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FOREWOR
the
1996
S um m er O l y m p ~ c
In Atld nta, Ge or g~ a, xemphfles a
p ro ject . The c~ t y f A t l a n t ~ ,
k.;g;; J
~ i i~g j t~ ju r i c t~on~ t hhe Olympic Pla nn~ ng ommtt -
tee, had to plan a nd execute a w ide variety of tasks,
including the cons truc tion of buildings to house
some of the events, arrange men t of housing and se-
c u r ~ t y or the athle tes, organization of a t ransporta-
tion system for millions of spectators, and
coo rdin atio n of all the athletic events. M ore than
2 athletic events had to be scheduled for the 15-
day period of the games. Logical constrain ts, such as
scheduling the sem ifinals before the finals and not
scheduling tw o events for the same time and place,
c o m p l ~ c a t e dhe plan ning process. Tradition s also
had to be considered, such as running the marathon
on the last day and e nsuring that swimming and
track a nd field did n ot g o on at the sam e t ime. Inter-
national
TV
networks also imposed conditions, such
as having events popula r with their h ome audiences
take place in prime time. O f course, not all events
could be run when the V networks wanted because
different countries in different time zones enjoy the
sam e sports. Only careful project scheduling and
control would enable the athlet ic events to take
place o n time an d ensu re the availability of the re-
sources to run them properly.
rojects like the
1996
Sum mer Olympics are unique operat ions with a f inite
life span. Generally, many interrelated activities must be scheduled and
monitored within strict time, cost, and performance guidelines.
In
this
chapter
we
conside r methods f or man aging comp lex projects. We begin with a
general introduction to the basic project manag emen t tools and some of th e ma n-
agerial aspects of project scheduling and con trol. We then explore the use of net-
work methods for managing projects and end with an assessment of their
limitations.
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MANAGING PROJECTS
The Olympic Committee is responsible for scheduling and controlling a large
project. We define a project as an interrelated set of activities that has
a
definite
start ing and ending point and that results in a unique prod uct o r service. Exam-
ples of large projects include constructing a building, ball park, road, dam, or oil
pipeline; renovating a blighted urba n area; developing a p roto type for a new air-
plane; introduc ing a new prod uct; organizing a state fair; an d redesigning the lay-
ou t of a plant o r office.
Project management is goal oriented: When the team accomplishes its as-
signed objectives,
i t disbands. Team members might move o n to ot her projects or
return to their regular jobs. The project manager must motivate and coordinate
the personnel assigned to the project to deliver the project on time. Complex
projects such as organization of the O lympic games involve thous and s of interre-
lated, often unique, activities. Thus the project manager may have difficulty
falling back on prior experience or established procedures. The personnel come
from diverse backgrounds and have many different skills. Furthermore, many
team members will n ot be associated with the project for its full duratio n. They
may view the project as disruptive to their regular work relationships and rou-
tines. Oth ers will experience conflicts in loyalty or in d em and s on their time be-
tween their projects and department supervisors. But, despite these potential
difficulties, workin g on projects offers substantial reward s: t he excitement of dy-
nam ic work, the satisfaction of solving challenging problem s, th e status
o mem-
bership on an eli te team, and the o pportunity to work with an d learn from other
skilled professionals.
Project managers must stay on top of their projects to meet schedules and
keep costs within budget. Unexpected problems can cause delays, requiring
rescheduling an d reallocation of resources-and often resulting in severe finan-
cial repercussions. For example, Microsoft announced a delay in the release of
Windows 95 because preli min ary testing results unexpec tedly uncovered bugsn
in the program. The problems had to be corrected before further tests could be
conducted. The delay dealt a blow t o third-party softw are developers, who also
had to delay the. release of their pro ducts. After the delay a nnouncem ent,
Mi-
crosoft's stock closed d ow n 2 poin ts on Nasdaq trading.
hat
tools
are
available to schedule and control projects
Frequently, m anag ers must ma ke quick decisions on th e basis of incomplete
information. Netw ork planning models can help project ma nag ers maintain con-
trol, giving them t he capability t o evaluate
the
time and cost implications of re-
source trade-offs . Gan tt charts have long been used t o schedule and control
projects For large proj ects, however, Ga ntt charts present difficulties:
They don't directly recognize
precedence relationships between activities, an d they d on't indicate which activi-
ties are crucial to completing t he project on time.
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Tw o network planning meth ods were developed in the 195 0s to deal with
som e of the shortcoming s of G an tt charts. Both methods look a t a project as a set
of interrelated activities that can be visually displayed in a network diagram,
which consists of nodes (circles) an d arcs (a rrow s) that depict the re lationships
between activities. Working with a network diagram, an analyst can determine
which activities,
if
delayed, will delay the en tire project.
The program evaluation and review technique (PERT) was created for the
U S Navy s Polaris missile project, w hich involved 30 00 sep arat e contra ctor s
an d suppliers. Because man y of the project s activities had never been per form ed
before, PERT w as developed t o handle uncertain time estimates. In retrospect,
PERT generally is credited with reducin g the project s com pletion time by at leas t
18 months .
J. E. Kelly of Remington-Rand and M. R Walker of Du Pont developed the
cri tical path metho d (CPM) as a m eans of scheduling maintenance shu tdow ns at
chemical processing plants. Because maintenance projects were routine in the
chemical industry, reasonably accurate time estimates for activities were avail-
ab le . Thus
CPM w s based on the assumption that project activity times can be
estjmated acc urately and d o not vary.
Although early versions of PERT and CPM differed in their treatment of
time estimates, today the differences between PERT and C PM ar e minor. Basical-
ly,
either approach can cope with uncertainty. For purposes of our discussion, we
simply refer to th em collectively as PERT/CPM .
NETWORK
METHODS
Managing a complex project requires identifying every activity to be undertaken
an d planning w hen each activity rnus.t begin and end t o complete the ove rall
project on time. Th e degree of difficulty in scheduling a complex pro ject depends
on the num ber o f activities, their required sequence, and their timing. Typically,
mana ging projects with n etw orks involves four steps:
1
describing the project,
2. diagramming the netwo rk,
3. estimating time of completion, and
4.
monitoring project progress.
Describing
the
Project
The project manager must first describe the project in terms that everyone in-
volved will understand. This description should include a clear statement of the
project s end point . For exam ple, the end poin t for a software development team
would be publication of the completed softwa re package. W ith the inpu t of the
team, the project manager must carefully define all project activities and prece-
dence relationships. An activity is the smallest unit of work
effort
consuming
both t ime and resources that the project manager can schedule and control .
A
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precedence relationship determines a sequence for undertaking activities; i t speci-
fies that one activity cannot start until a preceding activity has been completed.
For exam ple, brochures ann oun cing a conference for executives must first be de-
signed by the program committee (activity A) before they can be printed (activity
8 . In oth er wo rds, activity A must precede activity B.
Just wh at co nstitutes an activity will vary. For exam ple, suppose a divisional
vice-president is put in charge of a project to start m anuf acturin g a product i n a
for eign country. Her list of activities may include constr uct the plant. This
item ind icates th at completion of construction will have a major bearing on when
operations can begin. However, the construction supervisor's list of activities
must include a greater level of detail such as pou r fou ndatio n and wire for
electrica l service. In general, a manager's project description should reflect only
the level of detail that he or she needs in o rder t o ma ke scheduling and resource
allocatio n decisions.
Diagramming
the
etwork
Diag ram min g the project as a network requires establishing th e precedence rela-
tionships between activities. For complex projects this task is essential because
incorrect or omitted precedence relationships will result in costly delays. The
precedence relationships are represented by a network diagram, consisting of
nodes (circles) and arcs (ar row s) that depict the relationships between activities.
Two d ifferent approaches may be used to create a network diagram. The first ap-
proac h, the activity-on-arc (A OA ) netw ork, uses arcs to represent activities and
nodes to represent events. An event is the point at which one or more activities
are to be completed and one or more other activities are to begin. An event con-
sum es neither time nor resources. Because the
AOA
appro ach emphasizes activity
connection points, we say that it is
event oriented
Here , the precedence relation-
ships requ ire th at a n event not occur until all preceding activities have been com-
pleted. A convention used in AOA networks is to number events sequentially
from left to right.
T he second approach is the activity-on-node ( A O N ) network, in which the
nodes represent activities and the arcs indicate the precedence relationships be-
tween them. This approach is
activity oriented
Here, the precedence relation-
ships require that an activity not begin until all preceding activities have been
completed.
