ch05 - project.ppt

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CHAPTER 5CHAPTER 5

Project Scheduling Project Scheduling ModelsModels

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• A project is a collection of tasks that must be A project is a collection of tasks that must be completed in minimum time or at minimal cost.completed in minimum time or at minimal cost.

• Objectives of Project SchedulingObjectives of Project Scheduling– Completing the project as early as possible by Completing the project as early as possible by

determining the earliest start and finish of each activity.determining the earliest start and finish of each activity.– Calculating the likelihood a project will be completed Calculating the likelihood a project will be completed

within a certain time period.within a certain time period.– Finding the minimum cost schedule needed to complete Finding the minimum cost schedule needed to complete

the project by a certain date. the project by a certain date.

5.1 Introduction5.1 Introduction

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• A project is a collection of tasks that must be A project is a collection of tasks that must be completed in minimum time or at minimal cost.completed in minimum time or at minimal cost.

• Objectives of Project SchedulingObjectives of Project Scheduling– Investigating the results of possible delays in activity’s Investigating the results of possible delays in activity’s

completion time.completion time.– Progress control.Progress control.– Smoothing out resource allocation over the duration of Smoothing out resource allocation over the duration of

the project.the project.

5.1 Introduction5.1 Introduction

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• Tasks are called “activities.”Tasks are called “activities.” – Estimated completion time (and sometimes Estimated completion time (and sometimes costs) costs) are associated with each activity.are associated with each activity.– Activity completion time is related to the amount Activity completion time is related to the amount of of resources committed to it. resources committed to it. – The degree of activity details depends on the The degree of activity details depends on the

application and the level of specificity of data.application and the level of specificity of data.

Task DesignateTask Designate

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5.2 5.2 Identifying the Activities of a ProjectIdentifying the Activities of a Project

• To determine optimal schedules we need toTo determine optimal schedules we need to– Identify all the project’s activities.Identify all the project’s activities.– Determine the precedence relations among activities.Determine the precedence relations among activities.

• Based on this information we can develop managerial Based on this information we can develop managerial tools for project control.tools for project control.

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Identifying Activities, ExampleIdentifying Activities, Example

KLONE COMPUTERS, INC.KLONE COMPUTERS, INC.

• KLONE Computers manufactures personal computers.KLONE Computers manufactures personal computers.

• It is about to design, manufacture, and market the It is about to design, manufacture, and market the Klonepalm 2000 palmbook computer.Klonepalm 2000 palmbook computer.

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• There are three major tasks to perform:There are three major tasks to perform:– Manufacture the new computer.Manufacture the new computer.– Train staff and vendor representatives.Train staff and vendor representatives.– Advertise the new computer.Advertise the new computer.

• KLONE needs to develop a precedence relations KLONE needs to develop a precedence relations chart.chart.

• The chart gives a concise set of tasks and their The chart gives a concise set of tasks and their immediate predecessors.immediate predecessors.

KLONE COMPUTERS, INCKLONE COMPUTERS, INC

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Activity DescriptionA Prototype model designB Purchase of materials

Manufacturing C Manufacture of prototype model activities D Revision of design

E Initial production run

F Staff trainingTraining activities G Staff input on prototype models

H Sales training

Advertising activities I Pre-production advertising campaign

J Post-redesign advertising campaign


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From the activity description chart, we From the activity description chart, we can determine immediate predecessors can determine immediate predecessors for each activity.for each activity.

Activity A is an immediate predecessorof activity B, because it must be competedjust prior to the commencement of B.

A B


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Immediate EstimatedActivity Predecessor Completion Time

A None 90B A 15C B 5D G 20E D 21F A 25G C,F 14H D 28I A 30J D,I 45

Precedence Relationships Chart


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5.3 The PERT/CPM Approach for 5.3 The PERT/CPM Approach for Project SchedulingProject Scheduling

• The PERT/CPM approach to project scheduling The PERT/CPM approach to project scheduling uses network presentation of the project touses network presentation of the project to– Reflect activity precedence relationsReflect activity precedence relations– Activity completion timeActivity completion time

• PERT/CPM is used for scheduling activities such PERT/CPM is used for scheduling activities such that the project’s completion time is minimized.that the project’s completion time is minimized.

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KLONE COMPUTERS, INC. - ContinuedKLONE COMPUTERS, INC. - Continued

• Management at KLONE would like to schedule the activities Management at KLONE would like to schedule the activities so that the project is completed in minimal time.so that the project is completed in minimal time.

• Management wishes to know:Management wishes to know:– The earliest and latest start times for each activity which will not alter The earliest and latest start times for each activity which will not alter

the earliest completion time of the project.the earliest completion time of the project.– The earliest finish times for each activity which will not alter this date.The earliest finish times for each activity which will not alter this date.– Activities with rigid schedule and activities that have slack in their Activities with rigid schedule and activities that have slack in their

schedules.schedules.

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Earliest Start Time / Earliest Finish TimeEarliest Start Time / Earliest Finish Time• Make a forward pass through the network as follows:Make a forward pass through the network as follows:

– Evaluate all the activities which have no immediate predecessors. Evaluate all the activities which have no immediate predecessors. • The earliest start for such an activity is zero ES = 0.The earliest start for such an activity is zero ES = 0.• The earliest finish is the activity duration EF = Activity duration.The earliest finish is the activity duration EF = Activity duration.

– Evaluate the ES of all the nodes for which EF of all the Evaluate the ES of all the nodes for which EF of all the immediate predecessor has been determined. immediate predecessor has been determined.

• ES = Max EF of all its immediate predecessors. ES = Max EF of all its immediate predecessors. • EF = ES + Activity duration.EF = ES + Activity duration.

– Repeat this process until all nodes have been evaluatedRepeat this process until all nodes have been evaluated• EF of the finish node is the earliest finish time of the project.EF of the finish node is the earliest finish time of the project.

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Earliest Start / Earliest Finish – Earliest Start / Earliest Finish – Forward PassForward Pass

A90

B15

C5

F25

I30

G14

D20

E21

H28

J45

90,105

90,115

90,120

105,110

110,124115,129 129,149

149,170

149,177

120,165149,194

170

194

A

0,90

B

I

F

C

G D

E

H

J

177

194

EARLIEST FINISH

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Latest start time / Latest finish timeLatest start time / Latest finish time• Make a backward pass through the network as follows:Make a backward pass through the network as follows:

– Evaluate all the activities that immediately precede the finish node. Evaluate all the activities that immediately precede the finish node. • The latest finish for such an activity is LF = minimal project completion The latest finish for such an activity is LF = minimal project completion

time.time.• The latest start for such an activity is LS = LF - activity duration.The latest start for such an activity is LS = LF - activity duration.

