21
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
Slack
A
—
4
0
4
0
4
0
B
A
2
4
6
4
6
0
C
A
6
4
10
9
15
7
D
B
7
6
13
6
13
0
E
B
8
6
14
24
32
18
F
D
7
13
20
13
20
0
G
12
20
32
20
32
0
H
C
9
10
19
15
24
5
I
H
8
19
27
24
32
5
J
2
32
34
32
34
0
K
H
3
22
25
31
34
9
L
J, K
0
34
34
34
34
0
Critical path is A-B-D-F-G-J-L with a final project completion time of 34 days. The largest
activity slack is for E at 18 days.
42. Two international banks are integrating two financial processing software systems as a
result of their merger. Preliminary analysis and interviews with all parties involved resulted
in the following project information. The “systems integration team” for this project plans
a. Draw the project network.
22
b. Identify the critical path(s).
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
Slack
A
—
3
0
3
0
3
0
B
A
1
3
4
13
14
10
C
A
2
3
5
3
5
0
D
B, C
3
5
8
14
17
9
E
C
5
5
10
5
10
0
F
C
3
5
8
9
4
G
E
7
10
17
10
17
0
H
5
2
I
8
0
Critical path is A-C-E-G-I.
c. What is the total project completion time and total cost?
Completion time is 25 weeks.
43. You are assigned your first project and if things go well you should get promoted. The CEO
is aware of this project, so you have a chance for high visibility. You want to do a good job!
Your first task is to define the work break down structure by interviewing project activity
23
Crash cost/week = ($17,365 – $7,560)/(5 weeks – 2 weeks) = $9,805/3 = $3,268
44. You have collected the information in the table below for the project you are leading on
web page design and development. The times are in weeks.
a. What is are the total time of this project?
A
D
C
The critical path is A-B-D-E, a total of 12 weeks. Activity C has a slack time of 3 weeks (all
other activities are on the critical path)
b. What is the total normal cost of the project?
Normal cost is $200 + $300 + $200 + $500 + 500 = $1,700. This is the sum of the “normal
c. If you need to crash the project by one week, what activity would you crash to minimize
project costs and what is the revised completion time?
Crash costs per activity are as follows: A $200, B $300, C – cannot crash, D – $50, E $400.
45. You are a marketing major and have been working at a major consumer products company for the
past three years. Today, your boss walked into your office and said, “Polly, I want you to be the
Project Leader for our launch of Perfect Clean Home System. It must be successful; we have invested
over $100 million in its development.” After discussing the project for an hour, your boss left the
room. After several days of thinking about the project, panic began to overwhelm you. You realized
B
24
you should have paid more attention to your operations management course, and especially the
a. Draw the project network.
b. What is the critical path(s) and expected completion time for this project?
Activity
Normal
Time
Crash Time
Normal Cost
Crash Cost
Crash
cost/week
A
3
2
$14,000
$16,500
$2,500
B
6
6
$5,500
$5,500
—–
C
2
1
$8,000
$2,000
5
3
$3,500
E
2
1
$10,500
$1,500
7
6
$3,000
The critical path is A-F-G at 14 weeks
A
D
25
c. What activity has the largest amount of slack?
Immediate
Predecessor
Time
Earliest
Earliest
Latest
Latest
Activity
(days)
Start
Finish
Start
Finish
Slack
A
None
3
0
3
0
3
0
B
None
6
0
6
4
10
4
C
None
2
0
2
6
8
6
D
5
3
8
5
10
2
E
2
2
4
8
10
6
F
7
3
10
3
10
0
G
4
10
14
10
14
0
Activity C and E both have 6 weeks of slack
d. If activity D is delayed by two weeks, what are the revised critical path(s)?
There are now two critical paths: A-D-G and A-F-G at 14 weeks
e. What activity or activities would you crash to reduce the original project completion
time by two weeks at minimum crash costs?
f. What is the total project (normal plus crash) costs given your answer to the previous
question?
46. Office Automation, Inc. has developed a proposal for introducing a new computerized
office system that will improve word processing and interoffice communications for a
particular company. Contained in the proposal is a list of activities that must be
accomplished to complete the new office-system project. Information about the activities
is shown below.
Times are in weeks and costs are in thousands of dollars.
a. Draw the network for this project.
B
C
F
b. Develop a schedule for the project using normal times.
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
Slack
A
—
10
0
10
0
10
0
B
A
8
10
18
10
18
0
C
B
10
18
28
18
28
0
D
A
7
10
17
11
18
1
E
D
10
17
27
18
28
1
F
3
28
31
28
31
0
c. What is the critical path and the expected project-completion time?
