.17 days
A-D-G-H = 4 + 6.5 + 2.5 + 4.67 = 17.67 days
B-E-G-H = 5.5 + 2 + 2.5 + 4.67 = 14.67 days
Out of above , A-D-F-H has the longest duration and hence is the critical path and its duration 20.17 days is the project completion time
CRITICAL PATH = A-D-F-H
PROJECT COMPLETION TIME = 20.17 DAYS
Variance of the critical path
= Sum of variances of A ,D,F and H
= 1 + 1.17^2 +0.67^2 + 0.67^2
= 1 + 1.3689 + 0.4489 + 0.4489
= 3.2667
Therefore,
Standard deviation of the critical path = Sd = Square root ( 3.2667 ) = 1.807
Let the Z value corresponding to probability of completing the project in 22 days = Z1
Therefore,
Duration of critical path + Z1 x Standard deviation of critical path = 22
Or, 20.17 + 1.807.Z1 = 22
Or, 1.807.Z1 = 22 – 20.17 = 1.83
Or, Z1 = 1.83/1.807 = 1.012 ( 1.01 rounded to 2 decimal places )
Corresponding value of probability for Z1 = 1.01 from Normal distribution table = 0.84375
PROBABILITY OF COMPLETING THE PROJECT IN 22 DAYS = 0.8437
Time ( Days)
Activity
a
M
B
Expected time , e = ( a + 4.m +b)/6
Standard deviation, Sd = ( b - a)/6
A
1
4
7
4
1.00
B
3
5
10
5.5
1.17
C
2
5
8
5
1.00
D
4
6
11
6.5
1.17
E
1
2
3
2
0.33
F
3
5
7
5
0.67
G
1
2
6
2.5
0.83
H
2
5
6
4.67
0.67
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