OM5 C18 IM
11
G
B,C
9
6
15
9
18
H
F.G
3
18
21
18
21
Critical path is B-D-E-F-H. Project completion time is 21. We work one or more
CPM problems on the board, first setting up, then doing a forward pass, and next a
Gantt Chart:
There is very little slack in the schedule, so the project manager would want to ensure
that the activities remain on schedule. Instructors should ensure that students “see”
12. Suppose that some of the activities in the Environment Recycling, Inc. situation in
Problem 8 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. Find the total project completion time and lowest cost solution if the
state wants to complete the project three weeks early.
Activity Predecessor(s) Normal Time Crash Time Normal Cost Crash Cost
A — 5 4 $ 400 $ 750
Time 12345678910 11 12 13 14 15 16 17 18 19 20 21
A
B
C
D
E
F
G
H
OM5 C18 IM
12
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
A
—
5
0
5
0
5
B
A
8
5
13
5
13
C
A
7
5
12
6
13
D
—
6
0
6
16
22
E
B,D,C
8
13
21
13
21
F
D
3
6
9
22
25
G
D
3
6
9
22
25
H
E
4
21
25
21
25
I
F,G,H
6
25
31
25
31
Critical path is A-B-E-H-I. Completion time = 31. Total normal cost = $7,200.
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‘
250‘
F’
3′
2′
800‘
1000‘
200‘
G’
3′
2′
500‘
650‘
150‘
H’
4′
3′
400‘
600‘
200‘
I’
6′
5′
900‘
1300‘
400‘
A
B
C
F
D
E
G
H
I
OM5 C18 IM
13
Crashing options on the critical path (shown in red above)
Crash A by 1 week: $350
Crash B by 2 weeks: $400 ($200/week)
Crash E by 2 weeks: $500 ($250/week)
Crash H by 1 week: $200
Crash I by 1 week: $400
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:
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
A
—
5
0
5
0
5
B
A
6
5
11
6
12
C
A
7
5
12
5
12
D
—
6
0
6
15
21
E
B,D,C
8
12
20
12
20
F
D
3
6
9
21
24
G
D
3
6
9
21
24
H
E
4
20
24
20
24
I
F,G,H
6
24
30
24
30
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
A
—
5
0
5
0
5
B
A
7
5
12
5
12
C
A
7
5
12
5
12
D
—
6
0
6
15
21
E
B,D,C
8
12
20
12
20
F
D
3
6
9
21
24
G
D
3
6
9
21
24
H
E
4
20
24
20
24
I
F,G,H
6
24
30
24
30
OM5 C18 IM
14
By crashing E for 2 weeks we have:
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
A
—
5
0
5
0
5
B
A
7
5
12
5
12
C
A
7
5
12
5
12
D
—
6
0
6
13
19
E
B,D,C
6
12
18
12
18
F
D
3
6
9
19
24
G
D
3
6
9
19
22
H
E
4
18
22
18
22
I
F,G,H
6
22
28
22
28
The total cost of this option is $700. Students might try other sequences to seek a
13. The table below shows the crash times, and normal and crash costs for the
international bank systems integration project described in Problem 9. What is the
total project completion time and lowest-cost solution if the bank wants to complete
the project 2 weeks early?
Activity Predecessor Normal Time Crash Time Normal Cost Crash Cost
A — 3 1 $1,000 $ 8,000
OM5 C18 IM
15
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
A
—
3
0
3
0
3
B
A
1
3
4
9
10
C
A
2
3
5
3
5
D
B,C
3
5
12
10
17
E
C
5
5
10
5
10
F
C
3
5
8
9
12
G
E
7
10
17
10
17
H
E,F
5
10
15
12
17
I
D,G,H
8
17
25
17
25
Critical path is A-C-E-G-I. Completion time is 25 weeks. Total normal cost = $31,500
Activity‘
Normal’
Time‘
Crash’Time‘
Normal’Cost‘
Crash’Cost‘
Crash’
cost/week‘
A’
3′
1′
1000‘
8000‘
3500‘
B’
1′
1′
4000‘
4000‘
HH‘
C’
2′
2′
2000‘
2000‘
HH‘
D’
3′
1′
5000‘
6000‘
500‘
E’
5′
4′
2500‘
3800‘
1300‘
F’
3′
2′
1500‘
3000‘
1500‘
G’
7′
4′
4500‘
8100‘
1200‘
H’
5′
4′
3000‘
3600‘
600‘
A
C
B
F
D
E
G
H
I
OM5 C18 IM
16
I’
8′
5′
8000‘
18,000‘
3333‘
The solution can be easily seen as crashing G by 2 weeks for an additional cost of
$2,400. Activity D is the lowest cost per week crash but it is not on the critical path.
This is the optimal solution that can also be found using linear programming.
14. The table below shows estimates of activity times (weeks) for a project:
Most
Optimistic Probable Pessimistic
Activity Predecessor Time Time Time
A – 4 5 6
B A 2.5 3 3.5
C A 6 7 8
D B,C 5 5.5 9
E D 5 7 9
F D 2 3 4
G E,F 8 10 12
Suppose that the critical path is A-C-D-E-G. What is the probability that the project
will be completed within
a. 34.5 weeks?
b. 33 weeks?
c. 35 weeks?
Activity
Mean
Variance
A
5
0.111
B
3
0.028
C
7
0.111
D
6
0.444
E
7
0.444
F
3
0.111
G
10
0.444
H
8
1.778
The expected completion time for the critical path is 35; the variance along the
a. z = (34.5 – 35)/1.25 = -0.4. Probability = 0.3446
15. The table below shows estimates of the optimistic, most probable, and pessimistic
times for the situation described in Problem 10. What is the probability the project
OM5 C18 IM
17
will be completed in time for Kozar to begin marketing the new product within 27
months?