Critical path is A-B-C-F. Completion time is 31 weeks.
d. Assume the company wants to complete the project in 26 weeks. What crashing
decisions would you recommend for meeting the completion date at the least possible
cost? What is the added project cost to meet the 26-week completion time?
A
D
E
27
Total Crash
Crash Activities Number of Weeks Cost
A 2 $40 = ($70 – $30)/(10 – 8) = $20(2 weeks)
e. Develop an activity schedule for the crashed project using early and late start times, early
and late finish times and slack.
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
Slack
A
—
8
0
8
0
8
0
B
A
6
8
14
8
14
0
C
B
9
14
23
14
23
0
D
A
6
8
8
0
E
D
9
14
23
14
23
0
F
3
All activities are now critical.
47. You are in charge of the purchasing, installing, and testing a new 30-ton stamping machine
for a major automobile assembly line. It stamps out metal fenders and hoods for vehicles
prior to painting them. The goal is to complete the project on time and within budget or
you may not have this job next year—the pressure is on! The activities listed are for
installing and testing the machine, not the purchase price.
28
a. Draw the project network.
b. What is the earliest completion time for this project?
Critical path is A-D-F-G and a 13 day completion time.
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
Slack
A
None
2
0
2
0
2
0
B
None
1
0
1
1
2
1
C
A, B
5
2
7
3
8
1
D
4
2
6
2
6
0
2
7
9
8
1
4
6
6
0
G
3
0
c. What is the largest amount of slack that any activity has in this project?
The largest amount of slack is one day
d. What is the total normal cost of the project?
e. If your boss gives you $5,000 to speed up the original completion time (i.e., project
improvement dollars), what activity would you spend the improvement dollars on first?
Activity crash costs/day are as follows: A and B cannot be crashed, C = $3,000, D =
C
A
E
f. What activity or activities would you crash to complete the project two weeks early
from the original project completion time to minimize total crash costs?
Options
48. Suppose that some of the activities in the Environment Recycling, Inc. project (problem 40)
can be crashed. The table below shows the crash times and costs associated with
performing the activities at their original (normal) times and also for the crash times (all
times are in weeks). Find the total project completion time and lowest cost solution if the
state wants to complete the project three weeks early.
30
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
Slack
A
—
5
0
5
0
5
0
B
A
8
5
13
5
13
0
C
A
7
5
6
1
D
—
6
0
6
E
8
13
21
13
21
0
F
D
3
6
9
G
D
3
6
9
H
E
4
21
25
21
25
0
I
6
0
Critical path is A-B-E-H-I. Completion time = 31. Total normal cost = $7,200.
Crashing options on the critical path (shown in red):
Activity
Normal
Time
Crash Time
Normal Cost
Crash Cost
Crash
cost/week
A
5
4
$400
$750
$350
B
8
6
1800
2200
$200
C
7
6
800
1100
300
D
6
5
600
1000
400
E
8
6
1700
2200
3
2
800
1000
200
G
3
2
500
650
150
H
4
3
6
5
1300
Crash A by 1 week: $350
Crash B by 2 weeks: $400 ($200/week)
To reduce completion time to 28 weeks, we have multiple options. Typically, select the
activity with the smallest crashing rate first. Thus, we can choose B or H. Suppose we
choose B to crash for 2 weeks:
31
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
Slack
A
—
5
0
5
0
5
0
B
A
6
5
11
6
12
1
C
A
7
5
5
0
D
—
6
0
6
E
8
12
20
12
20
0
F
D
3
6
9
G
D
3
6
9
H
E
4
20
24
20
24
0
I
6
0
Note that this only reduces the project time by one week since we have a new critical path,
A-C-E-H-I with a completion time of 30. We would have gotten the same solution by
crashing B for only 1 week at a cost of $200 and which would have resulted in multiple
critical paths.
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
Slack
A
—
5
0
5
0
5
0
B
A
7
5
12
5
12
0
C
A
7
5
5
0
D
—
6
0
6
E
8
12
20
12
20
0
F
D
3
6
9
G
D
3
6
9
H
E
4
20
24
20
24
0
I
6
0
32
By crashing E for 2 weeks we have:
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
Slack
A
—
5
0
5
0
5
0
B
A
7
5
12
5
12
0
C
A
7
5
12
5
12
0
D
—
6
0
6
13
19
13
E
6
12
18
12
18
0
F
D
3
6
9
19
24
13
G
D
3
6
9
19
22
13
H
E
4
18
22
18
22
0
I
6
22
28
0
The total cost of this option is $700. Students might try other sequences to seek a better
solution. Another solution is
H 1 Week $200
49. The table below shows the crash times, and normal and crash costs for the international
bank systems integration project. What is the total project completion time and lowest–
cost solution if the bank wants to complete the project two weeks early?