Figure
1 shows the AOA and
AON
app roa che s for several activity rela-
tionships commonly encountered. In Fig.
l( a ) , activity S m ust be completed
before activity T which in tur n mu st be comple ted before activity U can be start-
ed. For example, in the AOA diagram, event 1 might
be
the start of the proj-
ect, an d event 2 might be the completio n of activity S. The arrows in the
AOA diagr am den ote both precedence an d the activity itself. T he arr ow for activ-
ity S starts from event 1 and ends at event 2 ~n dica ting hat the sequence of
events is from
1
to
2.
In the AON diagram, the arrows represent precedence rela-
tionships only. The direction of the arrows indicates the sequence of activities,
from
S
t o
T
t o U.
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F I G U R O and O N pproaches to ctivity Relationships
Figure l b ) shows tha t activities
S
and ca n be wo rke d simultaneously,
but bo th mus t be completed before activity U can begin.
In
Fig.
l c),both ac-
tivities and U ca nn ot begin until activity S has been comp leted. Multiple depen-
dencies also can be identified. Figure I d ) show s tha t and V cannot begin
until both S and have been completed.
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Sometimes the
AOA
app roach requi res the addi t ion
of
a ummy
activity
t o
clarify the preced ence relationships between tw o activities. Figure
l ( e ) s h o w s
an example
of
this situation. Activity
U
cannot begin until both
S
and T have
been completed; however V depends only on the completion of T. A d u m m y a c -
tivity, which ha s an activity time of zero and requires n o resources, must b e used
to clarify the precede nce between
T
an d V and between
S
a n d
T
and
U.
A
d u m m y
activity also is hsed when tw o act ivi ties have the sam e start ing and end ing nodes.
For example, in Fig. l ( f) , both activities
T
and
U
cannot begin until
S
has been
completed, and activity
V
can no t begin unti l both
T
a n d
U
have been completed.
T h e dumm y act ivi ty enables act ivi t ies
T
and
U
to have unique beginning nodes.
This dist inct ion is importan t for com puter prog rams because act ivit ies often are
identified by their beginning and ending nodes. Without dummy activities, activi-
t ies with identical beginning an d end ing nodes could not be different iated from
each other , which becomes important when the act ivi t ies have different t ime
requirements.
Example :Diagramming
a
Hospital Project
In the interest of better serving the public in Benjamin C ounty, St. Adolf s Ho spi-
tal has decided to relocate from Christofer to Northvil le, a large suburb that at
present has n o prim ary medical faci li ty. Th e move to N orthvil le wil l involve con-
struct ing a ne w hospital a nd making i t opera t ional . Jud y Kramer, executive direc-
tor of the b oard of St. Adolf7s, must prep are for a hearing, scheduled for nex t
week, before the C ent ra l Ohio Hospi ta l Board
(COHB)
on the proposed project.
T h e hearing will add ress the specif ics of the to tal project , including t ime an d co st
est imates for i ts com plet ion.
With the help of her staff, Kram er has identified ma jor project activities.
She also has specified the immediate predecessors (those activities that must be
completed before a part icular act ivi ty can begin) for each activi ty, a s sho wn in
the following table.
Immediate
Activity Description Predecessorfs
Selec t administrative and medical staff.
Select site and do site survey.
Select equipment.
Prepare final construction plans and layout.
Bring utilities to the site.
Interview applicants and
fill
positions in nursing,
support staff, maintenance, and security.
Purchase and take delivery o equipment.
Construct the hospital.
Develop an information system.
Install the equipment.
Train nurse s and support staff.
a.
Draw
the
AON
network diagram.
h
D r a w t h e
O
network diagram.
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Solution
a. The
AON
network for the hospital project, based on Kram er s 11 activi-
ties and their p recedence relationships, is sho wn in Fig.
2. It depicts ac-
tivities as circles, with a rro ws ind icating the sequence in which they are to
be performed. Activities
A
and
B
emanate from
a
start no de because they
have no immediate predecessors. Th e arrow s connecting activity to ac-
tivities
C, F,
and 1 indic ate tha t all three require completion of activity
before they can begin. Similarly, activity
B
must be completed before ac-
tivities D and E can begin, a nd so on. Activity K connects to a
finish
node
because n o activities follow it. Th e start a nd finish nod es
do
not ac t~ia l ly
represent activities. Th ey merely provide beginning and ending po ints for
the network.
R
E
A ON Network for the St. Adolf s Hospital Project
b. The
A O A
diagram is shown in Fig.
3
Event 1 is the start of the proj -
ect. Activities A and B have n o imm ediate predecessors; therefore the ar-
row s representing those activities both have event 1 as their base. Event 2
signals the completion of activity
A
As activities C, F, and I all require th e
cornpletioh of A, the arrows representing these activities leave the node
u
a
O A
Network
for
the St. Adolf s Hospital Proiect
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represenring event
2.
Similarly, the arrows for activities
L)
and
E
leave the
node for event 4, which signals the completion of activity B. The arrow
for activity G leaves event
3,
and event 6 is needed to tie activities G, H,
a n d E together because they must be completed before activity J can
begin.
Properly representing the relationship for activity K requires the use of
a dummy activity. Activities
I
and F both emanate from event 2, and both
mu st be completed before K can begin. Activities
I
and F will have the sam e
be g~ nn in g nd ending nodes unless a dum my activity is used. Hence event 7
signals the end of activity
I
and event 8 signals the end of activity F, with a
dummy activity joining them. Now
all
activities are uniquely defined, and
the netw ork show s tha t activities F, I and
J
must be comple ted before activ-
ity
K
can begin. Event 9 indicates the completion of the p roject.
Both the
AON
and the AOA ap proa ch can accurately rep resent all the activi-
ties and precedence relationships in a project. However, the AOA approach often
requires fewer nodes than the AON approach. In Example 1 the AON dia-
gram has
13
nodes whereas the A O A diagram has only 9. In contrast, the AON
appr oach doesn t need dum my activities. Regardless of the appro ach used, mod-
eling a large project as a network forces managers to identify the necessary activ-
ities and recognize the precedence relationships. I f this preplanning is skipped,
project often experiences unexpected delays.
In the remainder
of
ou r discussion o f PERT ICPM , we will use the
AON
con-
vention, although AOA diagrams also can be applied to all the ~ro ce du re s.
Estimating
Time
of C o m p l e t i o n
Project managers next must m ake time estimates fo r activities. When the sam e
type of project has been d on e many times before, time estimates are ap t to have a
higher degree of certainty an d are said to be deterministic estimates. I f a project
has never been done before, time estimates invoive uncertainty and are called
probabilistic estimates. For now, assume that the time estimates used in the
St
Adolf s Hos pital relocation pr oblem are determ inistic estimates. Figure
4
on
the next page show s the estimated time for each activity of t he St. ~ d o l f s roject.
Which activities determine
the
direction of an entire project
A
crucial aspect of project m anag eme nt is estimating the ti me of com pletion.
If each activity in relocating th e hospital were don e in sequence, w ith w ork pro-
ceeding on only on e activity at a t ime, the t ime of completion wo uld equal the
su m of the times for all th e activities, o r
175
weeks. However, Fig. 4 indicates
that some activities can be carried on simuItaneously. We call each sequence of
activities between th e project s st ar t and finish a path.
Figure 5
shows tha t the
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G u R E 4 etwork
for
St. Adolf s
Hospital Project, Showing
Activity Times
Finish
network describing the hospital relocation project has five paths: A-I-K,
A-F-K,
A-C-G-J-K, B-D-H-J-K, an d B-E-J-K. T h e critical pa th is the sequence of ac-
tivities between a project s start and finish th at takes the longest time to com-
plete. Thu s the activities along the critical path determine the completion time of
th e project; tha t is, if one of the activities on th e critical path is delayed, the entire
project will be delayed. Th e expected times for th e paths in the hospital project
network are
Path xpected Time wk)
A-F-K 8
A I K
A-C-G-J-K
7
ED H J K
9
BE J K
4
T he activity string RD-H-J-K is expected to take 69 weeks t o complete. As
the longest, it constitutes the critical path fo r the hospital project an d is show n in
., . . I Fig- 5.
As the critical path defines the com pletion time o
the
project, Judy Kramer
should focus on these activities in managing the project.-However, projects can
have m ore than o ne critical pat h. f activity
A, C,
o r G were
to
fall behind by two
wee ks, the strin g A-C-G-J-K wo uld be a second critical pa th . Consequently,
managers should
be
awa re that delays in activities not on the critical path could
caus e delays in the entire project.
