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2008 Prentice-Hall, Inc.
Chapter 13
To accompanyQuantitative Analysis for Management, Tenth Edition,
by Render, Stair, and Hanna
Project Management
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Learning Objectives
1. Understand how to plan, monitor, and controlprojects with the use of PERT and CPM
2. Determine earliest start, earliest finish, lateststart, latest finish, and slack times for eachactivity, along with the total projectcompletion time
3. Reduce total project time at the least totalcost by crashing the network using manualor linear programming techniques
4. Understand the important role of software inproject management
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Chapter Outline
13.1 Introduction
13.2 PERT/CPM
13.3 PERT/Cost13.4 Project Crashing
13.5 Other Topics in ProjectManagement
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Introduction
Most realistic projects are large and complex
Tens of thousands of steps and millions of dollarsmay be involved
Managing large-scale, complicated projectseffectively is a difficult problem and the stakes arehigh
The first step in planning and scheduling a projectis to develop the work breakdown structure
Time, cost, resource requirements, predecessors,and people required are identified for each activity
Then a schedule for the project can be developed
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Introduction
The program evaluation and review technique(PERT) and the critical path method(CPM) are twopopular quantitative analysis techniques to helpplan, schedule, monitor, and control projects
Originally the approaches differed in how theyestimated activity times
PERT used three time estimates to develop aprobabilistic estimate of completion time
CPM was a more deterministic technique They have become so similar they are commonly
considered one technique, PERT/CPM
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1. Define the project and all of its significantactivities or tasks
2. Develop the relationships among the activitiesand decide which activities must precede others
3. Draw the network connecting all of the activities4. Assign time and/or cost estimates to each activity
5. Compute the longest time path through thenetwork; this is called the critical path
6. Use the network to help plan, schedule, monitor,and control the project
Six Steps of PERT/CPM
The critical path is important since any delay inthese activities can delay the completion of the
project
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PERT/CPM
Given the large number of tasks in a project, it iseasy to see why the following questions areimportant
1. When will the entire project be completed?
2. What are the criticalactivities or tasks in theproject, that is, the ones that will delay the entireproject if they are late?
3. Which are the non-criticalactivities, that is, the
ones that can run late without delaying the entireprojects completion?
4. If there are three time estimates, what is theprobability that the project will be completed bya specific date?
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PERT/CPM
5. At any particular date, is the project on schedule,behind schedule, or ahead of schedule?
6. On any given date, is the money spent equal to,less than, or greater than the budgeted amount?
7. Are there enough resources available to finishthe project on time?
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General Foundry Example ofPERT/CPM
General Foundry, Inc. has long been trying toavoid the expense of installing air pollutioncontrol equipment
The local environmental protection group has
recently given the foundry 16 weeks to install acomplex air filter system on its main smokestack
General Foundry was warned that it will be forcedto close unless the device is installed in the
allotted period They want to make sure that installation of the
filtering system progresses smoothly and on time
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General Foundry Example ofPERT/CPM
Activities and immediate predecessors forGeneral Foundry
ACTIVITY DESCRIPTIONIMMEDIATEPREDECESSORS
A Build internal components
B Modify roof and floor
C Construct collection stack A
D Pour concrete and install frame B
E Build high-temperature burner C
F Install control system C
G Install air pollution device D, E
H Inspect and test F, G
Table 13.1
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Drawing the PERT/CPM Network There are two common techniques for drawing
PERT networks
Activity-on-node(AON) where the nodesrepresent activities
Activity-on-arc(AOA) where the arcs are used torepresent the activities
The AON approach is easier and more commonlyfound in software packages
One node represents the start of the project, onenode for the end of the project, and nodes foreach of the activities
The arcs are used to show the predecessors foreach activity
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General Foundry Example ofPERT/CPM
Network for General Foundry
A
Build InternalComponents
H
Inspectand Test
E
Build Burner
C
ConstructCollection Stack
Start
F
Install ControlSystem
Finish
G
Install PollutionDevice
D
Pour Concreteand Install Frame
B
Modify Roofand Floor
Figure 13.1
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Activity Times The time estimates in PERT are
Optimistic time(a) = time an activity will take if everythinggoes as well as possible. There
should be only a small probability(say, 1/100) of this occurring.
Pessimistic time(b) = time an activity would take assumingvery unfavorable conditions. Thereshould also be only a smallprobability that the activity will really
take this long.