– Evaluate the LF of all the nodes for which LS of all the immediate Evaluate the LF of all the nodes for which LS of all the immediate successors has been determined. successors has been determined.

• LF = Min LS of all its immediate successors. LF = Min LS of all its immediate successors. • LS = LF - Activity duration.LS = LF - Activity duration.

– Repeat this process backward until all nodes have been Repeat this process backward until all nodes have been evaluated.evaluated.

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B

F

C

A

I

E

DG HH28

166,194

JJ45

149,194

E21

173,19490,105

90,115

90,120

105,110

115,129 129,149

149,170

149,177

149,194

153,173146,166

194

129,149

0,90

129,149

D20

129,149129,149129,149129,149129,149129,149129,149

G14

115,129

I30

119,149

29,119

C5

110,115B1595,110

5,95F25

90, 115

0,90A90

Latest Start / Latest Finish –Latest Start / Latest Finish – Backward PassBackward Pass

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• Activity start time and completion time may be Activity start time and completion time may be delayed by planned reasons as well as by delayed by planned reasons as well as by unforeseen reasons.unforeseen reasons.

• Some of these delays may affect the overall Some of these delays may affect the overall completion date.completion date.

• To learn about the effects of these delays, we To learn about the effects of these delays, we calculate the calculate the slack timeslack time,, and form the and form the critical pathcritical path..

Slack TimesSlack Times

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– Slack time is the amount of time an activity can be delayed Slack time is the amount of time an activity can be delayed without delaying the project completion date, assuming no without delaying the project completion date, assuming no other delays are taking place in the project.other delays are taking place in the project.

Slack Time = LS - ES = LF - EF

Slack TimesSlack Times

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Critical activities

must be rigidly

scheduled

Activity LS - ES SlackA 0 -0 0B 95 - 90 5C 110 - 105 5D 119 - 119 0E 173 - 149 24F 90 - 90 0G 115 - 115 0H 166 - 149 17I 119 - 90 29J 149 - 149 0

Slack time in the Klonepalm 2000 Project

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• The critical path is a set of activities that have no slack,The critical path is a set of activities that have no slack,connecting the START node with the FINISH node.connecting the START node with the FINISH node.

• The critical activities (activities with 0 slack) form The critical activities (activities with 0 slack) form at least one critical path in the network.at least one critical path in the network.

• A critical path is the longest path in the network.A critical path is the longest path in the network.

• The sum of the completion times for the activitiesThe sum of the completion times for the activities on the critical path is the minimal completion time on the critical path is the minimal completion time of the project. of the project.

The Critical PathThe Critical Path

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B

F

C

A

I

E

DG HH28

166,194

JJ45

149,194

E21

173,19490,105

90,115

90,120

105,110

115,129 129,149

149,170

149,177

149,194

D20

0,90129,149

G14

115,129

I30

119,149

A90

C5

110,115B15

95,110

F25

90, 1150,90

The Critical PathThe Critical Path

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• We observe two different types of delays:We observe two different types of delays:

– Single delays.Single delays.

– Multiple delays.Multiple delays.

• Under certain conditions the overall project Under certain conditions the overall project completion time will be delayed. completion time will be delayed.

• The conditions that specify each case are presented The conditions that specify each case are presented next.next.

Possible DelaysPossible Delays

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• A delay of a certain amount in a A delay of a certain amount in a critical activitycritical activity, , causes the entire project to be delayed by the same causes the entire project to be delayed by the same amount.amount.

• A delay of a certain amount in a A delay of a certain amount in a non-criticalnon-critical activity activity will delay the project by the amount the delay exceeds will delay the project by the amount the delay exceeds the slack time. When the delay is less than the slack, the slack time. When the delay is less than the slack, the entire project is not delayed. the entire project is not delayed.

Single delaysSingle delays

24LS =119

A90

J45

H28

E21

D20

I30

G14

F25

C5

B15

ES=149

LS=173 DELAYED START=149+15=164

ES=90DELAYED START=90+15 =105

Activity E and I are each delayed 15 days.

THE PROJECT COMPLETION TIME IS NOT DELAYEDTHE PROJECT COMPLETION TIME IS NOT DELAYED

FINISHMultiple delays of non critical activities: Multiple delays of non critical activities:

Case 1: Activities on different pathsCase 1: Activities on different paths

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AA9090

90

BB1515

Gantt chart demonstration of the (no) effects on the project completion time when delayingactivity “I” and “E” by 15 days.

Activity I

FF2525

II3030

105

CC55

115

GG1414

129

DD2020

149

EE2121

HH2828

JJ

4545

194

194Activity E

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A90

B15

C5

F25

I30

G14

D20

E21

H28

J45

FINISH

ES=149

LS =173 DELAYED START=149+15 =164

ES=90DELAYED START =94

LS =95

THE PROJECT COMPLETION TIME IS NOT DELAYEDTHE PROJECT COMPLETION TIME IS NOT DELAYED

Multiple delays of non critical activities:Multiple delays of non critical activities:Case 2: Activities are on the same path, Case 2: Activities are on the same path,

separated by critical activities.separated by critical activities.

Activity B is delayed 4 days, activity E is delayed 15 days

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A90

B15

C5

F25

I30

G14

D20

E21

H28

J45

FINISH

DELAYED START= 109 + 4 =113;

ES= 90

DELAYED START =94DELAYED FINISH =94+15=109

LS =110

3 DAYS DELAYIN THE ENTIRE

PROJECT

Activity B is delayed 4 days; Activity C is delayed 4 days.

THE PROJECT COMPLETION TIME IS DELAYED 3 DAYSTHE PROJECT COMPLETION TIME IS DELAYED 3 DAYS

Multiple delays of non critical activities:Multiple delays of non critical activities:Case 2: Activities are on the same path, Case 2: Activities are on the same path,

no critical activities separating them.no critical activities separating them.

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5.4 A Linear Programming Approach to 5.4 A Linear Programming Approach to PERT/CPMPERT/CPM

• VariablesVariables– XXii = The start time of the activities for i=A, B, C, …,J = The start time of the activities for i=A, B, C, …,J– X(FIN) = Finish time of the projectX(FIN) = Finish time of the project

• Objective functionObjective function– Complete the project in minimum time.Complete the project in minimum time.