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F
5
Activity Paths
for
the Hospital
Project wi th th e Critical Path
Shown in eavy
l ines
Manually finding the critical path in this way is easy for small projects; how-
ever, compu ters must be used f or large, comp lex projects. C o~ np ut er s alculate
activity slack and prepare periodic repor ts for managers to m onitor progress.
Ac
tivity
slack
is the maximum length of time that an activity can be delayed without
delaying the entire project. Activities on the critical path have zero slack. Con-
stantly m onitoring the progress of activities with little or no slack enables m an-
agers to identify activities that need to be expedited to keep the project on
schedule. Activity slack is calculated from four times for each activity: earliest
start time, earliest finish time, latest start time, and latest finish time.
Earliest Start
and
Earliest Finish
Times.
The earliest start and earliest finish
times are obtained as follows.
The earliest finish time EF) of an activity eq uals its earliest sta rt time plus
its
expected
duration,
t
or
EF
ES
t.
Th e earliest sta rt
time ES)
for an activity is the earliest finish time
o
the
immediately preceding activity. For activities w ith m ore th an one preced ing
activity, ES is the latest of the earliest finish times of the preceding
activities.
To calculate the duration of the entire project, we determine the EF for the last
activity on the critical path.
xample 2:
alculating Eeadliest Start and Earliest Finish Times
Calculate the earliest sta rt and finish times fo r the activities in the hos pital pro j-
ect. Figure 5 con tain s the activity times.
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olution
We begin at the start
node
at time zero. Activities
A
a n d
B
have no
predecessors, s o the earliest sta rt times
for
these activities are also zero. T he earli-
est finish times for these activities are
EFA 0 1 2 1 2 a n d
EFB 0 9
9
Because the earliest sta rt time fo r activities
I
F a n d C is the earliest finish tim e of
activity
A,
ES, 12,
ESF
12, and
ES,
12
Similarly,
ESD
9
and
ESE 9
fter placing these ES values on the netw ork diagram see Fig.
we determine the EF times for activities I, F, C, D, and E:
E F D = 9 + 1 0 = 1 9 , and E F E = 9 + 2 4 = 3 3
i
F I G U R Network f o ~he Hos pitul Project Sh ow in g Earliest Start
and Earliest Finish Times
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Th e earliest sta rt time for activity G is the latest EF time of all immediately pre-
ceding activities, so
ESG EFc
ESH EFD
Activity
J
has several predecessors, so the earliest time activity
J
can begin
is
the latest of the EF tim es o f any of its prece din g activi ties: EF,, EF,, EF,. Thus
EF,
5 9 4 63. Similarly, ESK 6 and
EFK
6 3 6 69 . Because activi-
ty K is the last activity on the critical path , the earliest the project can be complet-
ed is week 69. The earliest start and finish times for all activities are shown in
Fig. 6.
Latest Start and Latest Finish Times.
To obta in th e latest star t an d latest finish
times, we must work backw ard from the finish node. We start by setting the lat-
est finish time of the project equal to t he earliest finish time of the la st activity on
the critical path.
Th e latest finish
t ime
LF) for an activity is the latest star t time of th e activ-
ity immed iately following it. For activities with m ore than on e activity fol-
lowing , LF is the earliest of th e latest sta rt tim es of those activities.
Th e latest start time
LS)
for an activity equals its latest finish time minus
its expected duration , t or LS
LF
t
Example
: Calculating Latest Start and Latest
Finish
Times
For the hospital project, calculate the latest sta rt an d latest finish times for each
activity fr om Fig. 6.
time fo r activity K is
LSK L K
69
6
63
If
activity
K
is to start no later than week 63, all its predecessors must finish no
later than that time. Consequently,
LF, 63, LFF 63 , an d LFJ 6 3
olution
We begin by setting the latest finish activity tim e of activity K at week
69,
its earliest finish time as determined in Example
2. Thus the latest s tart
Th e latest start times
for
these activities See Fig.
7 are
LSI 63 15 48,
LSp 63
10
53,
and
LSJ 63 4 5 9
After obtaining
U,,
we can calculate the latest start times for the immediate
predecessors of activity
J:
LS, 59 35 24,
LSH
59
40
19,
and
LS,
5 9 2 4
5
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F I G U R
Network
for
the Hospital
Project howing
Data Needed
for Act ivit y Slack Calculation
arliest
st rt
time
arllest f~n~sh~m e
Similarly
we
can now calculate latest st rt times f or activities C and D:
L S c = 2 4 - 1 0 = 1 4 and
L S D = 1 9 - 1 0 = 9
Activity A has more than one immediately following activity:
I
F
and
C.
The
earliest of the latest start times is 14 for activity C s o
Similarly activity B
has two
immediate followers
D
and
E. The
earliest
of
the
latest start times of these activities is 9 so
L S B = 9 - 9 ~ 0
This result implies th at activity B must be started im mediately if the p roject is to
be
completed by week
69.
Th e latest start and latest finish time s
for
all activities
are shown
in
Fig. 7.
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divitySlack
Information on slack is useful to project managers because i t
helps them make decisions regarding reallocation of resources. Activities with
zero sla ck a re on the critical path. Resources could be taken from activities with
slack a nd given to other activities tha t are behind schedule until the slack is used
up. Activity slack can be calculated in one of two ways for any activity:
S = L S - E S
or S = L F - E F
Example
:
Calculating
ctivity
Slack
Calculate the slack for the activities in the hospital project. Use the data in Fig.
7.
olution
We can use either starting times or finishing times. T he following table
show s the slack fo r each activity
LS ES.
Node Duration
S
LS Slack
Activities B
D
H
J
and are on the critical path because they have zero slack.
The slack at an activity depends on the performance of activities leading to
it. If the time fo r activity had to be
4
weeks instead of 2weeks the slack for
activities and G would be zero. Thus slack is shared among all activities on a
particular path.
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PROBABILISTIC
TIME ESTIMATES
ow
can uncertainty in time estimates be incorporated into
project planning
To this poin t, we have assumed that the time estimates for the project w ere cer-
tain. Many times, however, managers must deal with uncertainty caused by labor
shortages, weather, supply delays, or accidents.
To
incorporate uncertainty into
the netw ork model, probabilistic time estimate s can be used.
With the probabilistic approach, activity times are stated in terms
of
three
reaqona ble tim e estimates.
1. The optimistic time a) is the shortest time in which the activity can be
completed, if all goes exceptionally well.
2.
Th e most likely time m)s the probable time required t o perform the ac-
tivity.
3. Th e pessimistic time
b)
is the longest estimated time required t o perfo rm
an activity.
Calculating
Time
Statistics
With three time estimates-the optimistic time, the most likely time, and th e pes-
simistic time-the manager has enough info rm ation to estima te the probability
that the activity will be completed in the scheduled amount of time. To do so, the
manager must first calculate the mean and variance of a probability distribution
fo r each activity. In PERTICPM, each activity time is treated as thoug h it were a
random variable derived from a beta probability distribution. This distribution
can have various shapes, allowing the most likely time estimate
m)
o fall any-
where between the pessimistic b )
and optimistic
a)
ime estimates. The most
likely time estimate is the mode of the beta distribution, or the time with the
highest probability of occurrence. This conditio n isn t possible w ith the no rma l
distribution, which is symmetrical, as it requires the m ode t o be equidistant fr om
the end points of the distribution. Figure
8 shows the difference between the
tw o distributions.
Two other key assumptions are required. First, we assume that a m and b
can be estimated accurately. The estimates might best be considered values that
define reasonable time range for the activity duration, negotiated between the
manager and the employees responsible for the activities. Second,
we
assume
that
the standa rd deviation,
a,
of the activity time is one-sixth the rang e b a. Hence
the chance that actual activity times will fall below a or above
b
is slim. The as-
sumption makes sense becaase,
if
the activity time followed the n orm al distribu-
tion, six standard deviations would span approximately 99.74 percent of the
normai distribution.
Even with these assumptions, derivation of the mean and variance
of
each
activity s probability distribution is complex. Thes e derivations sh ow tha t the
mean of the beta distribution can be estimated by using the following weighted
average of the three time estimates:
t ,
a 4 m b
6
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F I G UR
Differences Between
Beta and N ormal
Distr ibutions for
Project nalysis
a rn b
Mean
Time
a) Beta distribu tion: The most likely time
m)
has the highest
probability and can be placed anywhere be tween the
optimistic
a)
and pe ssimistic b) imes.
m
Mean
Time
b) Normal dlstrib utlon : The mean and most likely times must
e
the same. fa and b are chosen to
be
u part, there is a
99 74
percent chance that the actual activity time w ill fall between them.