Most
Immediate Optimistic Probable Pessimistic
Activity Predecessors Time Time Time
A — 1 1.5 5
B A 3 4 5
C A 1 2 3
D B, C 3.5 5 6.5
E B 4 5 12
F C, D, E 6.5 7.5 11.5
G E 5 9 13
Using the 3-point estimates of times, we obtain:
Activity
Mean
Variance
A
2
0.44
B
4
0.11
C
2
0.11
D
5
0.25
E
6
1.78
F
8
0.69
G
9
1.78
Activity
Immediate
Predecessor
Time
(days)
Earliest
Start
Earliest
Finish
Latest
Start
Latest
Finish
A
—
2
0
2
0
2
B
A
4
2
6
2
6
C
A
2
2
4
5
7
A
B
C
E
D
F
G
OM5 C18 IM
18
D
B,C
5
6
11
7
12
E
B
6
6
12
14
20
F
C,D.E
8
12
20
12
20
G
E
9
20
29
20
29
Critical path is A-B- F-G; completion time is 29 months. Variance of the critical
path is 3.02. Assuming a normal distribution, the probability of completion in 26
z = (27 – 29)/sqrt(3.02) = -1.15
Probability = 0.125. There is only a small chance of meeting this deadline.
Case Teaching Note: Alternative Water Supply – Single Project
Introduction
Gordon Rivers, the City Manager of Saratoga, Florida and Jay Andrews, the project
manager for Major Design Corporation (MDC), must manage this project. “We need the
‘intake and transmission main’ designed, bid, and completed in 35 weeks. The City of
Saratoga has a future $2 million dollar federal grant riding on the project getting done on
time,” Mr. Rivers said. Jay nodded in agreement. Mr. Rivers continued by saying, “Jay,
the project needs to come in on-schedule and within the budget. Now take this schedule
back and figure out how we are going to do it.” Notice that the case does not provide
how long the current project takes (which is 40 weeks). Therefore, the student needs to
do a basic CPM analysis and compute the total normal costs, critical path, activity slacks,
and current project completion time. Then they must crash the project from 40 to 35
weeks and eventually two paths become critical and require the crashing of both CPs.
Project Description
The objective of the project is to design a fully functional surface water intake that is
protective of the environment, will last at least 30 years, and will have a low life cycle
cost (i.e. capital, maintenance, and energy consumption). For this type of project,
engineering design accounts for 20 percent of total project cost. The design stage is also
important because the decisions made during design lock in 80 percent or more of the life
cycle costs of the project. The case describes each project activity and provides the data
below.
Alternative Water Supply (AWS) – Single Project
OM5 C18 IM
19
Activity
ID
Description
Precedence
Regular
Time
(weeks)
Crash
Time
(weeks)
Normal
Cost
Estimate
Crash
Cost
Estimate
A
Conceptual Design
none
4
3
$30,000
$33,500
B
Preliminary Design
A
12
10
$52,000
$58,000
C
Final Design
B
19
16
$59,000
$76,000
D
Environmental Permit Application
Preparation
B
8
5
$48,000
$58,200
E
Environmental Permit Review and Approval
D
4
4
$38,000
$38,000
F
Building Permit Application Preparation
E
2
1
$35,000
$38,000
G
Building Permit Review and Approval
F
4
4
$6,000
$6,000
H
Property Acquisition
B
20
18
$90,000
$115,000
I
Bid Project
C, H
4
4
$6,000
$6,000
J
Construction Start (Dummy Activity)
G, I
0
0
$0
$0
Decisions and Analysis
Jay manages about a half dozen engineering projects at any one time so he asks you to
analyze this project for ways to complete the project in 35 weeks. Jay would like to meet
with you tomorrow to discuss the results of your analysis. To organize your analysis you
outline the following steps.
1. Draw the project network diagram and determine the normal time to complete the
project, activity slack times, the critical path(s), and total project costs (i.e.,
baseline your project) using Critical Path Method.
2. Determine the best way to crash the project to complete it in 35 weeks with
revised activity slack times, critical path(s), and total project costs. Provide
reasoning as to how all crashing decisions were made.
See the OM5 C18 Case TN Excel File for CPM diagrams from 40 to 35 weeks.
The total cost to achieve a 35 weeks project completion date is $404,167 by crashing A
by 1 week, B by 2 weeks, H by 2 weeks, and C by 1 week. Notice that two paths
eventually become critical paths and therefore both need to be crashed (H by 1 week and
OM5 C18 IM
20
recognize that more resources are required including moving resources from slack
3. Activity times with the greatest uncertainty are Activities D, E, and H. Describe
conceptually how you could model this uncertainty in activity times. (You do not
have the necessary data to actually do this numerically.)
Property acquisition (H) and environmental permitting (D and E) activities represent
the greatest uncertainty. PERT, of course, is one way to try to build uncertainty into
the analysis but it requires more data and assumptions. Ask your students, “Can
4. What are your final recommendations?
• Crash the project to 35 weeks so as not to lose the opportunity for $2 million
Other case issues
• A short discussion of the actual water resource project and its impact on the
environment might be in order. Economic, environment, and social sustainability
issues are involved in the project. Ask the students to briefly describe each.
• Postscript: The actual project was delayed about one year due to activities property
acquisition (H) delays. The city missed their opportunity for the federal funding grant
of $2 million dollars.