176
CH A P T ER 5 / S CH ED UL IN G T H E P R OJECT
9. When using AON networks, how does one indicate
an event such as a project milestone?
10. A probabilistic network has a critical path of 21 days
and a .95 probability of completing this path in 24
days. Therefore, the project has a .95 chance of being
finished by the end of the 24th day. True or False?
Briefly explain your answer.
11. “Not uncommonly, the Gantt chart is deceptive in its
apparent simplicity.” Briefly explain.
12. When activity times are known with certainty, the
critical path is defined as the set of activities on a
path from the project’s start event to its finish event
that, if delayed, will delay the completion date of the
project. Why must this definition be modified in situations where the activity times are not known with
certainty? Are there any dangers associated with not
modifying the definition?
DISCUSSION QUESTIONS
1. Should a PM manage critical path tasks differently
than noncritical path tasks?
2. How might you use the network approach to help
prepare cost estimates?
3. When would it be accurate to determine the probability of project completion by multiplying the probabilities of all the paths through the network together?
When would it not be accurate?
4. Reconsider Question 3. If this approach is not accurate, would the probability of completion considering
the critical path alone be more accurate? How might
you estimate the correct probability without resorting
to simulation?
5. Why do you think most PMs use MSP’s Gantt chart
format (see Figure 5-20) more commonly than the
network format?
6. Which of the linkages in precedence diagramming do
you think is most commonly used? Why?
7. In the calculation of variance for optimistic and
pessimistic activity duration estimates made at the
95 or 90 percent level, the denominator of the fraction that approximates the standard deviation of the
time distribution changes from the traditional
(b − a)/6 to (b − a)/3.3 for 95 percent and to (b −
a)/2.6 for 90 percent. Where did the 3.3 and the 2.6
come from?
8. Given all the estimating done to determine the duration of project activities, what does it mean to say
that “only after the fact do we know which path was
actually the critical path?”
9. It was noted that “the PM must manage the project
team as well as the project.” Explain why.
10. Why do you think scheduling has been the major
focus of effort throughout the history of project management rather than performance or budgeting?
EXERCISES
1.
2.
3.
Refer to the network in Figure 5-14. What is the probability that path a‐b‐c‐f will interfere with the promised project completion of 50 days? Recall that the
critical path, a‐b‐d‐g‐h, had a probability of .86 for a
50‐day completion. What is the probability that both
paths will be complete in 50 days?
Refer to Table 5-4 and Figure 5-14. Recalculate the
variance for each activity on the assumption that
the optimistic and pessimistic estimates were made
with a 95 percent probability. Recalculate the probability that the critical path will be completed
in 50 days.
Refer to Table 5-4 and Figure 5-14. Recalculate the
variance for each activity on the assumption that
the optimistic and pessimistic estimates were made
with a 90 percent probability. Again, recalculate the
likelihood that the critical path will be finished
in 50 days.
4.
Given the information in the following table:
Activity
a
b
c
d
e
f
g
h
i
(a)
(b)
(c)
(d)
(e)
Duration
Predecessor
4
6
4
2
4
5
3
4
2
None
a
a
c
b
b, d
c
f, g
e, h
Construct the network diagram.
Find each activity’s ES, EF, LS, and LF.
Identify all paths. Which path is the critical path?
Calculate the slack for each activity.
How long will it take to complete the project?
1 77
E XERC ISES
5.
In the following table, task durations are given in
weeks. The estimates were made at the 95 percent
level (see Section 5.2, Calculating Probabilistic
Activity Times subsection).
Activity
a
b
c
d
e
f
g
6.
Opt.
Normal
Pess.
—
—
a
a
b, c
d
e
2
3
4
4
4
3
3
4
5
5
6
5
4
5
6
9
7
10
7
8
8
(a) Find the expected time and variance for each task.
(b) Draw the network (either AOA or AON) and
find the path with the longest expected time.
What is the expected time to complete this path?
(c) Find the probability using the analytical approach
that the path with the longest expected time will
be completed in 23 weeks.
(d) Find the probability that the other paths will be
completed in 23 weeks.
(e) Using simulation, what is the probability that the
entire network will be completed in 23 weeks?
Given an auditing project with the following activities:
Activity
a. Add
b. Balance
c. Count
d. Deduct
e. Edit
f. Finance
g. Group
h. Hold
7.
Predec.
Std. Dev.
Critical
2
1
0
3
1
2
2
0
Yes
Yes
Yes
Yes
Yes
Duration
(wks)
2
3
4
2
1
6
4
2
Using the analytical approach find:
(a) The probability of completing this project in 12
weeks or less, as the client desires.
(b) The probability of completing this project in 13
weeks or less.
(c) The probability of completing this project in 16
weeks or less, the client’s drop‐dead date.
(d) The number of weeks required to assure a 92.5
percent chance of completion, as guaranteed by
the auditing firm.
Resolve the previous exercise using computer simulation and compare your answers. Explain any differences you observe. Which methodology do you have
more confidence in and why?
8.
9.
Referring to the previous two questions, modify your
simulation model to develop distributions for the slack
time of each path. What do these distributions tell
you?
Given the following information regarding a project
involving an initial public offering (IPO):
Activity
Duration
(weeks)
Preceding
Activities
3
1
3
4
4
5
2
—
—
a
a
b
b
c, e
3
f
a. Check feasibility
b. Determine funding
c. Find possible banks
d. Select two possibles
e. Interview two banks
f. Analyze funding costs
g. Determine chance of
success
h. Sign contract
(a) Draw the network.
(b) What is the critical path?
(c) When will the offering be available (end of
the project)?
(d) What is the effect on the project if activity e
(approvals) takes an extra week? Two extra weeks?
Three extra weeks?
10. Enter the following information into an Excel® spreadsheet. The time estimates were made at the 90 percent
level (see Section 5.2, Calculating Probabilistic
Activity Times subsection). All activity times
are in days.
Activity
a
b
c
d
e
f
g
h
i
j
k
Predec.
Opt.
Normal
Pess.
