CHAPTER 3
PROJECT MANAGEMENT
2.
a.
6
9
15
19
B (3)
0
19
E (4)
12
15
6
13
26
G (7)
15
19
19
26
6
A (6)
0
6
13
C (7)
6
15
15
D (2)
13
13
18
F (3)
15
16
b.
A-C-D-E-G, also shown in the network above as the bold path.
c.
26 weeks, 6+7+2+4+7.
d.
6 weeks, 15-9.
19
3.
a.
1
5
5
B (4)
1
7
7
9
D (2)
5
5
9
F (2)
7
8
11
G (2)
10
10
12
12
0
15
1
H (3)
A (1)
12
0
1
1
4
7
C (3)
4
b.
12
E (5)
7
7
12
A-B-D-E-H, also shown in the network above as the bold path.
21
15
Chapter 3
c.
15 weeks, 1+4+2+5+3.
d.
C, 3 weeks; F, 1 week; and G, 1 week.
4.
a.
3
5
5
11
B (2)
0
3
3
5
3
7
A (3)
0
E (6)
9
15
7
C (4)
3
13
F (6)
3
7
3
7
7
15
15
G (2)
13
7
D (4)
3
13
13
18
I (3)
15
15
18
10
H (3)
7
12
15
Note that G has both D and F as immediate predecessors. However, D is redundant because F has
D as an immediate predecessor.
b. A-C-F-G-I, and A-D-F-G-I.
c. B is not on a critical path and has slack of 4; therefore, do not shorten as it will not
change the project completion time. Shorten C, D, and G one week each. C and D are on
parallel critical paths, reducing them will only reduce project completion time by 1 week.
d. A-C-F-G-I; and A-D-F-G-I. Project completion time is reduced from 18 to 16 weeks.
6.
a.
22
Project Management
3
5
6
10
C (2)
4.83
0
F (4)
6.83
3
6.83
10.83
7
10.83
A (3)
.83
3.83
3
6
D (3)
G (3.83)
10.83
14.67
H (3.83)
Start
3.83
0
2
6.83
2
10.83
10.83
14.67
7
B (2)
0
7
E (5)
2
2
7
Delete ‘START’ node and the two arcs from it to ’A’ and ‘B’.
b. B-E-G-H
c. 14.67, 2.00+5.00+3.83+3.83
d. Variance of project completion time is found by adding the variances of activities on the
critical path.
Activity
Variance
B
[(3-1)/6]2 = .11
E
[(11-3)/6]2 = 1.78
G
[(6-1)/6]2 = .69
H
[(5-2)/6]2 = .25
Total
2.83
Z
(16  14.67)
= .79
2.83
P(T<16) = .7852
23
Chapter 3
9.
a.
3
5
Job No.
a
m
b
ET
2
1
2
3
4
3.00
.11
2
1
2
3
2.00
.11
3
4
5
12
6.00
1.78
4
3
4
11
5.00
1.78
5
1
3
5
3.00
.44
6
1
2
3
2.00
.11
7
1
8
9
7.00
1.78
8
2
4
6
4.00
.44
9
2
4
12
5.00
2.78
10
3
4
5
4.00
.11
11
5
7
8
6.83
.25
5
8
5 (3)
2 (2)
11
6
8
8
15
11
8 (4)
15
11
0
3
3
1 (3)
0
9
9
3 (6)
3
15
6 (2)
9
9
11
3
8
8
15
4 (5)
4
9 (5)
11
3
9
20
20
15
26.83
11 (6.83)
20
7 (7)
9
20
15
26.83
19
10 (4)
16
16
20
b. 1-3-6-8-9-11.
c. 26.83, 3.00+6.00+2.00+4.00+5.00+6.83
d. (1) No. Job 5 is not on the critical path; therefore, reducing it time by two days will not
reduce project completion time.
(2) No. Job 3 is on the critical path, but reducing it by two days shifts the critical path to
(1-4-7-10-11), only saving one day. Therefore, $1,000 is saved in completion time at a cost
of $1,500.
24
Project Management
e. Variance of project completion time is found by summing the variances of activities on the
critical path. (1-3-6-8-9-11), .11+1.78+.11+.44+2.78+.25=5.47.
Z
30  26.83
= 1.35
5.47
P(T>30) = .0885
11.
7
16
10
20
F (4)
B (3)
8
0
16
11
7
11
A (7)
0
20
16
20
D (5)
7
11
7
G (5)
16
11
20
16
C (4)
7
25
25
18
E (2)
11
18
20
a. A-C-D-F-G
b.
Activity
Normal
Time (NT)
Crash Time
(CT)
Normal Cost
(NC)
Crash Cost
(CC)
NT-CT
Cost/day to
expedite
A
7
6
$7,000
$8,000
1
$1,000
B
3
2
5,000
7,000
1
2,000
C
4
3
9,000
10,200
1
1,200
D
5
4
3,000
4,500
1
1,500
E
2
1
2,000
3,000
1
1,000
F
4
2
4,000
7,000
2
1,500
G
5
4
5,000
8,000
1
3,000
First, lowest cost activities to crash are A and E at $1,000 per day. E is not on the critical path,
therefore select A. Critical path remains the same. Second, lowest cost activity on the critical
path is C. Crash activity C. The critical path remains the same. Third, D and F are next lowest
cost activities on the critical path. Both have a cost of $1,500 per day. Select D then F or reverse
25
Chapter 3
the order (F then D). F can not be reduced by two day because it would cause E to become part
of a critical path.
Summary of steps to reduce project by four days:
Step
Activity to crash
Cost to crash
Days saved
1
A
$1,000
1
2
C
1,200
1
3
D (or F)
1,500
1
4
F (or D)
1,500
1
13.
15
21
D (6)
5
15
15
21
B (10)
5
0
15
5
13
A (5)
0
20
21
E (7)
5
5
25
G (4)
14
21
13
17
21
25
13
C (8)
6
14
F (4)
17
21
a. A-B-D-G, 25 weeks, 5+10+6+4.
b.
Activity
Normal
Time (NT)
Normal Cost
(NC)
Crash Time
(CT)
Crash Cost
(CC)
NT-CT
Cost/week
to expedite
A
5
$7,000
3
$13,000
2
$3,000
B
10
12,000
7
18,000
3
2,000
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Project Management
C
8
5,000
7
7,000
1
2,000
D
6
4,000
5
5,000
1
1,000
E
7
3,000
6
6,000
1
3,000
F
4
6,000
3
7,000
1
1,000
G
4
7,000
3
9,000
1
2,000
First, reduce D (lowest cost activity on the critical path) by one week. This adds an additional
critical path with activities C and E in it. Second, crash activity G by one week. Critical paths
remain the same. Third, crash activity A by one week at a cost of $3,000, which is the least
expensive.
Summary of activities crashed:
Step
Activity
Cost to crash
Weeks reduced
1
D
$1,000
1
2
G
2,000
1
3
A
3,000
1
Total cost
$6,000
15.
a.
Activity
A
B
C
D
E
F
G
H
Expected Time
Activity Variance
a  4m  b
6
 ba 


