pert handout

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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.


2/58

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.


3/58

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


4/58

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.


5/58

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.


6/58

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.


7/58

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


8/58

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


9/58

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.


10/58

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.


11/58

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


12/58

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


13/58

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.


14/58

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.


15/58

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


16/58

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.


17/58

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 )


18/58

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:


19/58

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


20/58

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.


21/58

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.


22/58

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


23/58

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.


24/58

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


25/58

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.


26/58

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


27/58

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


28/58

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.


29/58

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.


30/58

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


31/58

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):


32/58

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


33/58

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.


34/58

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)


35/58

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

./


36/58

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


37/58

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.


38/58

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


39/58

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?


40/58

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

pert handout

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