—
—
—
a
b
b
c
c
d, e
f, g
h
5
4
7
6
4
12
8
7
10
6
7
6
4
9
6
5
16
12
9
14
12
9
9
6
15
6
7
17
20
16
18
20
14
(a) Draw the network. (You may use MSP, or draw an
AOA or AON network by hand.)
(b) Using Excel®, calculate the expected time (TE)
and variance for each activity.
(c) Using the expected times, find the path with the
longest expected time. What is the expected
time to complete this path? (Use the analytic method.)
178
11.
12.
13.
14.
(d) Find the probability that the critical path will be
completed in 38 days or less.
(e) Are there any serious sources of merger problems?
What are they? Calculate the probability of finishing the project on or before day 38 when
merger considerations are included.
(f) Assume that the times in the table were made on
the 99+ percent level. Recalculate the activity
variances with this assumption and find the probability that the critical path will be complete in
38 days. (Note, the altered assumption will
change activity variance, but not the expected
activity durations.) Briefly explain the difference
in probabilities.
(g) How many days are required for the critical path
to have a .9 probability of completion?
Given the project in Exercise 5, simulate the completion of the project 1,000 times, assuming that the
activity times follow a normal distribution and that
the time estimates are made at the 95 percent level.
(a) Determine the probability of each path becoming
the critical path.
(b) What is the probability that the project is completed in 23 weeks?
(c) How do your answers compare with your answers
in Exercise 6?
The project referred to in Exercise 11 has been
partially completed. Task a required 4 weeks, task b
9 weeks, task c 4 weeks, and task d 5 weeks. Update
the simulation model you developed for Exercise 11
and calculate the probability that the project
will be finished in 23 weeks. Explain why the probability of completing the project in 23 weeks
has changed.
Given the project in Exercise 11, simulate the completion of the project 1,000 times, assuming that the
activity times are at the 99+ percent level and follow
a triangular distribution.
(a) Determine the probability of each path becoming
the critical path.
(b) What is the probability that the project is completed in 32 days? In 34 days? In 38 days?
(c) How do your answers compare with your answers
in Exercise 11?
*Additional task precedence information about the
Textbook Revision Project described in Exercise 10
in Chapter 3 is now available. JR and Mike are preparing to revise their project management textbook.
The project of revising the book begins with devel-
CH A P T ER 5 / S CH ED UL IN G T H E P R OJECT
oping a revision plan. Developing a revision plan
includes four activities that each take 1 week to
complete. First the authors go over the reviews provided by a sample of current adopters of the book.
The second task involves benchmarking other project management books. Benchmarking other books
can either be done in parallel with going over the
reviews or done after the reviews have been studied.
After both the reviews have been gone over and the
benchmarking is completed, the changes to make to
the book are decided on. The final step in developing the plan is allocating the work between the
authors. The work can be allocated once the authors
finalize the changes to be made to the book. After
the plan has been developed, the book is revised
which is expected to take 24 weeks. Once all the
chapters have been revised, the page proofs are
reviewed which is expected to take 8 weeks. After
the page proofs have been reviewed, the index for
the book is created which takes 2 weeks. Finally,
after the book has been revised, the supplements to
the book can be updated which takes 8 weeks.
Update the information previously entered to
include the precedence relationships among the
tasks. Assuming the project begins on March 7,
2016, when is its expected completion date?
15. *Given the information in the following table, draw
the AOA network. Using the same information,
enter the data into MSP assuming a 7‐day workweek. (To change the calendar in MSP from its
5‐day week default, click “Help,” type “calendar
change,” and follow directions.) Develop the appropriate AON network and Gantt chart. Using any
method you wish, find the critical path and critical
time for the network. Then find the slack for all
activities.
Activity
a
b
c
d
e
f
g
h
i
j
k
l
Predecessor
—
—
—
a
b
b
c
d, e
d, e
f, g, h
i
j
* This problem is included for instructors who supplement their course with additional content on Microsoft Project.
Duration
5 days
7
4
6
9
6
4
6
8
9
10
9
1 79
I NC IDENTS FO R D I SCU SSIO N
DISCUSSION EXERCISE
1.
The following activities were listed during a brainstorming session on product development. Find the
appropriate predecessor‐successor relationships and
then construct an AON network to reflect the project using the activity duration times given in the
information table. Assume a 5‐day workweek. Find
the critical path and time for the project. Find the
slack for all activities.
1. Organize the sales office: Hire sales manager. (6 weeks)
2. Hire sales personnel: The sales manager will
recruit and hire the salespeople needed. (4 weeks)
3. Train sales personnel: Train the salespeople hired
to sell the product to the distributors. (7 weeks)
4. Select advertising agency: The sales manager
will select the agency best suited to promote the
new product. (2 weeks)
5. Plan advertising campaign: The sales office and
advertising agency will jointly plan the advertising campaign to introduce the product to the
public. (4 weeks)
6. Conduct advertising campaign: The advertising
agency will conduct a “watch for” campaign for
potential customers. (10 weeks)
7. Design package: Have packaging engineer
design the package most likely to “sell.” (2 weeks)
8. Set up packaging facility: Prepare to package the
products when they are received from the manufacturer. (10 weeks)
9. Package initial stocks: Package stocks received
from the manufacturer. (6 weeks)
10. Order and receive stock from the manufacturer:
Order the stock from the manufacturer. The time
given includes the time for delivery. (13 weeks)
11. Select distributors: The sales manager will select
the distributors whom the salespeople will contact to make sales. (9 weeks)
12. Sell to distributors: Take orders from the distributors for the new product, with delivery
promised for the product‐introduction date. If
orders exceed stock, assign stock on a quota
basis. (6 weeks)
13. Ship stock to distributors: Ship the packaged
stock to distributors in accord with their orders or
quota. (6 weeks)
Question: What managerial problems and opportunities
do you see as a result of your work?
INCIDENTS FOR DISCUSSION
Springville Fire Department
Attack of the Killer Worm
The city of Springville is building a new fire station in
their city. The city is expanding and is in need of a second fire station closer to the newer areas of the city to
ensure shorter response times. The project manager and
the project team have been selected for the project. The
team is very interested in selecting the scheduling technique that will be used to follow the project through to
completion.
