CB2201 Quantitative Methods
Project Management (Solutions)
Q12.12
Sid Davidson is the personnel director of Babson and Willcount, a company that specializes in
consulting and research. One of the training programs that Sid is considering for the middle-level
managers of Babson and Willcount is leadership training. Sid has listed a number of activities that
must be completed before a training program of this nature could be conducted. The activities and
immediate predecessors appear in the following table:
ACTIVITY
A
IMMEDIATE PREDECESSORS
--
B
C
D
E
F
G
--B
A,D
C
E,F
Develop a network for this problem.
Solution
E
A
Start
B
C
D
G
Finish
F
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Q12.13
Sid Davidson was able to determine the activity times for the leadership training program. He
would like to determine the total project completion time and the critical path. The activity times
appear in the following table (see Problem 12.12):
ACTIVITY
TIME (DAYS)
A
B
C
D
E
2
5
1
10
3
F
G
6
8
35
Solution
S = 13
S=0
A
2
E
3
0
2
15
18
13
15
15
18
B
Start
5
S=0
S=0
S=0
D
10
G
8
0
5
5
15
18
26
0
5
5
15
18
26
Finish
S = 11
S = 11
C
1
F
6
0
1
1
7
11
12
12
18
Critical path is B-D-E-G
Total project completion time is 26 days
2
Q12.16
Monohan Machinery specializes in developing weed-harvesting equipment that is used to clear
small lakes of weeds. George Monohan, president of Monohan Machinery, is convinced that
harvesting weeds is far better than using chemicals to kill weeds. Chemicals cause pollution, and
the weeds seem to grow faster after chemicals have been used. George is contemplating the
construction of a machine that would harvest weeds on narrow rivers and waterways. The activities
that are necessary to build one of these experimental weed-harvesting machines are listed in the
following table. Construct a network for these activities.
ACTIVITY
A
B
C
D
E
F
G
H
IMMEDIATE PREDECESSORS
--A
A
B
B
C,E
D,F
Solution
C
G
A
D
Start
Finish
E
H
B
F
3
Q12.17
After consulting with Butch Rander, George Monhan was able to determine the activity times for
constructing the weed-harvesting machine to be used on narrow rivers. George would like to
determine ES, EF, LS, LF, and slack for each activity. The total project completion time and the
critical path should also be determined. (See Problem 12-16 for details). The activity times are
shown in the following table:
ACTIVITY
TIME (WEEKS)
A
B
C
6
5
3
D
E
F
G
H
2
4
6
10
7
Slack=0
C
Slack=0
A
0
0
Start
6
6
6
Slack=0
B
0
0
5
5
5
3
6
9
6
9
Slack=4
D
6
10
2
8
12
E
4
5
9
5
9
Slack=0
F
5
6
Critical paths are A-C-G and B-E-G.
Slack=0
G
9
9
10
19
19
Finish
H
11
12
7
18
19
Slack=1
6
11
12
Slack=1
Total project completion time is 19 weeks.
4
Q13.24
Bowman Builders manufactures steel storage sheds for commercial use. Joe Bowman, president of
Bowman Builders, is contemplating producing sheds for home use. The activities necessary to
build an experimental model and related data are given in the accompanying table.
(a). What is the project completion date?
(b). Formulate an LP problem to crash this project to 10 weeks.
Activity
Normal Time
A
B
C
D
E
F
G
3
2
1
7
6
2
4
Crash Time Normal Cost ($)
2
1
1
3
3
1
2
1000
2000
300
1300
850
4000
1500
Crash Cost ($)
Immediate Predecessors
1600
2700
300
1600
1000
5000
2000
A
B
C
D,E
Solution
(a)
Thus the project completion date is 14 weeks.
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(b) Now the project is crashed to 10 weeks, the LP formulation will be:
Solution
Activity
Normal
Time
Crash
Time
Normal
Cost ($)
Crash
Cost ($)
Immediate
Predecessors
Crash Cost
per week ($)
A
B
C
D
E
F
3
2
1
7
6
2
2
1
1
3
3
1
1,000
2,000
300
1,300
850
4,000
1,600
2,700
300
1,600
1,000
5,000
A
B
C
600
700
0
75
50
1,000
G
4
2
1,500
2,000
D,E
250
where Crash Cost per week =
Crash Cost - Normal Cost
Normal Time - Crash Time
Decision Variables:
Let Xi be the earliest finish time for activity i (where i = a, b, c,…, g)
Xstart be the start time for Project
Xfinish be the earliest finish time of project
Yi be the amount of time reduced / crashed for activity i
Objective Function:
Min Z = 600YA + 700YB + 0YC + 75YD + 50YE + 1,000YF + 250YG
Constraints:
Precedence relationship
For Activity A,
XA  Xstart + 3 – YA,
i.e. XA – Xstart + YA 
3
[rewrite the equation]
Similarly,
XB – Xstart + YB 
2
XC – Xstart + YC 
1
6
For Activity D,
XD  XA + 7 – YD,
i.e. XD – XA + YD 
7
Similarly,
XE – XB + YE 
6
XF – XC + YF 
2
XG – XD + YG  4
XG – XE + YG  4
Activity Crash Time Limit
YA  1; YB  1; YC  0; YD  4; YE  3; YF  1; YG  2
Project Completion
Xfinish – XG  0
Xfinish – XF  0
Xfinish  10 weeks
Non-negativity
Xi, Xstart, Xfinish, Yi ≥ 0
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