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Note that the most likely time has four times the weight of the pessimistic and
optimistic estimates.
The variance of the beta distribution for each activity is
a2
ibiaj
The variance, which is the standard deviation squared, increases as the dif-
ference between and a increases. This result implies that the less certain a
per-
son is in estimating the actual time for an activity, t he greater will be t h e
variance.
Example
: Calculating Means and Variances
Suppo se that Judy Krame r has arrived at the following time estimates for activity
(site selection and survey ) of the hospital project:
a 7 weeks, 8 weeks, and b
5
weeks
a. Calculate the expected time for activity and the variance.
b. Calculate the expected time and variance for the other activities
olution
a . The expected time for activity is
t
7 4(8)
15
5
- 9 weeks
6 6
Note that the expected time (9 weeks) doesn t equal th e mo st likely
time 8 weeks) for this activity. These times will be the same only when
the most likely time is equidistant from the optimistic and pessimistic
times. We calculate the variance for activity B as
b. The following table shows expected activity times and variances for the
activities listed in Kram er s project description. No te th at t he grea test un-
certainty lies with the time estimate for activity I, followed by the esti-
mates for activities E and G. The expected time for each activity will
prove useful in determ ining the critical path.
Time Estimates wk) Activity Statistics
Optimistic MostLikely Pessimistic Expected Variance
Activity
a) m) 4
Time
t,) a 2 )
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A n a l y z i n g P r o b ab i li t ie s
Because the time estimates for activities involve uncertainty, pro ject m anag ers are
interested in determining the probability of meeting the project completion dead-
line. To dev elop the probability distribution for the project co mpletion
time, we
assum e that the duration time of one activity doesn t depend on t ha t
of
any other
activity. This assumption enables us to estimate the mean and variance of the
probability distribution of the time duration of the entire project by sum ming the
duration times and variances of the activities along the critical path. However, if
one work crew is assigned two activities that can be done at the same time, the
activity times will be interdependent. In addition,
i
other paths in the network
have sm all am ou nts of slack, we should calculate the joint probability distribu-
tion fo r tho se p ath s as well. We discuss this point later.
Because of the assumption that the activity duration times are independent
random variables,
we
can make use of the central limit theorem, which states
tha t the sum
of
a grou p of independent, identically distributed ra nd om variables
approaches
a
normal distribution as the number of random variables increases.
T h e mean of the normal distribution is the sum of the expected activity times on
the path. In the case of the critical path, it is the earliest expected finish time for
the project:
T
C(Activity times on the critical pat h) Me an of norm al distribution
Similarly, because of the assumption of activity time independence, we use
the sum of the variances of the activities along the path as the variance of the
time distribution for that path. T hat is,
a
I ( ~ a r i a n c e sf activities on the critical path)
To an alyze probabilities of completing a project by a certain dat e using the
nor mal distribution, we use the z-transformation formula:
where
T
due date fo r the project
T earliest expected completion dat e for the project
The procedure for assessing the probability of completing any activity in a
pro ject by a specific date is similar to the on e just discussed. H oweve r, in stead o f
the critical path, we would use the longest time path of activities from the start
node
to the activity node in question.
Example :
alculating the probability
of
completing project by
Given Date
Calculate the probability that the hospital will become oper ation al in 7 2 weeks,
using (a ) he cri tical path an d ( b) path
A-C-G-J-K.
I olution
a. Th e critical path
B-D-H-J-K
has
a
length
of
69
weeks.
From
rhe
table in
Example
5 we obtain the variance of pa th B-D-H-J-K:
o
1 78 -f-
1.78 2 .78 5.44 t
0 11 11.89.
Next, we calculate th e z-value:
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Using the norm al distribution table in Appendix 2 , we find that the prob-
ability
is
abo ut 0.8 1 that the length of path B-D-H-J-K will be no greater
than 72 weeks. Because this pat11 is the critical path, rhrre is a 19 percent
probability that the project will take longer than 7 2 weeks. This probabil-
i t y
is sho wn graphically
in
Fig.
9.
Probability
of Com pleting the Hospital Project on
Schedtlle
Normal distribution
=
3 45
weeks
weeks is 0 1922
69 7
Project duration weeks)
b. Fro m the table in Example
5,
we determin e th at the sun1 of the activity
times on path A-C-G-J-K is 6 7 weeks and that a 0.1 1 2.78
7 1 5.44 0.1 1 15.5 5. Th e z-value is
Th e probability is abou t 0.90 rhat the leng th of pa th A-C-G-J-K will be
no greater than
72
weeks. However, this analy sis implies chat there is a
10
percent chance rhat t h ~ s ath will cause a delay in the project.
I t
also
demonstrates the importance of monitoring paths that have durations
close to th at of the critical path.
As Example 6 demonstrated, one or more network paths for a project
may be shorter th an th e critical path but have enough variance in activity time es-
timates to become the critical path sometime during the project. In the hospital
project, path A-C-G-J-K will become the critical path if its length equals or ex-
ceeds 69 weeks
or if
th e leng th of pa th B-D-H-J-K eq ua ls
7
weeks or less. Fig-
ure 1 0 sho ws the considerable overlap between the probability distributions
fo r these tw o path s. Com puti ng the probability th at pa th A-C-G-J-K will
become the critical pa th requires the estimation of the joint probability that path
A-C-G-J-K 69 weeks and rh at pa th B-D-H-J-K 6 7 weeks, a s indicated by
the shaded areas. The two paths are dependent on each other share comm on ac-
tivities), so the calculation of the joint proba bility requires co mp ute r simulation.
Nonetheless, close actention to activities A,
C,
a n d G , in add ition to activities B
D,
H,
J and
K
seems warranted. If a project has multiple critical paths, the criti-
cal path with the largest variance should be used in the denominator of the
z
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F I G U R E 10
Normaldistribution Normaldistribution
Probability
or path ED H J K:
Mean = 69 weeks;
Distributions for the
= 3.94 weeks
= 3.45 weeks
ritical Path and Next
Longest Path for the
B D H J K is
Hospital Project
i: .
;.i
::
,
7
69
roject duration weeks)
transfo rmatio n form ula. This approach allows the probability estimate to reflect
the correct amount of uncertainty in the project duration.
COST CONSIDERATIONS
How do project planning methods increase the potential to
control costs and provide better customer service
Keeping costs at acceptable levels almost always is as important as meeting
schedule dates. In this section we discuss the use of
PERT/CPM
methods to ob-
tain minimum-cost schedules.
Th e reality of project managem ent is tha t there are always time-cost trade-
offs. For example, a project often can be completed earlier than scheduled by hir-
ing more workers or running extra shifts. Such actions could be advantageous i
savings or additional revenues accrue from completing the project early. Total
project costs are the sum of direct costs, indirect costs, and penalty costs. These
costs are dependent either on activity times or on the project completion time.
Direct costs include labor, materials, and any other costs directly related to p roj-
ect activities. Managers can shorten individual activity times by using additional
direct resources such a s overtime, personnel, or equipment. Indirect costs include
administration, depreciation, financial, and other variable overhead costs that
can be avoided by reducing total project time. The shorter the duration of the
project, the lower the indirect costs will be. Finally, a project ~ ncurs enalty costs
i it extends beyond some specific date, whereas a bonus may be provided for
early completion. Th us a project man ager may consider
crashing,
or expediting,
som e activities to reduce overall project completion time a nd total project costs.
To assess wh ethe r crashin g som e activities would be beneficial - rom either
a cost or a schedule perspective-the manage r needs to kn ow th e follow ing times
and costs.
I .
T he n o r n ~ a l
ime
NT) s the time to complete the activity und er no rmal
condi t~ons .Norm al time equals the expected time t calculated earlier.
2. The normal cost
NC)
is the activity cost associated with the normal
time.
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3 Th e crash time (CT) s the shortest possible time to comp lete tke activity.
4.
The crash cost (CC) s the activity cost associated w ith t he crash time.
Our cost analysis is based on the assumption that direct costs increase linear-
l y as activity time is reduced from its normal time. This assumption implies that
fo r every week the activity time is reduced, direct costs increase
by
a proportional
am oun t. For examp le, suppose that the normal time for activity
C
in the hospital
project is 1 0 weeks and is associated with a direct cost
of
4000.