• ConstraintsConstraints– For each arc a constraint states that the start time For each arc a constraint states that the start time

of M must not occur before the finish time of its immediate of M must not occur before the finish time of its immediate predecessor, L.predecessor, L.

ML

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A Linear Programming ApproachA Linear Programming Approach

Define X(FIN) to be the finish time of the project. Define X(FIN) to be the finish time of the project. The objective then isThe objective then is

Minimize X(FIN)Minimize X(FIN)

While this objective function is intuitiveother objective functions provide moreinformation, and are presented later.

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X(FIN) X(FIN) XXEE + 21 + 21 X(FIN) X(FIN) XXHH + 28 + 28

X(FIN) X(FIN) XXJJ + 45 + 45 XXDD XXGG + 14 + 14 XXEE XXDD + 20 + 20 XXGG X XCC+ 5 + 5 XXH H X XDD + 20 + 20 XXGG XXFF+ 25 + 25 XXJ J X XD D + 20 + 20 XXII XXDD+ 90+ 90XXJJ X XII + 30 + 30 XXFF X XAA+ 90+ 90XXC C X XBB+ 15 + 15 XXDD X XGG+ 14 + 14

XXBB XXAA+ 90+ 90

G

C5

F25

All X s are nonnegative

Minimize X(FIN)ST


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Minimize XA+XB+…+XJ

This objective function ensures that the optimal X values are the earliest start times of all the activities. The project completion time is minimized.

Maximize XA+XB+…+XJ

S.T. X(FIN) = 194and all the other constraints as before.

This objective function and the additional constraint ensure that the optimal X values arethe latest start times of all the activities.


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5.5 Obtaining Results Using Excel5.5 Obtaining Results Using Excel

MEAN 194STANDARD DEVIATION* 0 * Assumes all critical activities are on one critical path

VARIANCE* 0 If not, enter in gold box, the variance on one critical path of interest.

PROBABILITY COMPLETE BEFORE =

Acitivty Node Critical ES EF LS LF SlackDesign A * 90 0 90 0 90 0Materials B 15 90 105 95 110 5Manufacture C 5 105 110 110 115 5Design Revision D * 20 129 149 129 149 0Production Run E 21 149 170 173 194 24Staff Training F * 25 90 115 90 115 0Staff Input G * 14 115 129 115 129 0Sales Training H 28 149 177 166 194 17Preprod. Advertise I 30 90 120 119 149 29Post. Advertise J * 45 149 194 149 194 0

CRITICAL PATH ANALYSIS

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5.6 Gantt Charts5.6 Gantt Charts

• Gantt charts are used as a tool to monitor and control the Gantt charts are used as a tool to monitor and control the project progress.project progress.

• A Gantt Chart is a graphical presentation that displays activities A Gantt Chart is a graphical presentation that displays activities as follows:as follows:– Time is measured on the horizontal axis. A horizontal bar is drawn Time is measured on the horizontal axis. A horizontal bar is drawn

proportionately to an activity’ s expected completion time.proportionately to an activity’ s expected completion time.– Each activity is listed on the vertical axis.Each activity is listed on the vertical axis.

• In an In an earliest time Gantt chartearliest time Gantt chart each bar begins and ends at the each bar begins and ends at the earliest startearliest startfinish the activity can take place.finish the activity can take place.

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Here‘s how we build Here‘s how we build an Earliest Time Gantt Chart an Earliest Time Gantt Chart for KLONEPALM 2000for KLONEPALM 2000

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Immediate EstimatedActivity Predecessor Completion Time

A None 90B A 15C B 5D G 20E D 21F A 25G C,F 14H D 28I A 30J D,I 45

AA9090

90

BB1515

FF2525

II3030

105

CC55

115

GG1414

129

DD2020

149

EE2121

HH2828

JJ

4545

194

194

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• Gantt chart can be used as a visual aid for tracking the Gantt chart can be used as a visual aid for tracking the progress of project activities.progress of project activities.

• Appropriate percentage of a bar is shaded to document the Appropriate percentage of a bar is shaded to document the completed work.completed work.

• The manager can easily see if the project is progressing on The manager can easily see if the project is progressing on schedule (with respect to the earliest possible completionschedule (with respect to the earliest possible completion times). times).

Gantt Charts- Gantt Charts- Monitoring Project ProgressMonitoring Project Progress

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AA9090

BB1515

FF2525

II3030

CC55

GG1414

DD2020

EE2121

HH2828

JJ

4545

194

194

135

Monitoring Project ProgressMonitoring Project Progress

The shaded bars representcompleted work BY DAY 135.

Do not conclude that the project is behind schedule.

Activity “I” has a slack and therefore can be delayed!!!

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• AdvantagesAdvantages.. – Easy to constructEasy to construct– Gives earliest completion date.Gives earliest completion date.– Provides a schedule of earliest possible start and finish times of Provides a schedule of earliest possible start and finish times of

activities.activities.

• DisadvantagesDisadvantages– Gives only one possible schedule (earliest).Gives only one possible schedule (earliest).– Does not show whether the project is behind schedule.Does not show whether the project is behind schedule.– Does not demonstrate the effects of delays in any one activity on theDoes not demonstrate the effects of delays in any one activity on the

start of another activity, thus on the project completion time.start of another activity, thus on the project completion time.

Gantt Charts – Gantt Charts – Advantages and DisadvantagesAdvantages and Disadvantages

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5.7 Resource Leveling and Resource 5.7 Resource Leveling and Resource AllocationAllocation

• It is desired that resources are evenly spread out It is desired that resources are evenly spread out throughout the life of the project.throughout the life of the project.

• Resource leveling methods (usually heuristics) are Resource leveling methods (usually heuristics) are designed to: designed to: – Control resource requirementsControl resource requirements– Generate relatively similar usage of resources over time.Generate relatively similar usage of resources over time.

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• A heuristic approach to “level” expenditures

– Assumptions• Once an activity has started it is worked on continuously until it is completed.• Costs can be allocated equally throughout an activity duration.

Step 1: Consider the schedule that begins each activity at its ES. Step 2: Determine which activity has slack at periods of peak spending.Step 3: Attempt to reschedule the non-critical activities performed during these peak periods to periods of less spending, but within the time period between their ES and LF.

Resource Leveling – A HeuristicResource Leveling – A Heuristic

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• Management wishes to schedule the project such Management wishes to schedule the project such that that – Completion time is 194 days.Completion time is 194 days.– Daily expenditures are kept as constant as possible.Daily expenditures are kept as constant as possible.

