Chapter 13 – Project Scheduling
G 10 8 5,000 8,000
If a deadline of 17 weeks is imposed, give the linear programming model for the crashing decision.
55. For the project represented below, determine the earliest and latest start and finish times for each activity as well as the
expected overall completion time.
Activity Duration ES EF LS LF Slack
A 4
B 3
C 4
D 2
E 5
F 2
G 5
H 6
Chapter 13 – Project Scheduling
56. Consider the following PERT/CPM network with estimated times in weeks. The project is scheduled to begin on May
1.
The three-time estimate approach was used to calculate the expected times and the following table gives the variance for
each activity:
Activity Variance Activity Variance
A 1.1 E 0.3
B 0.5 F 0.6
C 1.2 G 0.6
D 0.8 H 1.0
a. Give the expected project completion date and the critical path.
b. By what date are you 99% sure the project will be completed?
57. A project consists of five activities. Naturally the paint mixing precedes the painting activities. Also, both ceiling
painting and floor sanding must be done prior to floor buffing.
Optimistic Most Probable Pessimistic
Activity Time (hr) Time (hr) Time (hr)
Floor sanding 3 4 5
Floor buffing 1 2 3
Paint mixing 0.5 1 1.5
Wall painting 1 2 9
Ceiling painting 1 5.5 7
a. Construct the PERT/CPM network for this problem.
b. What is the expected completion time of this project?
c. What is the probability that the project can be completed within 9 hr.?
Chapter 13 – Project Scheduling
58. Consider a project that has been modeled as follows:
Activity Immediate Predecessors Duration (hr)
A — 7
B — 10
C A 4
D A 30
E A 7
F B,C 12
G B,C 15
H E,F 11
I E,F 25
J E,F 6
K D,H 21
L G,J 25
a. Draw the PERT/CPM network for this project and determine project’s expected completion time and its critical path.
b. Can activities E and G be performed simultaneously without delaying the minimum project completion time?
c. Can one person perform A, G, and I without delaying the project?
d. By how much can activities G and L be delayed without delaying the entire project?
e. How much would the project be delayed if activity G was delayed by 7 hours and activity L was delayed by 4 hours?
Explain.
Chapter 13 – Project Scheduling
59. Joseph King has ambitions to be mayor of Williston, North Dakota. Joe has determined the breakdown of the steps to
the nomination and has estimated normal and crash costs and times for the campaign as follows (times are in weeks).
Normal Crash Immediate
Activity Time Cost Time Cost Predecessors
A. Solicit Volunteers 6 $5,000 4 $ 9,000 —
B. Initial “Free” Exposure 3 $4,000 3 $ 4,000 —
C. Raise Money 9 $4,000 6 $10,000 A
D. Organize Schedule 4 $1,000 2 $ 2,000 A
E. Hire Advertising Firm 2 $1,500 1 $ 2,000 B
F. Arrange TV Interview 3 $4,000 1 $ 8,000 B
G. Advertising Campaign 5 $7,000 4 $12,000 C, E
H. Personal Campaigning 7 $8,000 5 $20,000 D, F
Joe King is not a wealthy man and would like to organize a 16-week campaign at minimum cost. Write and solve a linear
program to accomplish this task.
60. Marcy Fetter, a staff analyst at the Los Angeles plant of Computer Products Corporation, is assigned to the team that
is developing the process design for producing an RFID sensor. The corporate planning group in San Jose, California has
contacted her and has asked how confident the design group is about completing the project in 60 days. She has
developed these estimated time durations in days for the project:
Activity Immediate Predecessor
Activities Optimistic
Time (to) Most Likely Time (tm) Pessimistic
Time (tp)
A — 10 12 15
B A 6 10 14
C A 10 15 20
D A 9 9 18
E B 5 6 8
F C,E 10 12 13
G B 12 14 16
H B,D 18 21 24
I B,D 10 15 20
J F.G.H 8 10 14
a. Compute the expected time and variance for each activity.
b. Determine the critical path and the expected duration of the project.
c. What is the probability that the project will take longer than 58 days to complete?
d. Which path in the project network offers the greatest risk of overrunning a new deadline of 56 days?
Chapter 13 – Project Scheduling
61. A project has the following activities, precedence relationships, and time estimates in weeks:
Activity Immediate Predecessor
Activities Optimistic
Time (to) Most Likely Time (tm) Pessimistic
Time (tp)
Chapter 13 – Project Scheduling
A — 15 20 25
B — 8 10 12
C A 25 30 40
D B 15 15 15
E B 22 25 27
F E 15 20 22
G D 20 20 22
a. Compute the expected time and variance for each activity.
b. Determine the critical path and the expected duration of the project.
c. What is the probability that the project will take longer than 56 weeks to complete?
62. Three paths of a PERT network have these mean durations and variances in weeks:
Path Mean
Duration
Variance
1 45 2.75
2 44 5.50
3 46 1.20
Which path offers the greatest risk of overrunning a contract deadline of 48 weeks?