If
by crashing
activity C we can reduce its time to only 5 weeks at a crash cost of 70 00 , the
net time reduction is 5 weeks at a net cost increase of 30 00 . We assume tha t
crashing activity C costs 300015 60 0 per week-an assu mp tion of linear
marginal costs that is illustrated in Fig. 1 I . Thus, if activity C were expedited
by two weeks (i.e., its time reduced from
10
weeks to 8 weeks). the estimated di-
rect costs would be 400 0 2( 6 00 ) = 520 0. For any activity, the cost to c rash
an activity by one week
is
F I G U R E
Cost-Time
Relationships in Cost
Analysis
Cost to crash per week
CC
NC
NT
CT
Crash cost CC)
000
U
Linear cost assumption
Estimated costs for
I
I
4000
I
Normal cost
NC)
3
I
5 6 7 8 9 1 0 1 1
I I
Crash time) Normal time)
Time weeks)
Table 2 contains direct cost and time data and the
costs
of crashing
per
week
for the activities in the hospita l project.
The objective of cost analysis is to determine the project completion time
tha t minimizes total project costs. Suppose that project indirect costs are 80 00
per week. Suppose also that, after week
65, St. AdolfS incurs a penalty cost of
20,000 per week
if
the hospital isn't fu l ly operational. With a
critical
path
com
pletion time of 69 weeks, the hospital faces potentially large penalty costs. For
every week th at the project is shortened-to week 65-the hosp ital saves on e
week of penalty
and
indirect costs, or 28,000. For reductions beyond week
65.
the savings are only the weekly indirect costs of 8000.
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TABLE
2
Direct
ost
and ime Data
for the Hospital
Project
Maximum
Normal Normal Crash Crash Time Cost of
Time Cost Time Cost
Reduction Crashing per
Activity
NT)
NC) CT) cc) wk) Week
A 12
B 9
C 1
D 1
E 24
F 1
G
35
40
5
J 4
K
Totals
In determining the minimum-cost schedule, we start with the normal time
schedule and crash activities along the critical path, because the length of the crit-
ical path equals the length of the project. We want to determine how much we
can add in crash costs without exceeding the savings in indirect and penalty
costs. T he procedure involves the following steps.
Step
1
Determine the project's critical path(s).
Step 2
Find the cheapest activity or activities on the critical pat h(s) to
crash.
Step
3
Reduce the time for this activity until ( a ) it cannot be further re-
duced, (b) anoth er path becomes critical, o r ( c) the increase in d irect costs ex-
ceeds the savings that result from shortening the project. If more than one
path is critical, the time for an activity on eac h path may have t o be reduced
simultaneously.
Step 4. Repeat this procedure until the increase in direct costs is less than
the savings generated by shortening the project.
Example 7: Finding
a
Minimum ost Schedule
Determine the minimum-cost schedule for the hospital project. Use the informa-
tion in Table 2 and Fig. 7.
Solution Th e projected completion time of the project is 69 weeks. T he project
costs for tha t schedule are 1,992,000 in direct costs, 69 ( 8 ,00 0) 552 ,000 in
indirect costs, and (6 9 65)( 20,0 00) 80,00 0 in pena lty costs, for total proj-
ect costs of 2,624,000. Th e five path s in the netw ork h ave the following norma l
times.
A I K 33 weeks B-D-H-J-K: 6 9 weeks
A-F-K: 28 weeks RE-J-K
43
weeks
A-C-G-J-K: 6 7 weeks
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I f all activities on
A-C-G-J-K
were crashed, the path duration would be 47
weeks. C ras hin g all activities on ELD-H-J-K results in a du ra tio n of
56
weeks.
The normal times of
A-I-K, A-F-K,
and B-E-J-K are less than the minimum
times of the other two paths, so we can disregard those three paths; they will
never become critical regardless of the crashing we may do,
Stage 1
Step
1
The critical path is
B-D-H-J-K.
Step 2. T he cheapest activity to crash per week is J a t 1000, which is much
less than the savings in indire ct an d penalty c osts of
28,000
per week.
Step 3
Crash activity
J
to its limit of 3 weeks because the critical path re-
mains unchanged. The new expected path times are
A-C-G-J-K:
64
weeks
B-D-H-J-K: 66
weeks
The net savings are 3( 28,000) 3( 1000) 81,000. The total project costs
are now
2,624,000 81,000 2,543,000.
Stage 2
Step 1. The critical path is still B-D-H-J-K.
Step 2. T he chea pest activity to crash pe r week no w is D a t 2000.
Step
3
Crash
D
by tw o weeks. T he first week of reduction in activity D saves
28,000
because
it
eliminates a week of penalty costs, as well as indirect
costs. Crashing
D
by a second week saves only
8000
in indirect costs
be
cause, after week
65,
there are no more penalty costs. These savings still ex-
ceed the cost of crashing D by two weeks. The updated path times are
A-C-G-J-K:
64
weeks
B-D-H-J-K: 64
weeks
The net savings are
28,000 8000 2( 200 0) 32,000.
The total project
costs are now
2,543,000 32,000 2,511,000.
Step 2.
O u r alternatives are t o crash o ne of the following combinations of
activities-(A, B),
A , H), C,
B),
(C, H),
(G,
B),
(G,
H)--o r to crash activity
K, which is on both critical paths
(J
has already been crashed).
We
consider
only those alternatives for which the cost of crashing is less than the poten-
tial savings of
8000
per week. Th e only viable alternatives are
(C, 8
t a
cost of
7600
per week and
K
a t
4000
per week. We choose activity
K
to
crash.
Step 3 We crash activity
K
to the greate st exten t possible-a reduc tion of
one week ecause it is on both critical paths. The updated path times are
Stage 3
Step
1
After crashing D, we now have two critical paths.
Both
critical
paths must now be shortened to realize an y savings in indirect project costs.
If
one is shortened and the other isn't, the length of the project remains
unchanged.
A-C-G-J-K:
63
weeks
B-D-H-J-K:
63
weeks
The net savings are
8000 4000 4000.
The total project costs are
2,511,000 4000 2,507,000.
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Stage 4
Step 1 T he cr~ ti cal are B-D-H-J-K an d A-C-G-J-K.
Step
2
The only viable alternative at this stage is to crash activities B and C
simultaneously at a cost of 76 00 per week. Th is am ou nt is still less than the
saving s of 80 00 per week.
Step
3 .
Crash activities
B
and
C
by tw o weeks, th e limit f or activity
B
The
updated path times are
A-C-G-J-K: 61 weeks R-D-H-J-K: 61 weeks
Th e net savings are 2( 80 00 ) 2( 7600) 800. The total project costs are
2,507,000 800 2,506,200.
Any other combination of activities will result in a net increase in total proj-
ect costs b ecause the crash c osts exceed weekly indirect costs. T h e minimum-cost
schedule is
61
weeks, with
a
total cost of 2,506 ,200. To obtain this schedule, we
crashed activities
8 D, J,
and K to their limits and activity
C
to 8 weeks. The
other activities remain a t their normal times. This schedule costs 117 ,80 0 less
than the normal-time schedule.
RESOURCE LIMITATIONS
T he project m anag em ent app roach es discussed so far consider only activity times
in determining overall project duration and the critical path. An underlying as-
sumption in the use of PERTICPM is that sufficient resources will be available
whe n needed to com plete all project activities on schedule. Howeve r, developing
schedules without considering the load placed on resources can result in ineffi-
cient resource use and even cause project delays
if capacity limitations are
exceeded.
What is the effect of Limited resources on project duration?
For purpo ses of discussion, consider the project dia gram in Fig.
12. Each
of the five activities involves a certain amount of time a n d has a resource require-
ment. The critical path is A-B-E, and the total time to complete the project, ig-
noring resource
Project Diagram
Showing Resource
Requirements Activity
Times and Critical
Path
limitations, is nine days.
Start
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Although AON o r AOA network d iagrams are i~sefulor displaying an ent ire
pro ject an d show ing the precedence relationships between activities, they are n t
especially useful f or show ing the implications of resource requirements for a
schedule
of activities. Ga nt t cha rts are mo re helpful in this regard.
W e want
to generate a schedule that recognizes resource constraints, as well
as the precedence relationships between activities. Let s suppo se tha t we are lim-
i ted
to
a small number of workers pe r day.
A
very
useful approach
is
the following procedure, developed by Weist
1966).
1
Start with th e first day of th e project and sched ule as ma ny activities as
possible, considering precedence relarionships and resource limitations.
Continue with the second day, and so on, until all activities have been
scheduled.
2.
When several activities compete for the same resources, give preference
to the activities with the least slack, as determined with standard
PERTICPM met hods.
3
Reschedule noncritical activities, i possible, to free resources for critical
or nonslack activities.