• To perform this analysis cost estimates for each To perform this analysis cost estimates for each activity will be needed.activity will be needed.

Resource Leveling –Resource Leveling –KLONE COMPUTERS, Inc. - continuedKLONE COMPUTERS, Inc. - continued

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Total Total CostCost Time per

Activity Description (x10000) (days) DayA Prototype model design 2250 90 25B Purchase of materials 180 15 12C Manufacture of prototype 90 5 18D Revision of design 300 20 15E Initial production run 231 21 11F Staff training 250 25 10G Staff input on prototype 70 14 5H Sales traini ng 392 28 14I Pre-production advertisement 510 30 17J Post-production advertisement 1350 45 30

Total cost = 5,623

Resource Leveling –Resource Leveling –KLONE COMPUTERS, Inc. – cost estimatesKLONE COMPUTERS, Inc. – cost estimates

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Cumulative Daily Expenditure – Cumulative Daily Expenditure – Earliest Times vs. Latest TimesEarliest Times vs. Latest Times55

50

45

40

35

30

25

201510 5

20 40 60 80 100 120 140 160 180 200

Level Budget

Earliest Start-Earliest FinishBudget

Latest Start-Latest FinishBudget

FeasibleBudgets

Time

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45

40

35

30

25

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20 40 60 80 100 120 140 160 180 200

E

H

JJ

Cost Leveling

E

J

E

J

H

J

H

C

F A

B

F

I

I

H

27

15

44

30

Daily Expenditure of the ES Schedule55

I I I I I

25I

I39

45I

ES = 90I

LS = 110

22

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I

I

I

F

G

D

45

G

55

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201510 5

20 40 60 80 100 120 140 160 180 200

E

H

JJ

IE

J

E

J

H

JC

F A

B

F

I

I

H

25 27

22

15

44

3032

55

H

H

Cost Leveling

ID

I

F

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5.8 The Probability Approach to 5.8 The Probability Approach to Project Scheduling Project Scheduling

• Activity completion times are seldom known with Activity completion times are seldom known with 100% accuracy.100% accuracy.

• PERT is a technique that treats activity completion PERT is a technique that treats activity completion times as random variables.times as random variables.

• Completion time estimates are obtained by theCompletion time estimates are obtained by theThree Time Estimate approachThree Time Estimate approach

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• The TThe Three Time Estimatehree Time Estimate approachapproach provides provides completion time estimate for each activity.completion time estimate for each activity.

• We use the notation:We use the notation:

a = an optimistic time to perform the activity.a = an optimistic time to perform the activity.m = the most likely time to perform the activity.m = the most likely time to perform the activity.b = a pessimistic time to perform the activity.b = a pessimistic time to perform the activity.

The Probability Approach – The Probability Approach – Three Time EstimatesThree Time Estimates

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= the mean completion time =a + 4m + b

6= the standard deviation =

b - a6

Approximations for the mean and the Approximations for the mean and the standard deviation of activity standard deviation of activity completion time are based on the completion time are based on the BetaBeta distribution.distribution.

The Distribution, Mean, and Standard The Distribution, Mean, and Standard Deviation of an ActivityDeviation of an Activity

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To calculate the mean and standard deviation of the project completion time we make some simplifying assumptions.

The Project Completion Time Distribution - The Project Completion Time Distribution - AssumptionsAssumptions

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• Assumption 2Assumption 2– The time to complete one activity is independent of the time The time to complete one activity is independent of the time

to complete any other activity.to complete any other activity.• Assumption 3Assumption 3

– There are enough activities on the critical path so that the There are enough activities on the critical path so that the distribution of the overall project completion time can be distribution of the overall project completion time can be approximated by the normal distribution.approximated by the normal distribution.

The Project Completion Time Distribution - The Project Completion Time Distribution - AssumptionsAssumptions

• Assumption 1Assumption 1– A critical path can be determined by using the mean A critical path can be determined by using the mean

completion times for the activities. completion times for the activities. – The project mean completion time is determined solely by the The project mean completion time is determined solely by the

completion time of the activities on the critical path.completion time of the activities on the critical path.

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Mean = Sum of mean completion times along

the critical path.

The three assumptions imply that the overall project The three assumptions imply that the overall project

completion time is completion time is normallynormally distributed, the following distributed, the following

parameters: parameters:

The Project Completion Time DistributionThe Project Completion Time Distribution

Variance = Sum of completion time variances

along the critical path.

Standard deviation = Variance

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Activity Optimistic Most Likely PessimisticA 76 86 120B 12 15 18C 4 5 6D 15 18 33E 18 21 24F 16 26 30G 10 13 22H 24 18 32I 22 27 50J 38 43 60

The Probability Approach – The Probability Approach – KLONE COMPUTERSKLONE COMPUTERS

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• Management at KLONE is interested in Management at KLONE is interested in information regarding the completion time of the information regarding the completion time of the project.project.

• The probabilistic nature of the completion time The probabilistic nature of the completion time must be considered.must be considered.

The Probability Approach – The Probability Approach – KLONE COMPUTERSKLONE COMPUTERS

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AA = = [76+4(86)+120][76+4(86)+120]//6 = 906 = 90

= (120 - 76)= (120 - 76)//6 = 7.336 = 7.33 AA2 2 = (7.33) = (7.33)22 = 53.78 = 53.78

Activity A 90 7.33 53.78B 15 1.00 1.00C 5 0.33 0.11D 20 3.00 9.00E 21 1.00 1.00F 25 2.33 5.44G 14 2.00 4.00H 28 1.33 1.78I 30 4.67 21.78J 45 3.67 13.44

KLONE COMPUTERS KLONE COMPUTERS – – Finding activities’ mean and varianceFinding activities’ mean and variance

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• The mean times are the same as in the CPM The mean times are the same as in the CPM problem, previously solved for KLONE.problem, previously solved for KLONE.

• Thus, the critical path is A - F- G - D – J.Thus, the critical path is A - F- G - D – J.

– Expected completion time =Expected completion time = AA + +FF + +GG + +DD + +JJ=194.=194.