Most likely time(m) = most realistic time estimate tocomplete the activity
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Activity Times PERT often assumes time estimates follow a beta
probability distribution
Probability of 1 in 100
ofa
Occurring
Probability of 1 in 100ofb Occurring
Probability
Activity TimeMostLikelyTime
(m)
MostOptimisticTime
(a)
MostPessimisticTime
(b)Figure 13.2
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Activity Times To find the expected activity time(t), the beta
distribution weights the estimates as follows
6
4 bmat
To compute the dispersion or variance of activitycompletion time, we use the formula
2
6Variance
ab
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Activity Times Time estimates (weeks) for General Foundry
ACTIVITYOPTIMISTIC,a
MOSTPROBABLE,m
PESSIMISTIC,b
EXPECTEDTIME,t = [(a + 4m +b)/6]
VARIANCE,[(ba)/6]2
A 1 2 3 2 4/36
B 2 3 4 3 4/36
C 1 2 3 2 4/36
D 2 4 6 4 16/36
E 1 4 7 4 36/36
F 1 2 9 3 64/36
G 3 4 11 5 64/36
H 1 2 3 2 4/36
25
Table 13.2
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How to Find the Critical Path
We accept the expected completion time for eachtask as the actual time for now
The total of 25 weeks in Table 13.2 does not takeinto account the obvious fact that some of the
tasks could be taking place at the same time To find out how long the project will take we
perform the critical path analysis for the network
The critical pathis the longest path through the
network
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How to Find the Critical Path
General Foundrys network with expected activitytimes
A 2 C 2
H 2E 4
B 3 D 4 G 5
F 3
Start Finish
Figure 13.3
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How to Find the Critical Path
To find the critical path, need to determine thefollowing quantities for each activity in thenetwork
1. Earliest start time(ES): the earliest time an
activity can begin without violation of immediatepredecessor requirements
2. Earliest finish time(EF): the earliest time atwhich an activity can end
3. Latest start time(LS): the latest time an activitycan begin without delaying the entire project
4. Latest finish time(LF): the latest time an activitycan end without delaying the entire project
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How to Find the Critical Path
In the nodes, the activity time and the early andlate start and finish times are represented in thefollowing manner
ACTIVITY t
ES EFLS LF
Earliest times are computed as
Earliest finish time = Earliest start time
+ Expected activity timeEF = ES +t
Earliest start = Largest of the earliest finish times ofimmediate predecessors
ES = Largest EF of immediate predecessors
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How to Find the Critical Path
At the start of the project we set the time to zero
Thus ES = 0 for both A and B
Start
A t = 2ES = 0 EF = 0 + 2 = 2
B t = 3ES = 0 EF = 0 + 3 = 3
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How to Find the Critical Path
General Foundrys ES and EF times
A 2
0 2
C 2
2 4
H 2
13 15
E 4
4 8
B 3
0 3
D 4
3 7
G 5
8 13
F 3
4 7
Start Finish
Figure 13.4
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How to Find the Critical Path
Latest times are computed as
Latest start time = Latest finish time Expected activity time
LS = LFt
Latest finish time = Smallest of latest start timesfor following activities
LF = Smallest LS of following activities
For activity H
LS = LFt = 15 2 = 13 weeks
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How to Find the Critical Path
General Foundrys LS and LF times
A 2
0 2
0 2
C 2
2 4
2 4
H 2
13 15
13 15
E 4
4 8
4 8
B 3
0 3
1 4
D 4
3 7
4 8
G 5
8 13
8 13
F 3
4 7
10 13
Start Finish
Figure 13.5
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How to Find the Critical Path
Once ES, LS, EF, and LF have been determined, itis a simple matter to find the amount of slack timethat each activity has
Slack = LS ES, or Slack = LF EF From Table 13.3 we see activities A, C, E, G, and
Hhave no slack time
These are called critical activitiesand they are
said to be on the critical path The total project completion time is 15 weeks
Industrial managers call this a boundarytimetable
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How to Find the Critical Path
General Foundrys schedule and slack times
ACTIVITY
EARLIESTSTART,ES
EARLIESTFINISH,EF
LATESTSTART,LS
LATESTFINISH,LF
SLACK,LS ES
ONCRITICALPATH?