T h e intent o f this proc edure is to minimize total project time, subject to reso urce
constraints.
Gantt chart software may
be used
to
schedule each
step a market survey
project.
his
project has
three phases: plan prepart
implement. Specific
mi tie s are shown u nder
each phase. Some activit~es
can be executed
simtlltaneously whereas
others must be sequenced.
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Example Developing a Resource
Constrained Schedule
Generate a resource-constrained schedule for the project depicted in Fig.
12.
Assume th at only six worker s per day are available.
olution
Step
1
Schedule activity
A
first because all other activities depend on its
completion.
Step
2.
Th e choice is among activities B C, and
D
because their predeces-
sor has been scheduled. Activities C and D have slack, but activity
B
doesn t
because it s on the critical path. The refore schedule B next. So far, we have
committed five workers on day 1 and two workers on days 2-6.
\
Step
3.
We have a choice between activities C and D, but we mu st choose
next. It requires only four work ers per day, and we can schedule it on days
2
and
3
with ou t violating th e resource co nstrain t of six workers per day. Activ-
ity
D
requires six worker s per day, bu t we have already scheduled activity
B,
which needs two workers.
Step 4. Th e remaining activities to sched ule are D and E. We must schedule
D first because of precedence constraints. The resulting schedule is shown in
Fig.
13
F I G U R Resource Constrained Schedule
on resource requirements and time estimates is
A B D E.
Howeveq the usk of
the procedure will not always be s o succes shl. We can only say tha t it
will
gener-
ally produce solutions ciose to the o ptimum.
This schedule results in th e shor test project time possible under th e resource
I
constraints. Activity C can be delayed three days without delaying the compie-
tion of the project o r exceeding the resource constraints.
The
critical path based
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BENEFITS AND LfMlTATIONS OF PERTICPM SYSTEMS
PERTICPM systems offer a number of benefits to project managers. However,
they also have limitations.
Benef i t s
We have already discussed the benefits of network planning models for large,
com ple x projects. In summary, they include the following.
1. Considering projects as networks forces managers to organize the re-
quired data and identify the interrelationships between activities. This
process also identifies the data to be gathered a nd prov ides a fo rum for
managers of different functions to discuss the na tur e of the variou s activ-
ities and their requirements.
2.
PERTICPM
computer packages provide graphic displays of the project
diagram and progress reports.
3
Networks enable managers to estimate the completion time of the pro-
ject, which can be useful in plann ing ot he r events or in contrac tual neg o-
tiations with customers.
4 . Reports highlight the activities that are crucial t o comp leting th e project
on schedule. These reports can be updated periodically over the life of
the project.
5 Reports also highlight the activities that have slack, thereby indicating
resources that may be reallocated to more urgent activities.
6
Net work s enable managers to analyze cost-time trade-offs.
L i m i t a t i o n s
Let s no w tu rn t o the limitations of PERTICPM.
etwork
Diagrams Th e method s used in PERTICPM are based on the assump-
tion tha t project activities have clear beginning an d en din g points, tha t they are
independent of each other, and that the activity sequence relationships can
be
specified in a network diagram. In reality, t wo activities ma y overlap, or th e out-
co me of one activity may determine the time and resources required for another
activity. Also a network diagram developed at the start of a project may later
limit the project manager s flexibility to han dle cha ng ing situations. At tim es, ac-
tual precedence relationships can no t be specified b eforehand because of so me de-
pendencies between activities.
Control
second underlying assumption in PERTJCPM methods is that man-
agers should focus only on the activities along the critical path. However, man-
agers also must pay attention to near-critical paths, which couln become critical
if
the schedules of one o r more of th e activities slip. Project managers who over-
look
near-critical
p ths may
find their project s
completion date
slipping
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Time Estimates
third assumption-that uncertain activity time s follow the
beta distribution-has bro ug ht a variety of criticism. First, the fo rm ul as used to
calculate the m ean and v ariance of the beta dist ribution are only approx ima tion s
and are subject to errors of up to
10 percent fo r the m ean a nd
5
percent for the
variance. The se errors could give incorrect crit ical paths. Se cond, arriv ing a t ac-
curate time estimates for activities that have never been performed before is ex-
tremely diff icul t . Many project managers pefer to use a s ingle t ime est imate,
arguing that pessimistic t ime estimates often are inflated and vary far more from
the most l ikely time estimate than do the optimistic t ime estimates. In flated pes-
simistic t ime estimates build
a
cushion of slack into the schedule. Finally, the
choice
of the beta
distribution
is somewhat arbitrary and the
use
of another dis
t r ibut ion wo uld result in a different expected time and variance f or eac h activity.
Resource Limitations
four th assun~ pt ion f PERT/CPM is that sufficient re-
source s will be available wh en needed t o complete all project activities o n sched-
ule. However, managers should consider the load placed on resources
to
ensure
efficient resource use and avoid project delays caused
by
exceeding capacity. Net-
wo rk d iagrams don t s how th e implicat ions of resource l imitat ions for a sched ule
of
activities.
Although PERTICPM has shortcomings, i ts ski l l ful appl icat ion to project
management can significantly aid project managers in their work.
COMPUTERIZED PROJECT SCHEDULING
AND CONTROL
Computerized network planning methods are used extensively for projects in
government, construction, aerospace, entertainment, pharmaceuticals, util i t ies,
man ufacturing , and archi tectural engineering.
Managerial Pract ice
1
discusses the project scheduling software used in
a
large construction company.
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Managerial Practice
-
T
e M. W. Kellogg Com pa ny is one of the world's
leading engineering contr acto rs specializing in the
engineering design and construction of petroleum
and petrochemical facilities. Maintaining promised de-
livery dates of such large and complex facilities is a diffi-
cult task. Th e typical project involves 1 50 0 engineering
activities, 1 10 0 materials requisit ions and purchase o r-
ders, 4000 cost accounts, 150 project change notices,
and 400,000 work hours. project may cost anywhere
from $10 mill ion to 300 million, an d delays can cost
the custo mer m illions in lost revenues. T he Kellogg
Company may have up to
20
of these projects ongoing
at any point i n
time.
A sophisticated comp uter pac kage with CPM at the
core, called Artemis@,wa s purchased t o assist managers
with complex scheduling problems. When the comp any
gets a new job, the followin g tasks ar e perform ed.
master
schedule
is developed with CPM and ap-
proved by managem ent. This schedule contains the
completion times of the various comp onents of the
project and becomes a com mitment to the customer.
Derailed engineering an d pro urement schedules
are established, reviewed by ma nagem ent, a nd fi-
nalized. Approved budgets for each department
are broken down into cost accounts and integrated
with existing schedules and worklo ads.
Performance is tracked every two weeks by mea-
suring progress and actual hou rs used against the
baselines provided by
CPM.
Schedule updates are
distributed internally a nd to customers at least
once a mo nth.
Th e approach taken by the Kellogg Com pany provides
an early warning system fo r detection of slippage in the
schedule.
T h e M . W. Kellogg Com pany had to purchase a sophisticated softwa re pack-
age because of the com plexity of its scheduling problem s. Ho wever, with the a d-
vent of personal com puters, off-the-shelf project ma nage me nt softwa re ha s
become accessible to many companies. Large as well as small projects are rou-
tinely managed with the assistance of standard computerized scheduling pack-
ages. Software costs have come down, and the user interfaces are friendly.
Standard software programs may differ in terms o their output reports and may
include o ne or more of the following capabilities:
Gantt charts and PERTICPM diagrams
The graphics capabilities
of
software ~a c k a g e s llow for visual displays of project progress on G antt
charts and PERTICPM network diagrams. Most packages allow the user
to display portions of the network on the video mo nitor to analyze specif-
ic problems.
Project status
and
sum mar y reports
These reports include budget vari-
ance reports that co mpa re planned to actual expenses at any stage in the
project, resource histog ram s that graphically display the usage of a pa rtic-
ular resource over time, status reports for each worker by task performed,
and summ ary reports that indicate project progress to to p managem ent.
Tracking reports
These reports identify areas of concern such as the
percentage o f activity completion with respect to-time , budget, or lab or re-
sources. Most software packages allow multiple projects to be tracked at
the same time. This feature
is imp o r t an t
when resour es m u s t be shared
jointly by several projects.
Almost any project requiring significant resources will be aided by the use of
project management software. However, despite today's user-friendly packages,
extensive
employee
training might
be needed
for an
organization
to
benefit
ful ly
from these systems.