Chapter 13 – Project Scheduling
63. A project has the following activities, durations, costs, and precedence relationships:
Activity Present Duration
(Weeks) Accelerated Duration
(Weeks) Immediate
Predecessor Activities
Present
Cost
Accelerated
Cost
A 10 9 — $11,000 $15,000
B 15 13 — 20,000 25,000
C 10 6 A 9,000 20,000
D 20 18 A 25,000 30,000
E 15 10 C 20,000 35,000
F 17 15 B 20,000 30,000
G 12 10 B 15,000 25,000
H 9 8 D,F 12,000 18,000
I 7 6 G,H 10,000 15,000
Develop a cost-time trade-off analysis. Detail the steps that you would use to accelerate or crash the project to its
minimum duration at the lowest cost. Determine each step’s cost and the duration of the project.
Chapter 13 – Project Scheduling
64. National Oil Company (NATOCO) must plan the shutdown of its Houston refinery for routine preventive
maintenance. Each hour of downtime is lost production time and is very costly, so NATOCO wants the maintenance
project completed in 22 hours. The PERT network below shows the precedence relationships of the activities involved in
the project. The table gives the activity times and costs under normal operations and maximum crashing.
NATOCO wants to know the minimum cost of completing the maintenance project within the 22-hour period. Formulate
and solve a linear program that will yield this information.
Activity Normal
Time (hr) Normal
Cost Crash
Time (hr) Crash
Cost
A 2 $2,000 1.5 $3,000
B 4 3,000 3 3,500
C 1 1,500 1 1,500
D 4 5,300 2.5 8,000
E 6 5,400 5 7,000
F 10 6,000 8 9,000
G 8 4,800 5 9,900
H 2 2,800 1 2,900
I 5 4,500 4 5,000
J 12 6,000 6 9,600
K 7 7,000 4 9,700
L 11 8,800 9 9,200
Chapter 13 – Project Scheduling
M 4 1,000 1 7,000
Chapter 13 – Project Scheduling
Chapter 13 – Project Scheduling
65. A manufacturing company wants to accelerate a project it has underway for a new product in order to beat their
competitors to market. They used CPM to develop the project which has the following activities, durations (in weeks),
costs (in dollars), and precedence relationships:
Immediate Present Crashed
Activity Predecessors Duration Cost Duration Cost
a — 3 3300 2 3400
b — 5 4100 3 4500
c a 4 3700 3 3800
d b 5 5900 4 6000
e c 6 6700 4 6900
f c 4 4100 3 4150
g d,e 6 9100 3 9700
h f,g 5 2700 4 3000
a. How many paths are in the project network and what are their durations? Which path is critical?
b. What are the earliest and latest start and finish times for each activity?
c. What activity should be crashed first in order to accelerate the project?
d. What is the final project duration in weeks assuming it is crashed as far as possible?
e. What is total cost to crash the project as far as possible?
Chapter 13 – Project Scheduling
66. A startup company is planning to use a CPM network to turn a promising new technology into a product. They have
established a project whose manager developed these activities, precedence relationships, and activity durations (in days)
for the effort:
Immediate
Activity Predecessors Duration
a — 4
b — 5
c — 3
d a 2
e b 3
f c 1
g d,e,f 5
h d,e,f 4
i g 7
j h 3
k i 1
l j,k 2
m I 2
a. How many paths are in the network and what is each path’s duration? Which path is critical?
b. What are the earliest and latest start and finish times for each activity?
Chapter 13 – Project Scheduling
have established the following activities, precedence relationships, and time estimates in days:
Immediate Most
Activity Predecessors Optimistic Likely Pessimistic
a — 8 10 12
b — 6 7 9
c b 3 3 3
d a 10 15 20
e d 6 7 8
f c 9 10 11
g d 5 7 10
h e,f 14 15 16
a. Compute the expected duration and variance for each activity in the project.
b. Determine the expected duration of each path in the project network.
c. What is the standard deviation of the critical path?
d. What is the probability of completing the project within 50 days?
Chapter 13 – Project Scheduling
68. The production engineering group at the Singapore plant of a computer company has been put in charge of production
for a new printer model. The printer will be built via an assembly line. The plant manager must design the assembly line
for the printer. He has been advised that the assembly line must be ready in four weeks for a test run. He has decided to
use CPM and has identified these activities, determined their precedence relationships, and estimated durations:
Immediate Duration
Activity Predecessors (days)
a. organize layout team – 5
b. organize modification team – 3
c. design personnel jobs a 2
d. set machinery in place b 3
e. connect utilities c,d 7
f. modify conveyor system e 4
g. train personnel e 3
h. finishing touches f 5
i. run test lot g 2
a. How many paths are in the network and what is each path’s duration? Which path is critical?
b. What are the earliest and latest start and finish times for each activity?
Chapter 13 – Project Scheduling
Essay
69. Name at least three managerial situations where answers are provided by project management solutions.
70. Explain how and why all predecessor activities must be considered when finding the earliest start time.
71. Explain how and why all successor activities must be considered when finding the latest finish time.
72. Once the earliest and latest times are calculated, how is the critical path determined?
73. Why should projects be monitored after the critical path is found?
74. Suppose that, after analyzing a project network, the project manager finds the project duration
unacceptable. Discuss the options the manager might have for reducing the duration.