 6 
5.00
5.00
6.17
2.00
3.00
3.83
7.50
2.00
1
1 7/9
25/36
0
1 7/9
1/4
1 1/3
1/9
b.
27
2
Chapter 3
5
8
11.17
E (3.00)
8.17
0
5
G (7.50)
11.17
5
11.17
18.67
11.17
A (5.00)
C (6.17)
15
11.17
5
0
18.67
18.67
5
Start
F (3.83)
0
5
5
7
B (5.00)
7.84
20.67
11.17
14.84
18.67
H (2.00)
18.67
20.67
D (2.0)
12.84
12.84
14.84
Delete ‘START’ node and the two arcs from it to ’A’ and ‘B’.
c. Shown on diagram.
d. Shown on diagram.
e.
Z
D T

E
2
cp

19  20.67
 1.67

 .9384
25 1 1 1.779
1 1 
36 3 9
Look up that value in Appendix E and we see that there is 17 percent chance of completing the
project by that date.
16.
a.
Activity
A
B
C
D
E
F
G
H
Expected Time
Activity Variance
a  4m  b
6
 ba 


 6 
4.17
4.33
6.00
3.17
3.67
3.83
7.50
2.00
25/36
1 7/9
1
1/4
1 7/9
1/4
1 13/36
1/9
28
2
Project Management
I
5.00
4/9
b.
4.17
8.50
8.50
B (4.33)
8.50
8.50
4.17
19.67
G (7.50)
12.17
12.17
19.67
I (5.00)
D (3.17)
19.67
13.84
10.67
24.67
19.67
13.34
10.17
A (4.17)
0
12.17
E (3.67)
4.17
0
12.17
24.67
4.17
4.17
10.17
C (6.00)
4.67
17.17
13.34
17.17
F (3.83)
10.67
13.84
19.17
H (2.00)
17.67
17.67
19.67
c. Shown on diagram.
d. Shown on diagram.
e.
Z
D T

E
2
cp

26  24.67
1.33

 .54047
25 7 7 13 4 2.4608
1 1 1 
36 9 9 36 9
Look up that value in Appendix E and we see that there is 71 percent chance of completing the
project by that date. The probability it will take longer is 1-.71 or 29 percent.
f. All the other paths will create problems since there is only a .5 days of slack.
29