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C H A P T E R R E V I E W
Solved roblenl
An advertising project manager has developed the n etwork diagrams show n in Fig.
4
for
a new advertising campaign. In addition the manager has gathered the time informa-
tion
for
each activity as sho wn
in
the accompan ying table.
F i R
E
I
4
Network Diagrams for an dvertising Program
a) AON network
b)
AOA
network
a.
Calculate the expected time and variance for each activity.
b. Calculate the activity slacks and determine the critical path using the expected activity
times.
. c. W ha t is the probability of completing the project within 23 weeks?
Time Estimates wk)
Most Immediate
Activity Optimistic Likely Pessimistic Predecessor s)
Solution
a. Th e expected time
for
each activity is calculated
as
follows:
Activity Expected
Time wk)
Variance
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b We need to calculate the earliest start, latest star t, earliest finish, an d latest finish times
for eac h activity. Starting with activities A an d B we proceed from the beginning of the
network and move to the end, calculating the earliest star t an d finish times show n
graphically in Fig.
15
for the
AON
diagram):
F I G U R E
5
O Diagram with
Earliest Start
and
Earliest Finish Times
Activity Earliest Start
wk)
Earliest inish wk)
Based on expected times, th e earliest finish fo r the project is week 20, whe n activity G
has been completed. Using that as a target date, we can work backward through the
network, calculat ing the latest st art an d finish t imes shown graphically in Fig. 16):
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F I G U R E 16
A O N Dfagram with ll
tme
Estrrnates Needed
to
Calculate Slack
Start
Activity Latest Start wk) Latest Finish wk)
We now ca lculate the activity slacks and determine w hich activities are on th e critical
path
Start
Finish
Activity Critical
Activity Earliest Latest Earliest Latest Slack Path
A 0 0 4 0
4 0
8 0
4 0 No
0 0 0 0
5 5
5 5
0 0 Yes
C 5 5 5 5 9 0
9 0
0 0 Yes
D 4 0 8 0 16 0
20 0
4 0
No
E
9 0
9 0
15 5 15 5
0 0
Yes
F
5 5 6 5
14 5 15 5 O
No
G 15 5 15 5
20 0 20 0 0 5 Yes
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The paths, and their
total
expected
times
a n d varrances, are
Total
Expected Total
Path Time wk) Variance
A-D
4
2 16
1.00 t 1.78 2 78
A E G
4 6.5 4.5
15
1 OO
2.25 0.69 3.94
RC-E-G
5 . 5 3 . 5 6 . 5 4 . 5 = 2 0
0.69 0 .25 2 .25 0 .69=3.88
B-F-G
5.5
t
9 4.5
19
0.69 2 78 0.69 4.16
The critical path is
B-C-E-G,
with a total expected time
of
20 weeks. However, path
B F G
is 19 weeks and has a large variance. In [his solution we used the AON nota-
tion, show ing the start a nd finish rimes
wi th~n
he node circles. The same results can be
obtained with the AOA notation, except that the times typically are shown in
a
box
draw n near the arc (a rrow ). For example:
c. We first calc ulate the z-value:
Using Appendix 2, w e find th at the probabili ty
of
completing the project in
23
weeks
o r less is 0.9357. Because the length of path
ELF-G
is very close to that of the critical
path and has a large variance, i t might well become the crit ical path during t he project.
Solved
Problem 2
Your compa ny has just received a n o rder from a good custom er for a specially designed
electric motor. The contrac t s tates that, s tarring on th e thirteenth day from now, yo ur firm
will experience a penalty of 100 pe r day until the job is corripleted. Indirect pro ject c osts
am ou nt to 200 per day. Th e data on direct costs and activity precedence relationships are
given in Table 3
a. D ra w the project netwo rk diagram.
b. Wha t complet ion date would you recommend?
Solution
a . T h e AON network diagram, including normal activity t imes, for this procedure is
shown in Fig.
17.
Keep the following points in mind while constructing a n etw ork
diagram.
Always have srarr and finish nodes.
Try to avoid crossing paths t o keep
the
diagram simple.
Use only one ar ro w to directly connect any tw o nodes.
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TABLE Electric
Motor
Project Data
Normal Normal Crash Crash
Time Cost Time Cost Imm ediate
Activity days) ( ) days) ( ) Predeces:nr s)
None
None
None
A
B
C
D,
E
F,
F I G U R E
7
O N Diagram for the
Electric Motor Project
Put the activities with n o predecessors a t the left and point the a rr ow s from
left to right.
Use scratch paper and be prepared t o revise the diagra m several t imes before
you come up with a correct and uncluttered diag ram.
b. With these activity durations, the project will be completed in
19
days and incu r a
700
penalty for lateness. Determ ining a good completion da te requir es the use of the
minimum -cost schedule procedu re. Using the da ta in Table .3,you ca n dete rmin e the
maximum crash t ime reduction and crash cost per day for each activity. For example,
for activity
A
Max imum crash t ime Nor mal t ime Crash t ime
4
d a y s 3 days day
Crash cost
Crash cost Nor mal cost
CC NC 1300 1000
per day Normal t ime Crash t ime
NT
CT 4 days
3
d a y s
300
Crash Cost Maximum Time
Activity per Day ( ) Reduction days)
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Table
I contains a summ ary of the analysis and the resultant project duration an d
tot al cos t. T he critical path is C-F-H at 19 days-the longest path in the netw ork. Th e
cheapest of these activities to crash is H, which costs only an extra 100 per day to
crash . Doing so saves 200 + 100
=
300 per day in indirect and penalty costs. If
you c rash this activity two days (the maximum), the lengths o the paths are now
A-D-G-H:
15
days
B-E-G-i-1:
JSdays C-F-H:
17
days
Th e cri t ical path is still C-F-H. Th e next cheapest critical activity t o crash is F at 250
per day. You c an crash F only two days because at that point you will have three critical
paths. Fu rther redu ctions in project duration will require simu ltane ous crashing of more
than one activity
(D,
E a n d
F).
The cost to do so, 650, exceeds the savings,
300.
Con-
sequently, you shou ld stop. N ote that every activity is critical. The project costs are min-
imized whe n the completion date is day 15. However, there may be some goodwill costs
associated with disappointing
a
customer that wants delivery in -12 days.
,
TABLE 4 Project Cost nalysis
Resu lting Time Project Project Crash Total Total Total
Crash Critical Reduction Duration Direct Costs, Cost Indirect Penalty Project
Stage Activity Path s) days) days) Last Trial Added Costs Costs Costs
0
C-F-H 19
10,100
3,800 700
14,600
H C-F-H
2
17
10,100
200 3,400
500 14,200
2 F A-D-G-H 2
15
1 0,300 500
3,000 300
14,100
B-E-G-H
C F H
Problem
F I G U R
1 8
Network for the
A maintenance crew at the Woody Manufacturing C ompa ny must d o scheduled machine
maintenance in the fabricating department. A series of interrelated activities must
be
ac-
complished, requiring a different number of workers each day. Figure 8 shows the
project network , the numbe r of workers required, and th e activity t ime. Th e compa ny can
devote a m axi mu m of six maintenance w orkers per day to these activit ies.
Workers
required per day
./
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a . Use Weist s proced ure
to
find
a
new schedule, and draw a Ga ntt ch art for
i t
b.
Ho w long will the project tak e and which activities are critical?
Solution
a. Th e critical path of this project (disregarding the resource con strain t) is A-C-D-E at
weeks. Co nsequently, on ly activity
B
has slack. Figure
19
sho ws the schedule.
F
R E
9
Gantt Chart Schedule for the Maintenance
Project
Step
1
Schedule activity first on day
1
We canno t schedule any other activi-
ties until day
4
because of the resource constraint.
Step 2. Activities
B
and C are now tied. We schedule C next because
it
has no
slack.
Step
3
Activities
B
and
D
are tied. We choose
D
next because it has no slack. We
must start it on day 8 because of the resource constraint.
Step
4 We
must schedule activity
B
next because of its precedence relationship
to activity
E.
Step 5 Finally, we schedule
E
for days
12
and
13
It could n t be started earlier
because of the resource con straint.
b. The project will take
13
days, a nd every activity is critical. N o activity can be shifted
from its present schedule without violating
the
maintenance worker capacity limita-
tion.
Formula Review
1.
Star t and finish times:
ES = m x [EF
times of all activities immed iately preceding a ctivity]
EF
= ES
LS
LF
LF = min [ S imes of all activities immediately following activity]
2. Activity slack:
S = f i - E 5
o r
S = L F - E F
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3 .