The project manager, city manager, and chief of the fire
department have set the following criteria for the process
of selecting the scheduling technique: easy to use, shows
durations of tasks, shows milestones, can see the flow of
work, can see the sequence of events, can depict which
tasks can be undertaken at the same time, and can tell how
far tasks are from completion. The city manager favors the
Gantt chart, the chief likes PERT, and the project manager prefers CPM.
Lee Antoinio was the CIO of a large publishing house. Her
network administrator, Andy McPester, came to her one
sunny September afternoon and informed her that a worm
had attacked their network. It had shut down two of their
45 servers and had potential to harm the other servers as
well as the 323 personal workstations throughout the
enterprise. Andy knew that he could not report this problem to Lee without a recommended project plan for a solution. Andy and Lee had a good, trusting relationship, but
Lee insisted that Andy investigate all options and come to
her with the most viable solution to any problem.
Andy’s project plan outlined that it would take 30 to 45
minutes to touch each PC to protect them from the worm,
and longer if they were already infected. It would take
almost an hour to check the other servers and repair them.
Andy discussed the time estimates with his desktop staff.
Andy proposed paying six staff members overtime to work
4 hours after their normal business day for the next week
or two to check each personal workstation. He did not
Question: If you were the project manager, which
method would you use, and why?
180
CH A P T ER 5 / S CH ED UL IN G T H E P R OJECT
want to affect productivity of the company’s staff during
the day. Andy proposed repairing the servers the same way
with two additional staff members in the evenings. Lee felt
that the time estimates and costs were too high. She did
not think it would take that amount of time for each
machine to get repaired. She was concerned that some of
C
The steering team meeting held August 31 went quite
well. Fred felt that his team members had worked well
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Question: How would you recommend they reach a
consensus on the project plan?
A
Friendly Assisted Living Facility
Program Plan—5
ID
Andy’s staff may have overestimated the time needed. She
also was not sold on the notion that all the work would be
done at night, paying staff overtime.
S
E
together at determining the steps and the associated costs
of the program. The CFO presented the program budget
first, and then project team members presented their draft
project plans.
The COO presented the following project plan:
Task Name
Operational Implementation Plan
Management/Organization structure
Recruit & hire Executive Director
Interior design issues decided (furnishings, etc.)
Determine what was provided with lease and what was
furnished in some units
Determine budget for interior
Carpet and wall finish determined
Furniture and room layout
Facility and equipment needs defined
Staffing determined
Office space for physicians
Medical staffing needs determined & Director appointed
Food service
Menus selected
Waiting and service staffing needs determined
Additional equipment needs
Telecommunications services
Investigate phone service options
Certification/Accreditation requirements
Investigate requirements & timing of applicants with Dept.
of Health to open facility
Develop clinical and operational quality monitoring systems
Develop financial systems (billing, etc.)
Human resources
Work force management recommendations
Project plan for recruitment developed
Policies and procedures developed
Obtain ‘samples’ of assisted living policies & procedures
from other institutions
Investigate assisted living laws proposed in other states/
federal
Technology & information systems
Develop plan for technology access for residents (TV,
Cable, PC’s)
Investigate software/technology options for residents
Duration
Predecessors
87 days
17.4 wks
20 days
2 wks
10 days
2 wks
2 wks
4 wks
2 wks
4 wks
4 wks
45 days
8 wks
4 wks
4 wks
45 days
45 days
42 days
0 days
0 days
6 wks
79 days
6 wks
2 wks
60 days
4 wks
12 wks
Resource Names
CFO, Legal, VP Mktg
Splient
COO
59
59
COO
COO, Dr. Link
Dr. Link
Dr. Link
68FS—3wks
Chief Engineer
Legal
CFO
HR Director
HR Director
78
Legal
Legal
344 days
3 wks
CIO
12 wks
CIO
1 81
C AS E
The Chief Legal Counsel for the medical center presented his project plan. Fred had asked him to join the
ID
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Task Name
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Duration
Legal and Licensing Requirements
Research licensing requirements for residential care facility
Uniform accessibility standard compliance (# hndcp
accessible beds)
Investigate law firm and outline services
Prepare project plan for license
File license—by opening date
Curb‐cut approval from county (access to County Rd.)
Investigate corporate structure for assisted living
Determine Board of Trustee membership
Appoint Board of Trustees
Prepare draft Code of Regulation
Prepare document and filing of governance structure
Draft service agreement with Friendly Medical Center for
services provided
Lease issues
Research Long Term Care insurance requirements
Facility “rules” defined (i.e., smoking, firearms, pets,
financial planning)
Spell out changes for residents in moving from “light” to
“heavy” assisted
Lease template prepared
Review all marketing materials for compliance
The Vice President of Marketing presented her project
plan and stated that she and her staff were responsible
for every step in the plan. She was still working with her
staff to determine who does what. The Marketing VP
made it clear to the team that she needed 5 months for the
ID
team when it became apparent that there were significant
compliance and legal issues associated with this project.
Predecessors
154 days
38 days
2 wks
4 wks
2 wks
0 days
53 days
115 days
3 wks
4 wks
4 wks
12 wks
4 wks
110 days
12 wks
4 wks
Resource Names
Legal
39
40
Legal
Legal, CFO
44
45
46
47
Legal
48
Legal
50
6 wks
6 wks
154 days
52, 50, 51
Legal
marketing plan implementation to be able to meet the
occupancy requirements at start‐up. She restated that her
team must have this lead time to the completion of the
construction and furnishing phase of the Program.
Task Name
Marketing
Community mailing about construction project
Initial informational meetings
Friendly Medical Center volunteers
Community groups
Friendly Medical Center staff (all shifts)
Presentation prepared for Speaker’s Bureau
Provide updates to community
Inquiry log established
Groundbreaking ceremony—during National Hospital Week
Marketing plan developed and implemented
PR firm contracted
Marketing plan developed
Determine name and signage for facility
Hire Marketing Director
Marketing plan ready to implement
Implementation of marketing plan—5 months before facility ready, then ongoing
Duration
270 days
0 days
16 days
1 day
4 days
3 days
0 days
0 days
0 days
0 days
180 days
4 wks
8 wks
0 days
4 wks
0 days
20 wks
Predecessors
88
92
88
97
98
98
98, 100
101
182
CH A P T ER 5 / S CH ED UL IN G T H E P R OJECT
As Fred was explaining that the next job of the group
was to complete a final version of all project plans and firm
up the schedule of the Program, the Construction Project
Manager stated that it was his turn to present his broad
ID
1
2
3
4
5
6
7
8
9
10
11
project plan for construction of the facility. He also added
that he had a major scheduling issue to bring to the team.