33
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
Slack
A
—
3
0
3
0
3
0
B
A
1
3
4
13
14
10
C
A
2
3
5
3
5
0
D
B,C
5
8
9
E
C
5
5
10
5
10
0
F
C
3
5
8
9
4
G
E
7
10
17
10
17
0
H
E,F
5
10
15
12
17
2
Critical path is A-C-E-G-I. Completion time is 25 weeks. Total normal cost = $32,000.
Activity
Normal
Time
Crash Time
Normal Cost
Crash Cost
Crash
cost/week
A
3
1
1,000
6,000
$2500
B
1
1
4,000
4,000
—
C
2
2
2,000
2,000
—
D
3
1
5,000
6,000
E
5
4
2,500
3,800
1,300
3
2
2,000
3,000
1,000
G
7
5
4,500
2,000
H
5
4
3,000
3,800
8
5
8,000
3,000
Activity D is the lowest cost per week crash but it is not on the critical path.
34
Options:
1. crash A by 2 weeks for a cost of $5,000
2. crash G by 2 weeks for a cost of $4,000
The key is to ensure that no other path becomes critical as we crash. The lowest cost crash
option is #4 (E by 1 and G by 1 for a total crash cost of $3,300). Total project cost is now
$32,000 + $3,300 = $35,300.
50. Estimates of activity times (in weeks) for a project are as follows.
Suppose that the critical path is A-D-F-H. What is the probability that the project will be
completed within
a. 20 weeks?
b. 22 weeks?
c. 24 weeks?
Activity
Mean
Variance
A
5
0.111
B
3
0.028
C
7
0.111
6
0.444
E
7
0.444
3
0.111
0.444
8
1.778
The expected completion time for the critical path is 22; the variance along the critical
path is 2.44, so the standard deviation is 1.56. (See Appendix A that shows the
cumulative standard normal distribution areas for –3.9 Z to +3.9 Z.)
35
51. The table below shows estimates of activity times (in weeks) for a project:
Suppose that the critical path is A-C-E-F-G. What is the probability that the project will be
completed
a. within 33 weeks?
b. within 30 weeks?
c. in more than 31 weeks?
d. in 32 weeks?
Activity
Mean
Variance
A
5
0.111
B
3
0.028
C
7
0.111
D
6
0.444
7
0.111
3
0.444
0.444
H
8
1.778
Critical path activities are shown in red. The expected completion time for the critical path
A-C-E-F-G is 5+7+7+3+10 = 32; the variance along the critical path sums to 1.221, so the
standard deviation is 1.105. (See Appendix A that shows the cumulative standard normal
distribution areas.)
a. z = (33 – 32)/1.105 = 0.905. Probability = 0.817 (between 0.81594 and 0.81859)
(If students select 0.82894, they are using the z table incorrectly!)
52. You are responsible for wiring a smart 12-story building in downtown Chicago. The project
36
network and basic information is shown in the following table. The time is in weeks. Your
promotion depends on the successful completion of this project in 20 weeks.
a. Draw the project network and compute expected times and variances for each activity.
b. What is the critical path and expected time of completion?
The critical path is A-D-E at 18 weeks.
c. What activity has the largest slack time and how much is it?
d. What is the probability of completing the project in 20 weeks using PERT?
Project
Immediate
Optimistic
Most
Likely
Pessimistic
Expected
Activity
Activity
Predecessor(s)
Time
Time
Time
Time
Variance
A
none
2
5
14
6
4*
C
5
7
9
7
D
6
9
12
9
1*
A
C
E
37
The project has a 50 percent change of completion in 18 weeks. The sum of the variances
53. A competitor of Kozar International, Inc. has begun marketing a new instant-developing
film project. Kozar has had a similar product under study in its R&D department but has not
yet been able to begin production. Because of the competitor’s action, top managers have
asked for a speedup of R&D activities so that Kozar can produce and market instant film at
the earliest possible date. The predecessor information and activity time estimates in
months are shown in the following columns.
a. Draw the project network and compute expected times and variances for each activity.
b. Develop an activity schedule for this project, compute activity slack time, and define
the critical activities.
Using the 3-point estimates of times, we obtain:
Activity
Mean
Variance
A
2
0.444
B
4
0.141
C
2
0.141
D
5
0.250
E
6
1.778