Activi ty r ~ m etatist~cs:
t , m
(expected activity time )
6
c2
+I2
(variance)
4. z - t r a ~ ~ s f o r m a t i o normula:
z A/$ where T due d ate for the project
TE z(ex pecte d activity rimes on the cri tical path )
mean of normal distribution
r z(va rianc es of activities on the critical path )
5.
Project costs:
Crash cost Nor mal cost
CC
N
Crash cost per unit of time =
Nor mal t ime Crash t ime
N T CT
Highlights
Projects are un ique o pera tions having a fini te life span.
Netw ork plan ning can help in man aging a project . It in-
volves 1) describing the project as a set of interrelated
activities, 2) diagramming the network to show prece-
dence relat ionships, 3) estimating t ime of completion
by determining the cri t ical path, and
( 4 )
moni toring
project progress.
PERTCPM methods focus on the cr i t ica l path : the se-
quence of activi t ies requiring the greatest cun~ulative
am ou nt o f t ime fo r completion. Delay in cri tical activi-
ties will delay rhe entire pro ject. Uncerta inty in activity
ies can be recognized by securing three time estimates
each activity, then calcu lating expected activity times
nd variances. Activity t imes are assumed to follow a
eta distribution.
ERT ICPM meth ods can be used to assess the probabil-
of
finishing the project by a certain date or to find
l costs ar e linear.
large projects with m any activities, when fre-
t upda tes o r changes to the original project occur,
and when comparisons of actual versus planned t ime
and resource usage are needed.
The project duration may increase if sufficient resources
aren t available when needed. Weist s proc edu re is a use-
ful appr oach to deriving a project schedule subject to re-
source const ra in ts .
Criticisms of PERTJCPM meth ods focus o n th e validity
of
four assumptions in the network model. First, activi-
ties sometim es don t have clear beginning an d endin g
points. Second, near-critical paths m a y become critical
and affect project completion. Third, use of the beta dis-
tribution may not result in good est imates for the ex-
pected times and variances, and the underlying activity
t ime est imates may be inaccurate. Fourth, ignoring re-
source capacity limitations may result in inefficient re-
source use an d project delays.
Skillful use
of
PERTICPM can help managers (1) orga-
nize a project and identify activity interrelationships,
(2)
report progress, 3 ) estimate project completion t ime,
( 4 ) highlight critical activities, (5 ) identify slack activi-
t ies and beneficial reallocation of resources, and 6 )an -
alyze cost-time trade-offs.
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Ouest ions
I . Wh at cons t i tutes effect ive project manag enlent?
2-
Wh at inform at ion i s needed
to
const ruc t the ne twork
diagram for a projec t? Can any projec t be dia-
grammed a s a ne t work?
3
When a large project is mismanaged, it makes news.
Identify penalties associated with a m ismanaged proj-
ect in y ou r experience o r in recent headlines.
f
possi-
ble, identify the cause
of
the problem. For example,
were the problems caused by inaccurate t ime est i-
mates, changed scope, i rnplanned or improperly se-
quenced act ivi t ies, inadequate resources, or poor
management-la b or relat ions?
4. A certain advert ising agency is preparing a bid for a
prom otional cam paign of a type never before at tempt-
ed. The project comprises a large num ber of interrelat-
ed act ivi t ies. Explain how you would arrive at three
t ime est imates for each act ivity s o that y ou could use a
ne twork planning m odel to assess the chances tha t the
project can be completed wh en the sp on sor want s it .
5
Wh y was the beta distribution chosen ove r the normal
clistribution for PERTICPM analyses?
6 Why is the critical path of such importance in project
management? C an i t change during the course of the
project? If so, wh y?
7. When determining the probabil i ty of complet ing a
project within a certain amo un t of t ime, wha t assump-
t ions a re you making Wh at role do the lengths and
variances of paths other than the cri t ical path play in
such an analysis
8. Suppose t lwt your company has accepted a project of
a type that i t has completed many t imes before. Any
activity can be expedited with an increase in costs.
The re are weekly indirect costs, and there is a weekly
penalty
i f
project complet io n exten ds beyond a certain
date . Ident i fy the da ta tha t you would need and ex-
plain the analyt ic process that you wou ld use to deter-
mine a minimum-cost schedule . What assumpt ions
would you make in do ing such an ana lys is?
9.
Explain the usefulness of a slack-sorted list of
activities.
10
Suppose tha t you are t ryrng to convince managemen t
that metho ds such a s PERTIC PM wou ld be useft11 to
them. Sonie of the man agers h ave voiced the fol lowing
cortcerns. Prepare a brief re spon se to each of these
concerns.
@
a. There is
a
t endency for technic ians to handle the
opera t ion of PERTKPM ; t hus management wi l l
not use it often.
b. It puts pressure on managers because everyone
know s where the c r i t ica l pa th i s . Manag ers of ac-
t ivi t ies along the cri t ical path are in the spotl ight ,
a n d if their act ivi t ies are delayed, the costs of the
delays a re on thei r shou lders .
c . T he int roduct ion of network planning techniques
may requi re new commu nica t ion channels and sys-
tems procedures.
the following problems, ne twork d ~a gr am s an be
a . D r a w th e n e tw o r k d ~ a g r a m .
in t h e AOA o r AON format. Your instructor wil l
b. Calculate the cri t ical path for this project .
P
J
te which is preferred.
c. H o w much slack is in act ivi t ies G, H, a n d
I
Consider th e fol lowing da ta fo r a project .
2. Th e fol lowing inform at ion i s kno wn ab ou t a projec t .
a . Dra w t he ne t work d i agram for t h i s p ro jec t.
Activity Time Imm ediate
b. Determine the c r i ti ca l pa th a nd projec t dura t ion.
Activity days) Predecessor s)
Activity Time Immed iate
Activity
days)
Predecessor s)
A
4
B
A
C A
D
E
4
B,
F
2
E
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3 A
project has the following precedence relationships
an d activity times.
Activity Time Imm ediate
Activity wks) Predecessor s)
a . Dr aw the ne twork diagram.
b. Calcula te the slack for each activity. Wh ich activi-
ties are on the critical pa th?
Th e fol lowing information is avai lable ab ou t a project .
Activity Time Imme diate
Activity days) Predecessor s)
a. Dra w the network diagram.
b. Find the critical path .
5 The fol lowing information has been gathered for a
project.
lmmediate
Activity Activity Time wk) Predec essor s)
a . Dra w the network diagram.
b
Calculate the slack for each activity and determine
the crit ical path. H ow long will t he project take ?
6
Consider the following project in form ation.
lmmediate
Activity Activity Time wk) Predec esso r s)
a.
Dra w the network d iagram for this project .
b. Specify the critical pa th( s).
c. Calculate the total slack for activities
A
and D.
d. Wh at happens to the slack for
D
if
A
takes five
days?
7
Recently, you were assigned to manage a project for
your company. You have constructed a network dia-
gram depicting the various activities in the project
(Fig.
20).
In addit ion, you have asked various man-
agers and subordinates to est imate the amou nt of t ime
that they would expect each of the activities to take.
Their responses ar e shown in th e fol lowing table.
Time Estimates days)
Activity Optimistic MostLikely Pessimistic
A
5 11
B
4
8 11
C
D 4
E
4
1 0
F u E 2
ON
Project Diagram
a. What is the expected complet ion t ime of the
project?
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b. W ha t is the probab ility of completing the project
r ~
1 days?
c. Wh at is the probability of completing the project
in 17 days?
8 In Solved Problem 1, estim ate the probability th at the
noncritical path
&F-G
will take m ore than 2 0 weeks.
Hint S u b t r a c t f r o m 1 . 0 0 t h e ~ r o b a b i l i t ~hat
B-F-G
will ta ke 2 0 weeks or less.
9. Consider the following data for a project never before
attempted by your company.
Expected Immediate
Activity Time, te wk) Predecessor(s)
A
5
B
C A
D
B
E C, D
F
7
D
a. Dra w the network diagram for this project.
b. Identify the critical path and estimate the project's
duration.
c. Calculate the slack for each activity.
The director of continuing education a t Bluebird Uni-
versity has just approved the plann ing f or a sales-
training seminar. Her administrative assistant has
identified the various activities that must be done and
their relationships to each other, as shown in Table
5
lmmediate
ctivity Description Predecessor(s)
A Design brochure and cou rse
announcement.
B Identify prospective teac hers .
Prep are detailed outline of
course.
D Se nd brochure an d student
applications.
E
Se nd teacher applications. I
F
Sele ct teacher for course. C
E
G Accept stu
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