The Construction Project Manager presented the following broad project plan for facility construction.
Task Name
Duration
Construction & Furnishing
Facility construction
Phase 1 ‐ Foundation & excavation (basement/1st floor slab)
Phase 2 ‐ Structure (steel/framing)
Phase 3 ‐ Enclosure (masonry/windows/roof)
Phase 4 ‐ Interiors (drywall/ceiling/flooring/case goods)
First 45 (light assisted) units ready to prepare for occupancy
First 45 units ready for residents
Remaining 57 units (light & heavy) ready to prepare for occupancy
Construction complete
Building ready for residents
The construction PM proceeded to explain that the
scheduling constraints that the Board of Trustees gave the
team were not feasible. The Board wanted construction to
begin immediately after the elections in November and to
be ready for occupants by June. The contractor did not
want to begin the project at the beginning of winter. The
first phases of the project plan detailed work that needed
to be completed outside. If the weather was bad, the construction PM knew the schedule would be affected. The
construction project manager also pointed out that the
schedule created by the contractor was designed around a
40‐hour, 5‐day workweek. If the building project began in
369 days
329 days
95 days
113 days
134 days
234 days
0 days
8 wks
0 days
0 days
8 wks
Predecessors
3FS—60 days
3
3
6FS—5 wks
7
6
9
10
November, the estimated project duration would be
increased by 1 to 2 months, during which time some construction crewmen would have to be paid, thereby increasing the building cost.
The PM recommended that construction begin in
February or March of the following year, which would give
the facility a shorter build time and a lower cost. The
budget and project duration submitted were based on a
March 1 start date. He stated that the construction phase
of the project did not need to hold up the other members
of the Program team—they could begin their work on
their projects anytime.
QUESTIONS
1.
2.
3.
Draw a Gantt chart for the construction phase of the
program. What is the completion date if construction
starts in March? What is the completion date of the
project if construction is started in November?
Why is it not possible to meet the scheduling constraints set by the Board? What is your recommendation to handle the scheduling problem?
When will the program be completed based on your
recommendation?
C
A
NutriStar
NutriStar produces a line of vitamins and nutritional supplements. It recently introduced its Nutri‐Sports Energy Bar,
which is based on new scientific findings about the proper
4.
5.
Develop a Gantt Chart of the Marketing Plan and
Implementation Phase of the Program. Determine the
start date of the Marketing Plan project in order to meet
your recommended facility ready for occupancy date.
What is the next step the team members must take in
order to complete their project plans?
S
E
balance of macronutrients. The energy bar has become
extremely popular among elite athletes and other people
who follow the diet. One distinguishing feature of the Nutri‐
Sports Energy Bar is that each bar contains 50 milligrams of
1 83
QUES T IO NS
eicosapentaenoic acid (EPA), a substance strongly linked to
reducing the risk of cancer but found in only a few foods,
such as salmon. NutriStar was able to include EPA in its
sports bars because it had previously developed and patented
a process to refine EPA for its line of fish‐oil capsules.
Because of the success of the Nutri‐Sports Energy Bar
in the United States, NutriStar is considering offering it in
Latin America. With its domestic facility currently operating at capacity, the President of NutriStar has decided to
investigate the option of adding approximately 10,000
square feet of production space to its facility in Latin
America at a cost of $5 million.
The project to expand the Latin American facility
involves four major phases: (1) concept development,
(2) definition of the plan, (3) design and construction,
and (4) start‐up and turnover. During the concept development phase, a program manager is chosen to oversee all
four phases of the project and the manager is given a
budget to develop a plan. The outcome of the concept
development phase is a rough plan, feasibility estimates for
the project, and a rough schedule. Also, a justification for
the project and a budget for the next phase are developed.
In the plan definition phase, the program manager selects
a project manager to oversee the activities associated with
this phase. Plan definition consists of four major activities
that are completed more or less concurrently: (1) defining
the project scope, (2) developing a broad schedule of
activities, (3) developing detailed cost estimates, and
Activity
(4) developing a plan for staffing. The outputs of this phase
are combined into a detailed plan and proposal for management specifying how much the project will cost, how long it
will take, and what the deliverables are.
If the project gets management’s approval and management provides the appropriations, the project progresses
to the third phase, design and construction. This phase
consists of four major activities: (1) detailed engineering,
(2) mobilization of the construction employees, (3) procurement of production equipment, and (4) construction
of the facility. Typically, the detailed engineering and the
mobilization of the construction employees are done concurrently. Once these activities are completed, construction of the facility and procurement of the production
equipment are done concurrently. The outcome of this
phase is the physical construction of the facility.
The final phase, start‐up and turnover, consists of four
major activities: pre‐start‐up inspection of the facility,
recruiting and training the workforce, solving start‐up problems, and determining optimal operating parameters (called
centerlining). Once the pre‐start‐up inspection is completed,
the workforce is recruited and trained at the same time that
start‐up problems are solved. Centerlining is initiated upon
the completion of these activities. The desired outcome of
this phase is a facility operating at design requirements.
The following table provides optimistic, most likely,
and pessimistic time estimates for the major activities.
Optimistic Time
(months)
Most Likely Time
(months)
Pessimistic Time
(months)
3
12
24
1
0.25
0.2
0.2
2
0.5
0.3
0.3
12
1
0.5
0.6
2
8
0.5
1
3
12
2
3
6
24
4
12
0.25
0.25
0
0
0.5
0.5
1
1
1
1
2
4
A: Concept Development
Plan Definition
B. Define project scope
C. Develop broad schedule
D. Detailed cost estimates
E. Develop staffing plan
Design and Construction
F. Detailed engineering
G. Facility construction
H. Mobilization of employees
I. Procurement of equipment
Start‐up and Turnover
J. Pre‐start‐up inspection
K. Recruiting and training
L. Solving start‐up problems
M. Centerlining
QUESTIONS
1.