– The project variance =The project variance =AA22 + +FF

22 + +GG22 + +DD

22 + +JJ22 = 85.66 = 85.66

– The standard deviation = = 9.255 The standard deviation = = 9.255

KLONE COMPUTERS KLONE COMPUTERS – – Finding mean and variance for the critical pathFinding mean and variance for the critical path

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• The probability of completion in 194 days =The probability of completion in 194 days =

P(X 194) = P(Z194 - 194

9.255 ) ( ) .P Z 0 0 5

The Probability Approach – The Probability Approach – Probabilistic analysisProbabilistic analysis

194

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• An interval in which we are reasonably sure the completion date An interval in which we are reasonably sure the completion date lies islies is z0.025


.95

• The interval is = 194 1.96(9.255) [175, 213] days.• The probability that the completion time lies in the interval

[175,213] is 0.95.

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XZ

1940

• The probability of completion in 180 days =

P(X 180) = P(Z -1.51) = 0.5 - 0.4345 = 0.0655

180-1.51

0.0655


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• The probability that the completion time is longer than 210 days =

0.0418=0.458-0.5=1.73)P(Z=210)P(X

XZ

1940

.4582

2101.73

?0.0418


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XZ

1940

• Provide a completion time that has only 1% chance to be exceeded.

X0

2.33

0.01.49


P(XX0) = 0.01, or P(Z X0 – )/ = P(ZZ0) = .01

P(Z 2.33) = 0.01; X0=+Z0 =194 + 2.33(9.255) = 215.56 days.

There is 99% chance that the projectis completed in 215.56 days.

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NORMDIST(194, 194, 9.255, TRUE)

NORMINV(.025, 194, 9.255)NORMINV(.975, 194, 9.255)

NORMDIST(180, 194, 9.255, TRUE)

1 - NORMDIST(210, 194, 9.255, TRUE)

NORMINV(.99, 194, 9.255)

The Probability Approach – The Probability Approach – Probabilistic analysis with a spreadsheetProbabilistic analysis with a spreadsheet

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The Probability Approach – The Probability Approach – Critical path spreadsheetCritical path spreadsheet

MEAN 194STANDARD DEVIATION* 9.255629 * Assumes all critical activities are on one critical path

VARIANCE* 85.66667 If not, enter in gold box, the variance on one critical path of interest.

PROBABILITY COMPLETE BEFORE 180 = 0.065192

Acitivty Node Critical ES EF LS LF SlackDesign A * 90 7.333333 53.77778 0 90 0 90 0Materials B 15 1 1 90 105 95 110 5Manufacture C 5 0.333333 0.111111 105 110 110 115 5Design Revision D * 20 3 9 129 149 129 149 0Production Run E 21 1 1 149 170 173 194 24Staff Training F * 25 2.333333 5.444444 90 115 90 115 0Staff Input G * 14 2 4 115 129 115 129 0Sales Training H 28 1.333333 1.777778 149 177 166 194 17Preprod. Advertise I 30 4.666667 21.77778 90 120 119 149 29Post. Advertise J * 45 3.666667 13.44444 149 194 149 194 0

CRITICAL PATH ANALYSIS

63

The Probability Approach – The Probability Approach – critical path spreadsheetcritical path spreadsheet

MEAN 189STANDARD DEVIATION* 9.0185 * Assumes all critical activities are on one critical path

VARIANCE* 81.33333 If not, enter in gold box, the variance on one critical path of interest.

PROBABILITY COMPLETE BEFORE 180 = 0.159152

Acitivty Node Critical ES EF LS LF SlackDesign A * 90 7.333333 53.77778 0 90 0 90 0Materials B * 15 1 1 90 105 90 105 0Manufacture C * 5 0.333333 0.111111 105 110 105 110 0Design Revision D * 20 3 9 124 144 124 144 0Production Run E 21 1 1 144 165 168 189 24Staff Training F 14 0.666667 0.444444 90 104 96 110 6Staff Input G * 14 2 4 110 124 110 124 0Sales Training H 28 1.333333 1.777778 144 172 161 189 17Preprod. Advertise I 30 4.666667 21.77778 90 120 114 144 24Post. Advertise J * 45 3.666667 13.44444 144 189 144 189 0

CRITICAL PATH ANALYSISA comment – multiple critical paths

In the case of multiple critical paths (a not unusual situation), determine the probabilities for each critical path separately using its standard deviation.

However, the probabilities of interest (for example, P(X xcannot be determined by each path alone. To find these probabilities, check whether the paths are independent.

If the paths are independent (no common activities among the paths), multiply the probabilities of all the paths: [Pr(Completion timex) = Pr(Path 1x)P(Path 2x)…Path kx)]

If the paths are dependent, the calculations might become very cumbersome, in which case running a computer simulation seems to be more practical.

64

5.9 Cost Analysis Using the5.9 Cost Analysis Using the Expected Value Approach Expected Value Approach

• Spending extra money, in general should decrease Spending extra money, in general should decrease project duration.project duration.

• Is this operation cost effective?Is this operation cost effective?

• The expected value criterion is used to answer this The expected value criterion is used to answer this question.question.

65

• Analysis indicated:Analysis indicated:

– Completion time within 180 days yields an additional profit Completion time within 180 days yields an additional profit of $1 million.of $1 million.

– Completion time between 180 days and 200 days, yields Completion time between 180 days and 200 days, yields an additional profit of $400,000.an additional profit of $400,000.

– Completion time reduction can be achieved by Completion time reduction can be achieved by additional training.additional training.

KLONE COMPUTERS - KLONE COMPUTERS - Cost analysis using probabilitiesCost analysis using probabilities

66

• Two possible activities are considered for training.Two possible activities are considered for training.

– Sales personnel training:Sales personnel training:• Cost $200,000; Cost $200,000; • New time estimates are a = 19, m= 21, and b = 23 days.New time estimates are a = 19, m= 21, and b = 23 days.

– Technical staff training:Technical staff training: • Cost $250,000; Cost $250,000; • New time estimates are a = 12, m = 14, and b = 16.New time estimates are a = 12, m = 14, and b = 16.

Which option should be pursued?


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• Evaluation of spending on Evaluation of spending on sales personnel training.sales personnel training.– This activity (H) is not critical.This activity (H) is not critical.– Under the assumption that the project completion time is Under the assumption that the project completion time is

determined solely by critical activities, this option determined solely by critical activities, this option should not be considered further.should not be considered further.

• Evaluation of spending on Evaluation of spending on technical staff training.technical staff training.– This activity (F) is critical.This activity (F) is critical.– This option should be further studied as follows:This option should be further studied as follows:

• Calculate expected profit when not spending $250,000.Calculate expected profit when not spending $250,000.• Calculate expected profit when spending $250,000.Calculate expected profit when spending $250,000.• Select the decision with a higher expected profit.Select the decision with a higher expected profit.