A 0 2 0 2 0 Yes
B 0 3 1 4 1 No
C 2 4 2 4 0 Yes
D 3 7 4 8 1 No
E 4 8 4 8 0 Yes
F 4 7 10 13 6 No
G 8 13 8 13 0 Yes
H 13 15 13 15 0 Yes
Table 13.3
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How to Find the Critical Path
General Foundrys critical path
A 2
0 2
0 2
C 2
2 4
2 4
H 2
13 15
13 15
E 4
4 8
4 8
B 30 3
1 4
D 43 7
4 8
G 58 13
8 13
F 3
4 7
10 13
Start Finish
Figure 13.6
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Probability of Project Completion
The critical path analysishelped determine theexpected project completion time of 15 weeks
But variation in activities on the critical path canaffect overall project completion, and this is a
major concern If the project is not complete in 16 weeks, the
foundry will have to close
PERT uses the variance of critical path activities
to help determine the variance of the overallproject
Project variance = variances of activitieson the critical path
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Probability of Project Completion
From Table 13.2 we know that
ACTIVITY VARIANCE
A 4/36
B 4/36
C 4/36D 16/36
E 36/36
F 64/36
G 64/36
H 4/36
Hence, the project variance is
Project variance = 4/36 +4/36 +
36/36 +64/36 +
4/36 =112/36 = 3.111
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Probability of Project Completion
We know the standard deviation is just the squareroot of the variance, so
We assume activity times are independent andtotal project completion time is normally
distributed
varianceProjectdeviationstandardProjectT
weeks1.76113.
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Probability of Project Completion
Probability distribution for project completiontimes
Standard Deviation = 1.76 Weeks
(Expected Completion Time)
15 Weeks
Figure 13.7
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Probability of Project Completion
The standard normal equation can be applied asfollows
T
ZcompletionofdateExpecteddateDue
570weeks1.76
weeks15weeks16.
From Appendix A we find the probability of0.71566 associated with thisZ value
That means there is a 71.6% probability thisproject can be completed in 16 weeks or less
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Probability of Project Completion
Probability of General Foundry meeting the 16-week deadline
0.57 Standard Deviations
Time15
WeeksFigure 13.8
16Weeks
Expected Time is 15 Weeks
Probability(T 16 Weeks)is 71.6%
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What PERT Was Able to Provide PERT has been able to provide the project
manager with several valuable pieces ofinformation
The projects expected completion date is 15weeks
There is a 71.6% chance that the equipment willbe in place within the 16-week deadline
Five activities (A, C, E, G, H) are on the criticalpath
Three activities (B, D, F) are not critical but havesome slack time built in
A detailed schedule of activity starting andending dates has been made available
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Sensitivity Analysis andProject Management
The time required to complete an activity canvary from the projected or expected time
If the activity is on the critical path, thecompletion time of the project will change
This will also have an impact on ES, EF, LS, andLF times for other activities
The exact impact depends on the relationshipbetween the various activities
A predecessor activityis one that must beaccomplished before the given activity can bestarted
A successor activityis one that can be startedonly after the given activity is finished
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Sensitivity Analysis andProject Management
Impact of an increase (decrease) in an activitytime for a critical path activity
ACTIVITY TIMESUCCESSORACTIVITY
PARALLELACTIVITY
PREDECESSORACTIVITY
Earliest start Increase (decrease) No change No change
Earliest finish Increase (decrease) No change No change
Latest start Increase (decrease) Increase (decrease) No change
Latest finish Increase (decrease) Increase (decrease) No change
Slack No change Increase (decrease) No change
Table 13.4
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PERT/COST Although PERT is an excellent method of
monitoring and controlling project length, it doesnot consider the very important factor of projectcost
PERT/Costis a modification of PERT that allows amanager to plan, schedule, monitor, and controlcost as well as time
Using PERT/Cost to plan, schedule, monitor, andcontrol project cost helps accomplish the sixthand final step of PERT
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Planning and Scheduling Project Costs:Budgeting Process
The overall approach in the budgetingprocess of a project is to determine howmuch is to be spent every week or month
This can be accomplished in four basicbudgeting steps
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Four Steps of the Budgeting Process1. Identify all costs associated with each of the
activities then add these costs together to getone estimated cost or budget for each activity
2. In large projects, activities can be combined into
larger work packages. A work packageis simplya logical collection of activities.