2.
Draw a network diagram for this project. Identify all
paths through the network diagram.
Simulate the completion of this project 1,000 times,
assuming that activity times follow a betaPERT
3.
distribution. Estimate the mean and standard deviation of the project completion time.
Develop a histogram to summarize the results of your
simulation.
184
4.
CH A P T ER 5 / S CH ED UL IN G T H E P R OJECT
Calculate the probability that the project can be
completed within 30 months. What is the probability
that the project will take longer than 40 months?
What is the probability that the project will take
between 30 and 40 months?
C
A
Launching E‐Collar
Pet Technologies Inc. has completed the development of a
revolutionary pet collar code named e‐collar. E‐collar is
designed for dogs and allows pet owners to identify the
location of their dog using their smartphones. In addition
to purchasing the e‐collar, the pet owners must download
a free app for their smartphone and pay a small monthly
fee that is yet to be determined in order to connect the e‐
collar to a cellular network.
In the past, Pet Technologies has used the services of
Ad Jungle to develop its advertising campaigns. Ad Jungle
is located in the same city as Pet Technologies and is a
boutique advertising agency with 25 full‐time employees
and a similar number of people it contracts with on an ad
hoc basis. Pet Technologies has reached out to Ad Jungle
and has requested an estimate for the cost of developing an
advertising campaign for the e‐collar.
Ad Jungle begins the process of developing an ad campaign by first meeting with the client to learn about the
product to be advertised. Based on what is learned, Ad
Jungle works with the client to define the target audience
for the advertising campaign. Because the target audience
may be different than its current customer base, market
research is often required. Historical data suggests that
defining the target market typically takes 80 hours (2 weeks).
In some cases, determining the target audience is fairly
routine and can be done in as little as a day and a half
(12 hours in total) while in other cases much less is known
about who will use the product and defining the target
market can take as long as 4 weeks (160 hours). Also, the
number of people working on the task at any one time varies as does their hourly bill rate. Given this, the average
amount the customer is billed for the work to determine
the target audience typically averages $170 per hour. In
some cases, however, the hourly rate was as low as $100 per
hour and in other cases as high as $450 per hour.
Once the target audience has been defined, alternative
concepts for the advertising campaign are developed. A
key aspect of developing the concept is deciding on the
tone the campaign will use such as humorous, dramatic, or
informative. Concept development is a highly creative
5.
What path has the longest expected time? What
is the probability that this path will be the
critical path?
S
E
process and often requires several iterations between Ad
Jungle and its clients. On average, concept development
takes 2 weeks (80 hours) but has been completed in as little as 1 week (40 hours) or as long as 5 weeks (200 hours).
Only one person from Ad Jungle works on concept development, and typically these employees are billed out at
$125 per hour. However, the least experienced employee is
billed out at $75 per hour and the most experienced person is billed out at $200 per hour.
Once the client signs off on the concept, the development of the creative collateral (brochures, print ads,
scripts for TV commercials, and so on) and determining
the mix of media are done concurrently. The amount of
time it takes to develop the creative collateral depends on
how many media channels will be used (e.g., print, TV,
online), how many people are assigned to work on developing the creative collateral, and how much involvement
the client wants to have in the process. Typically, developing the creative collateral requires 3 weeks (120 hours) but
has been done in as little as 1 week (40 hours) and on
other occasions required 6 weeks (240 hours). Furthermore,
the number of people and their bill rates varies from project to project. On average, the hourly bill rate for developing the collateral is $275 but has been as low as $145
and as high as $425.
Determining the mix of media is almost always done by
the same person at Ad Jungle. This person is currently
billed out at $150 per hour. The amount of time it takes to
determine the mix of media that will be used ranges from
half a day (4 hours) to 2 days (16 hours), but it typically
takes 1 day (8 hours).
Once the collateral has been developed and the mix of
media determined, the final step is production where the
advertising materials are developed. The production of
the advertising materials is outsourced to firms, and Ad
Jungle assigns one account manager to oversee the work.
The account managers are billed out at an hourly rate of
$200. On average, 80 hours of the account manager’s time
is needed to oversee production. In some cases, as little as
30 hours of the account manager’s time was required,
while in other cases as many as 160 hours were needed.
1 85
B IB LIOG RA PHY
QUESTIONS
1.
2.
Draw a network diagram for this project. Identify all
the paths through the network diagram.
Simulate the completion of this project 1,000 times,
assuming that activity times and costs follow a triangular distribution. Estimate the mean and standard
deviation of the project completion time. Also, estimate the mean and standard deviation of the amount
the client would be billed for the project.
3.
4.
What is the probability the project can be done for
less than $100,000? What is the probability the project will take between 350 and 450 hours?
Based on the results of your simulation analysis, how
would you respond to Pet Technologies’ request for
an estimate of the project costs and duration?
BIBLIOGRAPHY
De Meyer, A., C. H. Loch, and M. T. Pich. “Managing
Project Uncertainty: From Variation to Chaos.” MIT
Sloan Management Review, Winter 2002. (This article provides a useful classification of uncertainty faced in projects
and provides specific suggestions for managing each type
of uncertainty.)
Goldratt, E. M. Critical Chain, Great Barrington, MA,
North River, 1997.
Hulett, D. T. “Project Schedule Risk: Monte Carlo
Simulation or PERT?” PM Network, February 2000.
(Hulett comes to the same general conclusion that we do.
Simulation is superior to the statistical methods of PERT
for complex problems. In stating his case, however, he fails
to note that for a PM to use simulation effectively, the PM
should understand the statistics of simulation.)
Kamburowski, J. “New Validations of PERT Times.”
Omega, International Journal of Management Science, Vol.
25, No. 3, 1997.
Keefer, D. L., and W. A. Verdini. “Better Estimation of
PERT Activity Time Parameter.” Management Science,
September 1993.
Lawrence, J. A., Jr., and B. A. Pasternak. Applied
Management Science, New York: Wiley, 1998. (This book
has solution techniques for finding the critical path and
time for a network using Excel’s® Solver.)