68

– Case 1: Do not spend $250,000 on training.Case 1: Do not spend $250,000 on training.• Let X represent the project’s completion time.Let X represent the project’s completion time.

• Expected gross additional profit = E(GP) =Expected gross additional profit = E(GP) =P(X<180)($1 million) + P(180<X<200)($400,000) + P(X>200)P(X<180)($1 million) + P(180<X<200)($400,000) + P(X>200)(0).(0).

• Use Excel to find the required probabilities:Use Excel to find the required probabilities:P(X<180) = .065192; P(180<X<200) = .676398; P(X>200) =.25841P(X<180) = .065192; P(180<X<200) = .676398; P(X>200) =.25841

• Expected gross additional profit = .Expected gross additional profit = ..065192(1M)+.676398(400K)+ .25841(0) = $335,751.20.065192(1M)+.676398(400K)+ .25841(0) = $335,751.20


69

– Case 2: Spend $250,000 on training.Case 2: Spend $250,000 on training. • The revised mean time and standard deviation estimates for activity The revised mean time and standard deviation estimates for activity

F are:F are:FF= (12 + 4 (14) + 16)/6 = 14= (12 + 4 (14) + 16)/6 = 14FF= (16 -12)/6 =0.67= (16 -12)/6 =0.67FF

= 0(.67)= 0(.67)22 =0.44 =0.44

• Using the Excel PERT-CPM template we find a new critical path Using the Excel PERT-CPM template we find a new critical path

(A-B-C-G-D-J), with a mean time = 189 days, and a standard (A-B-C-G-D-J), with a mean time = 189 days, and a standard

deviation of = 9.0185 days. deviation of = 9.0185 days.


70

– The probabilities of interest need to be recalculated. The probabilities of interest need to be recalculated. From Excel we find:From Excel we find:

• P(X < 180) = .159152; P(X < 180) = .159152; • P(180 < X < 200) = .729561P(180 < X < 200) = .729561• P(X > 200) = .111287P(X > 200) = .111287

– Expected Gross Additional Revenue = Expected Gross Additional Revenue = P( X<180)(1M)+P(180<X<200)(400K)+P(X>200)(0) P( X<180)(1M)+P(180<X<200)(400K)+P(X>200)(0) = = .159152(!M)+.159152(!M)+ .729561(400K)+.729561(400K)+ .111287(0).111287(0) = $450,976.40= $450,976.40


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The expected net additional profit =The expected net additional profit =

450,976-250,000 = $200,976 < $335,751450,976-250,000 = $200,976 < $335,751

Expected additional net profit whenspending $250,000 on training

Expected profit withoutspending $250,000 on training

Conclusion: Management should not spend money on additional training of technical personnel.


72

5.10 Cost Analyses Using 5.10 Cost Analyses Using The Critical Path Method (CPM)The Critical Path Method (CPM)

• The critical path method (CPM) is a deterministic The critical path method (CPM) is a deterministic approach to project planning.approach to project planning.

• Completion time depends only on the amount of Completion time depends only on the amount of money allocated to activities.money allocated to activities.

• Reducing an activity’s completion time is called Reducing an activity’s completion time is called “crashing.”“crashing.”

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• There are two crucial completion times to consider There are two crucial completion times to consider for each activity.for each activity.– Normal completion time (TNormal completion time (TNN).).

– Crash completion time (TCrash completion time (TCC), the minimum possible ), the minimum possible completion time.completion time.

Crash timeCrash timeCrash costCrash cost

• The cost spent on an activity varies between– Normal cost (CN). The activity is completed in TN.

– Crash cost (CC). The activity is completed in TC.

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Crash timeCrash timeCrash costCrash cost – – The Linearity AssumptionThe Linearity Assumption

• The maximum crashing of activity completion time The maximum crashing of activity completion time is Tis TC C – T– TNN..

• This can be achieved when spending CThis can be achieved when spending CNN – C – CCC..• Any percentage of the maximum extra cost Any percentage of the maximum extra cost

(C(CNN – C – CCC)) spent to crash an activity, yields the spent to crash an activity, yields the same percentage reduction of the maximum time same percentage reduction of the maximum time savings (Tsavings (TC C – T– TNN). ).

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Time

Cost ($100)

20

18

1614

1210

8

642

5 10 15 20 25 30 35 40 45

NormalCN = $2000TN = 20 days

Add to the normal cost...

…and save oncompletion time

Add more to the normal cost...

CrashingCC = $4400TC = 12 days

…and save more on completion time Add 25% of the extra

cost...

… to save 25% of the max. time reduction

Total Cost = $2600Job time = 18 days

A demonstration A demonstration of the Linearity of the Linearity

AssumptionAssumption

76

Marginal Cost = Additional Cost to get Max. Time Reduction

Maximum Time reduction= (4400 - 2000)/(20 - 12) = $300 per day

M = E

R

Crash time/ Crash costCrash time/ Crash cost - -The Linearity AssumptionThe Linearity Assumption

77

• If the deadline to complete a project cannot be met If the deadline to complete a project cannot be met using normal times, additional resources must be using normal times, additional resources must be spent on crashing activities.spent on crashing activities.

• The objective is to meet the deadline at minimal The objective is to meet the deadline at minimal additional cost.additional cost.

Crashing activities – Crashing activities – Meeting a Deadline at Minimum CostMeeting a Deadline at Minimum Cost

78

• Baja Burrito (BB) is a chain of Mexican-style fast Baja Burrito (BB) is a chain of Mexican-style fast food restaurants.food restaurants.

• It is planning to open a new restaurant in 19 It is planning to open a new restaurant in 19 weeks.weeks.

• Management wants to Management wants to – Study the feasibility of this plan,Study the feasibility of this plan,– Study suggestions in case the plan cannot be finished Study suggestions in case the plan cannot be finished

by the deadline.by the deadline.

Baja Burrito Restaurants – Baja Burrito Restaurants – Meeting a Deadline at Minimum CostMeeting a Deadline at Minimum Cost

79

Baja Burrito Restaurants –Baja Burrito Restaurants –

Without spending any extra money, the restaurant will open in 29 weeks at a normal cost of $200,000.

When all the activities are crashedto the maximum, the restaurantwill open in 17 weeks at crash costof $300,000.