3. Convert the budgeted cost per activity into acost per time period by assuming that the cost ofcompleting any activity is spent at a uniform rate
over time4. Using the ES and LS times, find out how much
money should be spent during each week ormonth to finish the project by the date desired
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Budgeting for General Foundry
The Gantt chart in Figure 13.9 illustrates thisproject
The horizontal bars shown when each activity willbe performed based on its ES-EF times
We determine how much will be spent on eachactivity during each week and fill these amountsinto a chart in place of the bars
The following two tables show the activity costs
and budgeted cost for the General Foundryproject
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Budgeting for General Foundry
Gantt chart General Foundry project
A
B
C
D
E
F
GH
Activity
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Week
Figure 13.9
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Budgeting for General Foundry
Activity costs for General Foundry
ACTIVITY
EARLIESTSTART,ES
LATESTSTART,LS
EXPECTEDTIME,t
TOTALBUDGETEDCOST ($)
BUDGETEDCOST PERWEEK ($)
A 0 0 2 22,000 11,000
B 0 1 3 30,000 10,000
C 2 2 2 26,000 13,000
D 3 4 4 48,000 12,000
E 4 4 4 56,000 14,000
F 4 10 3 30,000 10,000
G 8 8 5 80,000 16,000
H 13 13 2 16,000 8,000
Total 308,000
Table 13.5
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Budgeting for General Foundry
Budgeted cost for General Foundry
Table 13.6
WEEK
ACTIVITY 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 TOTAL
A 11 11 22
B 10 10 10 30
C 13 13 26
D 12 12 12 12 48
E 14 14 14 14 56
F 10 10 10 30
G 16 16 16 16 16 80
H 8 8 16
308
Total per week 21 21 23 25 36 36 36 14 16 16 16 16 16 8 8
Total to date 21 42 65 90 126 162 198 212 228 244 260 276 292 300 308
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Budgeting for General Foundry
It is also possible to prepare a budget based onthe latest starting time
This budget will delay the expenditure of fundsuntil the last possible moment
The following table shows the latest start budgetfor the General Foundry project
The two tables form a budget range
Any budget can be chosen between these two
values depending on when the company wants toactually spend the money
The budget ranges are plotted in Figure 13.10
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Budgeting for General Foundry
Late start budgeted cost for General Foundry
Table 13.7
WEEK
ACTIVITY 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 TOTAL
A 11 11 22
B 10 10 10 30
C 13 13 26
D 12 12 12 12 48
E 14 14 14 14 56
F 10 10 10 30
G 16 16 16 16 16 80
H 8 8 16
308
Total per week 11 21 23 23 26 26 26 26 16 16 26 26 26 8 8
Total to date 11 32 55 78 104 130 156 182 198 214 240 266 292 300 308
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Budgeting for General Foundry
Budget rangesfor GeneralFoundryBudget Using
Earliest StartTimes, ES
Budget UsingLatest StartTimes, LS
$300,000
250,000
200,000
150,000
100,000
50,000
0
TotalBudgetedCost
Weeks
| | | | | | | | | | | | | | |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 13.10
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Monitoring and ControllingProject Costs
Costs are monitored and controlled to ensure theproject is progressing on schedule and that costoverruns are kept to a minimum
The status of the entire project should be
checked periodically The following table shows the state of the project
in the sixth week
It can be used the answer questions about the
schedule and costs so far
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Monitoring and ControllingProject Costs
Monitoring and controlling budgeted cost
ACTIVITY
TOTALBUDGETEDCOST ($)
PERCENT OFCOMPLETION
VALUE OFWORKCOMPLETED($)
ACTUALCOST ($)
ACTIVITYDIFFERENCE($)
A 22,000 100 22,000 20,000 2,000B 30,000 100 30,000 36,000 6,000
C 26,000 100 26,000 26,000 0
D 48,000 10 4,800 6,000 1,200
E 56,000 20 11,200 20,000 8,800
F 30,000 20 6,000 4,000 2,000
G 80,000 0 0 0 0
H 16,000 0 0 0 0
Total 100,000 112,000 12,000
Table 13.8 Overrun
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Monitoring and ControllingProject Costs
The value of work completed, or the cost to datefor any activity, can be computed as follows
The activity difference is also of interest
Value of work
completed
=(Percentage of work complete)
x (Total activity budget)
Activity difference = Actual cost
Value of work completed
A negative activity difference is a cost underrunand a positive activity difference is a cost overrun
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Project Crashing Projects will sometimes have deadlines
that are impossible to meet using normalprocedures
By using exceptional methods it may bepossible to finish the project in less timethan normally required
However, this usually increases the cost
of the project Reducing a projects completion time is
called crashing
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Project Crashing Crashing a project starts with using the normal
timeto create the critical path
The normal costis the cost for completing theactivity using normal procedures
If the project will not meet the required deadline,extraordinary measures must be taken
The crash timeis the shortest possible activitytime and will require additional resources
The crash costis the price of completing theactivity in the earlier-than-normal time
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Four Steps to Project Crashing1. Find the normal critical path and identify
the critical activities
2. Compute the crash cost per week (or
other time period) for all activities in thenetwork using the formula
Crash cost/Time period = Crash cost Normal costNormal time Crash time
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Four Steps to Project Crashing3. Select the activity on the critical path
with the smallest crash cost per weekand crash this activity to the maximumextent possible or to the point at whichyour desired deadline has been reached
4. Check to be sure that the critical pathyou were crashing is still critical. If thecritical path is still the longest paththrough the network, return to step 3. Ifnot, find the new critical path and returnto step 2.