Leach, L. “Schedule and Cost Buffer Sizing: How to
Account for the Bias between Project Performance and
Your Model.” Project Management Journal, June 2003. (This
article identifies a variety of sources of schedule and cost
biases and provides recommendations for coping with them.)
Liberatore, M. J. “Project Schedule Uncertainty Analysis
Using Fuzzy Logic.” Project Management Journal, December
2002. (Our discussion in this chapter addresses schedule
uncertainty from both a probability theory approach and
using computer simulation. This article demonstrates the
use of fuzzy logic for assessing project schedule uncertainty.)
McMahon, C. S. “Using PERT as an Approximation of
the Fuzzy‐Project Network Analysis.” IEEE Transactions
on Engineering Management, May 1993.
Mantel, J. R., and S. J. Mantel, Jr. Project Management:
A Managerial Approach, 8th ed., Hoboken, NJ: John
Wiley, 2012.
Pritsker, A. A. B. “Gert Networks.” The Production
Engineer, October 1968.
Ruskin, A. M. “Using Unders to Offset Overs.” PM
Network, February 2000.
C
H
A
P
T
E
R
6
Allocating Resources to the Project
Trade-Offs
In this chapter we consider the problem of allocating physical and human resources to
projects.* The physical and human resources are granted to and used by the project in
order to meet the project’s objectives. The amount of resources that can be allocated, of
course, depends on the timing of the allocation as well as on the total supply of resources
available for allocation. Mainly, resource allocation concerns how we allocate specific,
limited resources to specific activities (or projects) when there are competing demands
for the same limited resources.
Projects compete with each other for the same resources in two different ways. First,
consider a resource that is limited but is not consumed when used, the services of a
specific technical specialist for instance. The problem here is which project gets to use
the resource first and which must wait. Second, consider a resource that is limited and is
consumed when used, a specific chemical reagent for instance. In this case, the second
project may have to wait until more of the reagent can be purchased and delivered. In
both cases, the project that must wait may suffer a schedule delay that makes it late. Just
as projects may compete for resources, different activities of the same project may compete. Two or more concurrent activities might require the same personnel, or equipment,
or even work space. One activity will be given priority, and the other(s) must wait.
In order to manage resources in such a way as to optimize the use of a limited supply,
trade‐offs must be made. The interaction of project scheduling and resource scheduling
is clear, but we will examine several different solutions to the allocation problem. Those
include the Critical Path Method (CPM), Goldratt’s “critical chain” (1997), and many different priority rules for allocating scarce resources. The primary cause of concern is resource
scarcity. If some resources (including time) were not scarce, the resource allocation problem would be concerned solely with profit maximization—a relatively easy problem.
*With few exceptions, we will not make a distinction between human and nonhuman resources in this chapter. We need not distinguish between them in order to consider the allocation problem. The tasks of administering
the human and nonhuman resources are quite different, of course.
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6 .1 EX P E DI TI NG A PR OJ ECT
Trade-Offs
6.1
1 87
In Chapter 5 we evaluated project durations solely in terms of time. A project was
either on time or not. Now we must also consider when and for what purposes scarce
people, equipment, material, and facilities are used. The PM’s performance is judged
by the skill with which the trade‐offs of time, resources, and scope are managed, so the
PM must make constant use of cost/benefit analysis. There are countless questions to
be answered. “If we come in late on this project, we face a $1,000 per day penalty. How
much project slack do we need and what resources at what costs are required to get it?”
“If I hire Cheatham Engineering Associates as design consultants, can I improve project
performance by 3 percent without extending the project’s due date?” “Adding project
slack and hiring a consultant require monetary resources that could be used for other
things. Are these the best uses for the dollars?”
At times, the PM is asked to take on a project in which there are the usual time,
budget, and scope goals, but which also constrain the trade‐offs that the PM may wish to
make if required to help the project meet its most important goals. For example, some projects are time constrained and must be completed by a fixed time. In such cases, resources
(and possibly performance) are variable. Some projects are resource constrained and cannot
go over budget or use more than a fixed amount of a specific resource. In these cases, time
(and possibly performance) is variable. Occasionally, a senior manager suffers from a case
of the micromanagement virus and fixes time, cost, and scope, thereby leaving the PM
with no flexibility whatsoever. Such projects are certain to fail unless the micromanager
has been profligate with the firm’s resources, which is highly unlikely for micromanagers.
The fault actually lies with the PM who accepts command of such a project. (For those
who are thinking that such a PM may find him‐or herself without a job following a refusal
of an assignment, we would note the senior manager in question is insuring that the PM
will fail. Do you want to work for someone who will not allow you to succeed?)
We will start our tour through the wilds of resource allocation by reconsidering the
problem of dealing with a pointy‐haired boss who insists that a project be completed in
much less time than its expected duration.
EXPEDITING A PROJECT
The unreasonable boss problem in Chapter 5, Section 5.2 could be used as our example
here, but a smaller problem will help avoid unnecessary arithmetic. Our problem is set in
a deterministic world rather than in a probabilistic one, for the same reason. (Please
remember that in reality all projects are carried out under conditions of uncertainty.)
Finally, we must also take note of an assumption usually adopted when activities are
scheduled, as we did in Chapter 5. That assumption is that all estimates of task duration,
whether deterministic or probabilistic, are based on normal or standard resource loadings.
The Critical Path Method
In traditional PERT/CPM, the rules of “standard practice” apply and the normal task
duration estimate is made with the normal or standard‐practice resource usage. Then a
second estimate, referred to as the crash duration, is made based on the resources required
to expedite the task. More resources of the type already used might be added; more workers and shovels if there is a ditch to be dug. On the other hand, the technology used to
dig the ditch might be totally altered, utilizing a backhoe or a Ditch Witch®, for example.
When making estimates for crashing, it is important to make sure that the resources
required to crash the project are, in fact, available. Using a machine to dig the ditch in
three hours instead of the 3 days required for a worker with a shovel is dependent on
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CH A P T ER 6 / A L L OCAT IN G RESOU RC ES T O T H E P ROJE CT
Risk
the fact that the machine is available and can be on site when needed. (Of course, the
warning about resource availability applies equally to normal resource requirements as
well as to crash requirements.) There are times when the PM may expedite activities that
have little or no impact on the network’s critical time, such as when the resources used
must be made available to another project. It is important to remember that when we
change technology, we may also be changing the level of risk in carrying out the activity.