Determined by the PERT.xls template

80

A

D

C

B

E

F G

I

H

Baja Burrito Restaurants –Baja Burrito Restaurants –Network presentationNetwork presentation

L

O

J NM

K

P

81

Baja Burrito Restaurants –Baja Burrito Restaurants –Marginal costsMarginal costs

R = TN – TC = 5 – 3 = 2E = CC – CN = 36 – 25 = 11M = E/R = 11/2 = 5.5

82

Baja Burrito Restaurants –Baja Burrito Restaurants –HeuristicHeuristic SolutionSolution

• Three observations lead to the heuristic. – The project completion time is reduced only when

critical activity times are reduced.– The maximum time reduction for each activity is limited.– The amount of time a critical activity can be reduced

before another path becomes critical is limited.

• Small crashing problems can be solved heuristically.

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• Linear Programming ApproachLinear Programming Approach– VariablesVariables

XXjj = start time for activity j. = start time for activity j.YYj j = the amount of crash in activity j. = the amount of crash in activity j.

– Objective FunctionObjective FunctionMinimize the total additional funds spent on crashing activities.Minimize the total additional funds spent on crashing activities.

– ConstraintsConstraints• No activity can be reduced more than its Max. time reduction.No activity can be reduced more than its Max. time reduction.• Start time of an activity takes place not before the finish time of all its Start time of an activity takes place not before the finish time of all its

immediate predecessors. immediate predecessors. • The project must be completed by the deadline date D.The project must be completed by the deadline date D.

Baja Burrito Restaurants –Baja Burrito Restaurants –Linear ProgrammingLinear Programming

84

Min 5.5YA+10YB+2.67YC+4YD+2.8YE+6YF+6.67YG+10YH+5.33YI+12YJ+4YK+5.33YL+1.5YN+4YO+5.33YP

Minimize total crashing costs

Baja Burrito Restaurants –Baja Burrito Restaurants –Linear ProgrammingLinear Programming

85


Maximum time-reductionconstraints

H

F

E

D

C

B

A

0.5

2.0

0.5

0.5

2.5

1.0

1.5

Y

YYYYYY

Y G 1.5

……..

19FINX )(

ST

Meet the deadline

Linear ProgrammingLinear Programming

86

XA


Activity canstart only after all the Predecessors are completed.84

A

D

C

B

E

F G

I

H

Baja Burrito Restaurants –Network presentation

L

O

J NM

K

P

XBXA+(5 – YA)

BXB

A

-YA

XA+5-YA

AXA+5

BXB

BXB

BXB


BXB

87

84

A

D

C

B

E

F G

I

H

Baja Burrito Restaurants –Network presentation

L

O

J NM

K

P

XBXA+(5 – YA)XCXA+(5 – YA)XDXA+(5 – YA)XeXA+(5 – YA)XFXA+(5 – YA)XBXB+(1 – YB)XFXC+(3 – YC)XGXF+(1 – YF)

X(FIN)XN+(3 – YN)X(FIN)XO+(4 – YO)X(FIN)XP+(4 – YP)


Activity canstart only afterall the predecessorsare completed.

……..


88

Baja Burrito Restaurants –Baja Burrito Restaurants –Deadline SpreadsheetDeadline Spreadsheet

TOTAL PROJECT COST 248.75 PROJECT NORMAL COST 200COMPLETION TIME 19 PROJECT CRASH COST 300

ACTIVITY NODECompletion

TimeStart Time

Finish Time

Amount Crashed

Cost of Crashing Total Cost

Revisions/Approvals A 3 0 3 2 11 36Grade Land B 1 3 4 0 0 10Purchase Materials C 1.5 3 4.5 1.5 4 22Order Equipment D 2 3 5 0 0 8Order Furniture E 4 12.5 16.5 0 0 8Concrete Floor F 0.5 4.5 5 0.5 3 15Erect Frame G 4 5 9 0 7.87637E-11 20Install Electrical H 2 9 11 0 0 12Install Plumbing I 2.5 9 11.5 1.5 8 21Install Drywall/Roof J 2 11.5 13.5 0 0 10Bathrooms K 2 13 15 0 0 8Install Equipment L 1.5 13.5 15 1.5 8 22Finish/Paint Inside M 1.5 15 16.5 1.5 8 18Tile Floors N 2.5 16.5 19 0.5 0.75 6.75Install Furniture O 2.5 16.5 19 1.5 6 14Finish/Paint Outside P 4 13.5 17.5 0 0 18

CRASHING ANALYSIS

89

• Baja Burrito has the policy of not funding more than 12.5% Baja Burrito has the policy of not funding more than 12.5% above the “normal cost” projection.above the “normal cost” projection.

Crash budget = (12.5%)(200,000) = 25,000Crash budget = (12.5%)(200,000) = 25,000

• Management wants to minimize the project completion time Management wants to minimize the project completion time under the budget constraint.under the budget constraint.

Baja Burrito Restaurants –Baja Burrito Restaurants –Operating within a fixed budgetOperating within a fixed budget

90

5.5YA + 10YB + 2.67YC + 4YD + 2.8YE + 6YF + 6.67YG + 10YH + 5.33YI + 12YJ

+ 4YK + 5.33Y L+ 1.5YN + 4YO + 5.33YP 25

The crash funds become a constraint Minimize

5.5YA + 10YB + 2.67YC + 4YD + 2.8YE + 6YF + 6.67YG + 10YH + 5.33YI + 12YJ + 4YK + 5.33Y L+ 1.5YN + 4YO + 5.33YPThe completion time becomes the objective functionX(FIN) 19

Minimize X(FIN)

The other constraints of the crashing model remain the same.

Baja Burrito Restaurants –Baja Burrito Restaurants –Operating within a fixed budgetOperating within a fixed budget

91

Baja Burrito Restaurants –Baja Burrito Restaurants –Budget SpreadsheetBudget Spreadsheet

TOTAL PROJECT COST 225 PROJECT NORMAL COST 200COMPLETION TIME 23.3125 PROJECT CRASH COST 300

ACTIVITY NODECompletion

TimeStart Time

Finish Time

Amount Crashed

Cost of Crashing

Total Cost

Revisions/Approvals A 5 0 5 0 0 25Grade Land B 1 5 6 0 0 10Purchase Materials C 1.5 5 6.5 1.5 4 22Order Equipment D 2 5 7 0 0 8Order Furniture E 4 16.3125 20.3125 0 0 8Concrete Floor F 1 6.5 7.5 0 0 12Erect Frame G 4 7.5 11.5 0 0 20Install Electrical H 2 12 14 0 0 12Install Plumbing I 2.5 11.5 14 1.5 8 21Install Drywall/Roof J 2 14 16 0 0 10Bathrooms K 2 14 16 0 0 8Install Equipment L 1.5 16 17.5 1.5 8 22Finish/Paint Inside M 2.8125 17.5 20.3125 0.1875 1 11Tile Floors N 3 20.3125 23.3125 0 0 6Install Furniture O 3 20.3125 23.3125 1 4 12Finish/Paint Outside P 4 19.3125 23.3125 0 0 18

CRASHING ANALYSIS

92

7.11 PERT7.11 PERT//COSTCOST

• PERT/Cost helps management gauge progress PERT/Cost helps management gauge progress against scheduled time and cost estimates.against scheduled time and cost estimates.