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General Foundry Example
General Foundry has been given 14 weeksinstead of 16 weeks to install the new equipment
The critical path for the project is 15 weeks
What options do they have?
The normal and crash times and costs are shownin Table 13.9
Crash costs are assumed to be linear and Figure13.11 shows the crash cost for activity B
Crashing activities Band A will shorten thecompletion time to 14 but it creates a secondcritical path
Any further crashing must be done to both criticalpaths
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General Foundry Example
Normal and crash data for General Foundry
ACTIVITY
TIME (WEEKS) COST ($) CRASHCOST PERWEEK ($)
CRITICALPATH?NORMAL CRASH NORMAL CRASH
A 2 1 22,000 23,000 1,000 Yes
B 3 1 30,000 34,000 2,000 No
C 2 1 26,000 27,000 1,000 Yes
D 4 3 48,000 49,000 1,000 No
E 4 2 56,000 58,000 1,000 Yes
F 3 2 30,000 30,500 500 No
G 5 2 80,000 86,000 2,000 Yes
H 2 1 16,000 19,000 3,000 Yes
Table 13.9
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General Foundry Example
Crash and normal times and costs for activity B
NormalCost
CrashCost
Normal
Crash
ActivityCost
Time (Weeks)
$34,000
$33,000
$32,000
$31,000
$30,000
| | | |
0 1 2 3
Normal TimeCrash Time
Crash Cost/Week =Crash Cost Normal Cost
Normal Time
Crash Time
= = $2,000/Week$4,000
2 Weeks
$34,000 $30,000
3 1=
Figure 13.11
P j C hi i h
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Project Crashing withLinear Programming
Linear programming is another approachto finding the best project crashingschedule
We can illustrate its use on GeneralFoundrys network
The data needed are derived from thenormal and crash data for General
Foundry and the project network withactivity times
P j C hi i h
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General Foundrys network with activity times
Figure 13.12
A 2
H 2
B 3 D 4 G 5
FinishStart
C 2
E 4
F 3
P j C hi i h
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The decision variables for the problem are
XA
= EF for activity A
XB
= EF for activity B
XC
= EF for activity C
XD
= EF for activity D
XE
= EF for activity E
XF
= EF for activity F
XG = EF for activity GXH
= EF for activity H
Xstart = start time for project (usually 0)
Xfinish = earliest finish time for the project
P j t C hi ith
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The decision variables for the problem are
Y= the number of weeks that each activity iscrashed
YA
= the number of weeks activity A is crashed
and so forth
The objective function is
Minimize crash cost = 1,000YA + 2,000YB + 1,000YC+ 1,000Y
D+ 1,000Y
E+ 500Y
F
+ 2,000YG
+ 3,000YH
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Crash time constraintsensure activities arenot crashed more thanis allowed
YA 1YB
2
YC 1
YD
1
YE 2YF
1
YG 3
YH
1
This completionconstraint specifiesthat the last eventmust take place beforethe project deadline
Xfinish 12
This constraintindicates the project is
finished when activityHis finished
XfinishXH
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Constraints describing the network have the formEF time EF time for predecessor + Activity time
EF EFpredecessor + (tY), or
XXpredecessor + (tY)
For activity A, XAXstart + (2YA) or XAXstart + YA 2For activity B, XBXstart + (3YB) or XBXstart + YB 3For activity C, XCXA + (2YC) or XCXA + YC 2For activity D, XDXB + (4YD) or XDXB + YD 4
For activity E,XE
XC+ (4
YE) or
XE
XC +
YE 4For activity F, XFXC+ (3YF) or XFXC + YF 3
For activity G, XGXD + (5YG) or XGXD + YG 5For activity G, X
GX
E+ (5Y
G) or X
GXE + YG 5
For activity H, XHXF + (2YH) or XHXF + YH 2For activity H, XHXG + (2YH) or XHXG + YH 2
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Solution to crashing problem using Excel Solver
Program 13.1
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Other Topics in Project Management
Subprojects For extremely large projects, an activity may
be made of several smaller subactivities whichcan be viewed as a smaller project or
subproject of the original Milestones
Major events in a project are often referred toas milestonesand may be reflected in Gantt
chartsand PERT charts to highlight theimportance of reaching these events
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Other Topics in Project Management
Resource Leveling Resource levelingadjusts the activity start
away from the early start so that resourceutilization is more evenly distributed over time
Software There are many project management software
packages on the market for both personalcomputers and larger mainframe machines
Most of these create PERT charts and Ganttcharts and can be used to develop budgetschedules, adjust future start times, and levelresource utilization