Finally, we must remind ourselves that some tasks cannot be crashed. One must not
assume that because it takes one woman 9 months to carry and bear a child that nine
women can accomplish the same result in 1 month.
Consider the project described in Table 6-1. There is a set of activities, predecessors, normal task duration estimates, crash duration estimates, and estimates for normal cost and crash cost. One crash duration is marked with a single asterisk. For this
activity, the task may be carried out in normal time or crashed 1 day at a time. Another
activity is marked with a double asterisk. In this case, the duration must be one or the
other; it cannot be broken down to 1‐day segments. Activities are charged at the “cost
per day” (activity slope) increments shown in the last column. A given activity may
have only two or three technically feasible durations. If an activity cannot be split into
1‐day segments, the cost is as indicated. The “slope” information for non‐or‐partially
segmented activities is normally given in the slope chart. Activity slope is computed
as follows:
slope
crash cost normal cost
crash time normal time
When crashing a project, starting with the normal schedule for all project activities,
crash selected activities, one at a time, to decrease project duration at the minimum
additional cost. To crash a project, follow two simple principles: First, focus on the critical path(s) when trying to shorten the duration of a project. Crashing a noncritical activity will not influence project duration. Second, when shortening a project’s duration,
select the least expensive way to do it.
Given these guides, consider the network shown in Figure 6-1(a) that was constructed from the data in Table 6-1. It is easier to illustrate the impact of crashing on an
activity‐on‐arrow (AOA) network than on an activity‐on‐node (AON) network, so we
use that approach here. Also, we use dummy activities in this case not to illustrate precedence but to show time durations and slack on the time axis.
As indicated in Table 6-1, activity d can be partially crashed for $30, but it is not on
the critical path and will not shorten the project. Activity e involves a technological
discontinuity and must take either 3 days to complete at $10 or 1 day at $80. In general,
the impact of having such a technological discontinuity is that the best solution for
Table 6-1 An Example of a Normal/Crash Project
Activity
Precedence
Duration, Days
(norm, crash)
Cost (norm,
crash)
a
—
3, 2
$ 40, 80
40/−1 = −40
b
a
2, 1
20, 80
60/−1 = −60
Slope ($/day)
c
a
2, 2
20, 20
—
d*
a
4, 1
30, 120
90/−3 = −30
e**
b
3, 1
10, 80
−70 (2 days)
* Partial crashing allowed
**Partial crashing not allowed
1 89
6 .1 EX P E DI TI NG A PR OJ ECT
b
Normal Schedule
8 Days, $120
e
c
a
d
1
2
3
4
5
6
7
8 Days
6
7
8 Days
5
6
7
8 Days
5
6
7
8 Days
(a)
b
c
a
7-Day Schedule,
$160
e
d
1
2
3
4
5
(b)
e
b
6-Day Schedule,
$220
c
a
d
1
2
3
4
(c)
b
5-Day Schedule,
$260
e
c
a
d
1
2
3
4
(d)
e
b
4-Day Schedule,
$350
a
c
d
(e)
Figure 6-1 A PERT/CPM example of crashing a project, AOA network.
crashing n days might not be part of the best solution for crashing n + 1 days. Rather, it
may be best to crash the activity with the technological discontinuity at n + 1 days and
not crash another activity that could be crashed for n days. This situation is illustrated in
the discussion that follows.
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CH A P T ER 6 / A L L OCAT IN G RESOU RC ES T O T H E P ROJE CT
The network’s critical path is a‐b‐e, the project duration is 8 days, and the normal
total cost is $120, as illustrated in the network of Figure 6-1(a). The decision about
which activities to crash depends on how much we need to reduce the duration of the
project. To reduce the total network duration by 1 day, we must reduce the time required
by one of the activities along the critical path. Inspecting Table 6-1 to see which critical
activity can be reduced at the least cost, we find it is activity a, which adds $40 to the
project’s current cost of $120. Activity b could be crashed at an added cost of $60 or we
could even crash e 2 days for an additional cost of $70. Of course, crashing e would only
shorten the project duration by 1 day because when e is shortened, the path a‐d‐dummy,
7 days long, becomes the critical path and does not allow the project to be shortened to
6 days. Of the three options, crashing a is the lowest cost and therefore preferable, see
Figure 6-1(b). Notice that crashing a also shortens a‐d‐dummy and a‐c‐dummy by 1 day.
Suppose the project must be crashed by 2 days. What are the options? Reconsidering
Table 6-1 and Figure 6-1(a), we see that we could crash activity e for 2 days ($70), but
path a‐d‐dummy (7‐days’ duration) must also be crashed at least 1 day. We choose d
($30/day) because it is cheaper than a ($40). The total cost of crashing is $100, and the
total project cost is $120 + $100 = $220. Alternatively, we could crash a and b, also for a
cost of $100 ($40 + $60). Arbitrarily, we choose the latter option [Figure 6-1(c)].
Now suppose we wanted to crash the project by 3 days, from the original 8 days down
to 5 days. Clearly e must be crashed by 2 days, costing $70, and a or b by a day. We choose
a, the cheapest, for an additional $40. This leaves d to be crashed by 1 day for another
$30, resulting in a total crashing cost of $140 and a project cost of $120 + $140 = $260
[Figure 6-1(d)]. Note that we did not crash b this time, as we did for 6 days. This is due
to the technological discontinuity in activity e.
Last, let us consider crashing the project by 4 days down to a project duration of
4 days. Since we crashed e, the technological discontinuity, to reach a 5‐day duration, all
the remaining activities can be incrementally crashed. Thus, we can simply inspect
Figure 6-1(d) to see what else needs incremental crashing to reduce the project by another
day. Notice in Figure 6-1(d) that a‐b‐e and a‐d‐dummy are both critical paths. Only b and
d can still be crashed so we crash each by 1 day for an additional cost beyond the 5‐day
schedule of Figure 6-1(d) of $60 + $30 = $90 for a total project cost of $260 + $90 = $350
[Figure 6-1(e)]. Note that c is now critical; therefore, all paths are critical. Since the critical paths a‐b‐e and a‐c are at their full extent of crashing, the project duration cannot be
further reduced, even though activity d could be crashed another day. Thus, Figure 6-1(e)
is not the all‐crash network, although it equals the all‐crash time schedule of 4 days.
