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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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    Project Crashing withLinear Programming

    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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    Project Crashing withLinear Programming

    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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    Project Crashing withLinear Programming

    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

    P j t C hi ith

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    Project Crashing withLinear Programming

    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

    P j t C hi ith

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    Project Crashing withLinear Programming

    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

    P j t C hi ith

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    Project Crashing withLinear Programming

    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