• PERT/Cost is based on analyzing a segmented PERT/Cost is based on analyzing a segmented project. Each segment is a collection of work project. Each segment is a collection of work packages.packages. PROJECT

Work Package 1Activity 1Activity 2



93

– Once started, a work package is performed Once started, a work package is performed continuously until it is finished.continuously until it is finished. – The costs associated with a work package are spreadThe costs associated with a work package are spread evenly throughout its duration.evenly throughout its duration.

Work Package - AssumptionsWork Package - Assumptions

94

• For each work package determine:For each work package determine:– Work Package Forecasted Weekly cost =

Budgeted Total Cost for Work Package Expected Completion Time for Work Package (weeks)

– Value of Work to date = p(Budget for the work package)where p is the estimated percentage of the work package completed.

– Expected remaining completion time = (1 – p)(Original Expected Completion Time)

Monitoring Project progressMonitoring Project progress

95

• Completion Time AnalysisCompletion Time AnalysisUse the expected remaining completion time estimates, Use the expected remaining completion time estimates, to revise the project completion time.to revise the project completion time.

• Cost OverrunCost Overrun//Underrun AnalysisUnderrun AnalysisFor each work package (completed or in progress) calculateFor each work package (completed or in progress) calculate

Cost overrun =Cost overrun = [Actual Expenditures to Date] - [Value of Work to [Actual Expenditures to Date] - [Value of Work to

Date].Date].

Monitoring Project progressMonitoring Project progress

96

• A project may be found to be behind schedule, and or A project may be found to be behind schedule, and or experiencing cost overruns. experiencing cost overruns.

• Management seeks out causes such as:Management seeks out causes such as:– Mistaken project completion time and cost estimates.Mistaken project completion time and cost estimates.

– Mistaken work package completion times estimates and cost Mistaken work package completion times estimates and cost

estimates.estimates.

– Problematic departments or contractors that cause delays.Problematic departments or contractors that cause delays.

Monitoring Project Progress – Monitoring Project Progress – Corrective ActionsCorrective Actions

97

• Possible Corrective actions, to be taken whenever needed. Possible Corrective actions, to be taken whenever needed.

– Focus on uncompleted activities.Focus on uncompleted activities.

– Determine whether crashing activities is desirable.Determine whether crashing activities is desirable.

– In the case of cost underrun, channel more resources to problem In the case of cost underrun, channel more resources to problem

activities.activities.

– Reduce resource allocation to non-critical activities.Reduce resource allocation to non-critical activities.

Monitoring Project Progress – Monitoring Project Progress – Corrective ActionsCorrective Actions

98

• Tom Larkin is running for Mayor.Tom Larkin is running for Mayor.

• Twenty weeks before the election the campaign Twenty weeks before the election the campaign

remaining activities need to be assessed.remaining activities need to be assessed.

• If the campaign is not on target or not within budget, If the campaign is not on target or not within budget, recommendations for corrective actions are required.recommendations for corrective actions are required.

TOM LARKIN’s MAYORAL CAMPAIGNTOM LARKIN’s MAYORAL CAMPAIGN

99Work packages to focus on

Work Package Expenditures ($) StatusA Hire campaign staff 2,600 FinishB Prepare position paper 5,000 FinishC Recruit volunteers 3,000 FinishD Raise funds 5,000 FinishE File candidacy papers 700 FinishF Prepare campaign material 5,600 40% completeG Locate/staff headquarter 700 FinishH Run personal campaign 2,000 25% completeI Run media campaign 0 0% complete

MAYORAL CAMPAIGN –MAYORAL CAMPAIGN – Status Report Status Report

100

– The remaining network at the end of week 20.The remaining network at the end of week 20.

I9

Finish

F7.8

H15

(1-p)(original expected completion time)=(1-0.25)(20)=15

20+15=35

20+7.8=27.8

27.8+9=36.8 .8

The remaining activities are expected to take 0.8 weeks longer than the deadline of 36 weeks.

MAYORAL CAMPAIGN –MAYORAL CAMPAIGN – Completion Time AnalysisCompletion Time Analysis

101

Budgeted ValuesWork Total Total Percent EstimatedActual CostPackage Time Cost Completed Value Value OcerrunA 4 2000 100% 2,000 2,600 600B 6 3,000 100% 3,000 5,000 2000C 4 4,500 100% 4,500 3,000 -1500D 6 2,500 100% 2,500 5,000 2,500E 2 500 100% 500 700 200F 13 13,000 40% 5200 5,600 400G 1 1,500 100% 1,500 700 -800H 20 6,000 25% 1,500 2,000 -500I 9 7,000 0% 0 0 0

Total 40,000 20,700 24,600 3,900

Estimated work value to date=(13,000)(0.40)=$5,200

Cost overrun = 5600 - 5200 = 400

MAYORAL CAMPAIGN –MAYORAL CAMPAIGN –Project Cost ControlProject Cost Control

102

MAYORAL CAMPAIGN –MAYORAL CAMPAIGN –Results SummaryResults Summary

• The project is currently .8 weeks behind scheduleThe project is currently .8 weeks behind schedule• There is a cost over-run of $3900.There is a cost over-run of $3900.• The remaining completion time for uncompleted work The remaining completion time for uncompleted work

packages is:packages is:– Work package F: 7.8 weeks,Work package F: 7.8 weeks,– Work package H: 15 weeks,Work package H: 15 weeks,– Work package I: 9 weeks.Work package I: 9 weeks.

• Cost over-run is observed in Cost over-run is observed in – Work package F: $400,Work package F: $400,– Work package H: $500.Work package H: $500.

103

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ch05 - project.ppt

Documents

project control

introductiona project

earliest completion

scheduling activities

minimal time

minimum time

projects activities

projects completion