Whether all this crashing is worthwhile is another matter. On the cost side, Figure 6-2
shows the time/cost relationship of crashing the project. On the benefit side, some projects have penalty clauses that make the parent organization liable for late delivery—and
sometimes bonuses for early delivery. Starting at the right (all‐normal) side of Figure 6-2,
note that it becomes increasingly costly to squeeze additional time out of the project.
Charts such as the one shown in Figure 6-2 are useful to the PM in exercising control
over project duration and cost. They are particularly helpful in dealing with senior managers who may argue for early project completion dates with little understanding of the
costs involved. Similarly, such data are of great benefit when clients plead for early delivery. If the client is willing to pay the cost of crashing, or if the firm is willing to subsidize
the client, the PM can afford to listen with a sympathetic ear. (While we advise the PM
to ignore overhead costs over which he or she has no control, it should be noted that
indirect costs are often altered when a project is crashed.)
One final note on crashing projects. The same method is used when the task durations are probabilistic, that is, using three-time estimates. In this case, optimistic, most
likely, and pessimistic activity duration estimates are made for the “normal” resource
loading and new optimistic, most likely, and pessimistic duration estimates must be made
1 91
6 .1 EX P E DI TI NG A PR OJ ECT
400
All crash
a + b +2d + 2e
Cost ($)
300
a + d +2e – b
200
a+b
a
All normal
100
0
2
4
6
8
10
Total Duration (Days)
Figure 6-2 CPM crash cost‐duration history.
for crash resource loading. The PM should remember that the variance of both the normal and crash activity times largely depends on the technology used to accomplish the
activity in question. Thus the variance of the normal activity time may be quite different
from the variance of the crash time. The project budget can be determined in exactly the
same way. The solution to project duration and resource cost levels can be reached by
using the standard analytical method used in the last chapter, or by simulation, also
described in Chapter 5.
Crashing a Project with Excel
*
In this section we demonstrate how spreadsheets can greatly facilitate the task of choosing which activities to crash such that the project can be completed by a specified time.
To illustrate this, we use the previous example with one minor change. Namely, we
assume that partial crashing is allowed for all activities that can be crashed. The network
for the example and the spreadsheet developed to solve the problem are shown in
Figures 6-3 and 6-4, respectively.
3
e
b
1
a
c
2
5
d
4
Figure 6-3 AOA network from Figure 6‐1(e) with node labels added.
*This section is intended for readers that have a background in linear programming and Excel’s Solver. This
section can be skipped without loss of continuity.
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CH A P T ER 6 / A L L OCAT IN G RESOU RC ES T O T H E P ROJE CT
Figure 6-4 Spreadsheet model to find optimal crashing plan.
At the top of the spreadsheet shown in Figure 6-4, columns A through F contain the
information given in the problem. In column G, the crash cost per day (slope) is calculated by dividing the incremental cost of crashing the activity as much as possible by the
maximum number of days the activity can be shortened. For example, the formula =(E2‐
F2)/(C2‐D2) was entered in cell G2 and then copied to other cells in column G (except
for activity c which cannot be shortened).
Column H contains a formula to calculate the maximum number of days an activity can be crashed by subtracting the crash time from the normal time. For example,
the formula =C2‐D2 was entered in cell H2 and then copied to the other cells
in column H.
Column I corresponds to one of our decision variables, namely, the amount to crash
each activity. In column J, the cost of partially crashing an activity is calculated on the
basis of the amount of time the activity is to be crashed as determined in column I. For
example, the formula =I2*G2 was entered in cell J2 and copied to the other cells in
column J. The total crashing cost is calculated in cell J7 as the sum of the cost of crashing
the individual activities in cells J2:J6.
In column K, the actual time to complete each activity is calculated by subtracting
the amount the activity is crashed (column I) from its normal time (column C). For
example, the formula =C2‐I2 was entered in cell K2 and then copied to the other cells
in the column.
The middle of the spreadsheet shown in Figure 6-4 (cells B11:B14) are for the other
decision variables needed. More specifically, cells B11:B14 correspond to the event times
for each of the nodes in the network diagram. (Node 1 is excluded because we assume it
occurs at time zero.) As you will see, we need these decision variables to preserve the
precedence relationships shown in the network. For example, we need to make sure that
node 4 does not occur until after node 2 occurs, plus the time it take to complete activity d.
We now demonstrate how Excel’s Solver can be used to determine which activities
to crash so that the entire project is completed with 5 days at the minimum cost. To
begin, we select Solver from the Data ribbon (note that Solver is an Excel add‐in and
must be added before it can be used). The Solver Parameters dialog box is now displayed
(see Figure 6-5). Our objective is to minimize the total crash cost which is calculated in
cell J7. To specify this, we enter cell J7 in the Target Cell box and then select the Min
1 93
6 .1 EX P E DI TI NG A PR OJ ECT
Figure 6-5 Completed Solver Parameters dialog box.
option button. Next, we tell Excel which cells it can change in order to find the solution
with the least total crashing cost. In the spreadsheet shown in Figure 6-4, the values
that can be changed are the amount of time each activity is crashed (cells I2:I6) and the
time when each event occurs (cells B11:B14). Thus, these ranges are entered in the By
Changing Variable Cells box. Note that these two separate ranges are separated by a
comma in the By Changing Variable Cells box (see Figure 6-5).
The last task is to enter the constraints for the problem. Perhaps the most obvious
constraint is that we want to complete the project within 5 days. Since node 5 (cell B14)
corresponds to the event of the project being completed, we can specify this constraint
as follows:
B14
5
Another important set of constrains that needed is to make sure that we don’t crash an
activity more than the maximum number of days that is can be crashed. For example, to
ensure activity a is not crashed more than it can be physically crashed, we could enter the
constraint I2
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