8-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Project Management Chapter 8.

8-1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Project Management

Chapter 8

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■The Elements of Project Management

■CPM/PERT Networks

■Probabilistic Activity Times

■Formulating the CPM/PERT Network as a Linear Programming Model

Chapter Topics

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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■ Network representation is useful for project analysis.

■ Project examples: Construction of a building, organization of a conference, installation of a computer system etc.

■ Networks show how project activities are organized and are used to determine time duration of projects.

■ Network techniques used are:

▪ CPM (Critical Path Method)

▪ PERT (Project Evaluation and Review Technique)

■ Developed independently during late 1950’s.

Overview


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Elements of Project Management

■Management is generally perceived as concerned with planning, organizing, and control of an ongoing process or activity.

■Project Management is concerned with control of an activity for a relatively short period of time after which management effort ends.


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Project Planning

■Objectives

■Project Scope

■Contract Requirements

■Schedules

■Resources

■Control

■Risk and Problem Analysis


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Project Planning

■All activities (steps) of the project should

be identified.■The sequential relationships of the

activities (which activity comes first, which follows, etc.) is identified by precedence relationships.

■Steps of project planning:■Make time estimates for activities,

determine project completion time.■Compare project schedule objectives,

determine resource requirements. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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Elements of Project ManagementProject Scheduling■ Project Schedule : Timely completion of

project.

■ Schedule development steps:1. Define activities, 2. Sequence activities,3. Estimate activity times,4. Construct schedule.

■ Gantt chart and CPM/PERT techniques can be useful.

■ Computer software packages available, e.g. Microsoft Project.


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Elements of Project ManagementGantt Chart (1 of 2)

■ Popular, traditional technique, also known as a bar chart -developed by Henry Gantt (1914).

■ Used in CPM/PERT for monitoring work progress.

■ A visual display of project schedule showing activity start and finish times and where extra time is available.

■ Suitable for projects with few activities and precedence relationships.


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Elements of Project ManagementGantt Chart (2 of 2)

Figure 8.4 A Gantt chart



■ A branch reflects an activity of a project.

■ A node represents the beginning and end of activities, referred to as events.

■ Branches in the network indicate precedence relationships.

■ When an activity is completed at a node, it has been realized.

The Project NetworkCPM/PERT

Activity-on-Arc (AOA) Network


■ Time duration of activities shown on branches.

■ Activities can occur at the same time (concurrently).

■ A dummy activity shows a precedence relationship but reflects no passage of time.

■ Two or more activities cannot share the same start and end nodes.

The Project NetworkConcurrent Activities

Figure 8. 7 A Dummy Activity

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The Project NetworkHouse Building Project Data

No. Activity Activity Predecessor Duration (Months)

1. Design house and - 3 obtain financing

2. Lay foundation 1 2

3. Order Materials 1 1

4. Build house 2, 3 3

5. Select paint 2, 3 1

6. Select carpet 5 1

7. Finish work 4, 6 1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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The Project NetworkAOA Network for House Building Project

Figure 8.6 Expanded Network for Building a House Showing Concurrent Activities


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The Project NetworkAON Network for House Building Project

Activity-on-Node (AON) Network A node represents an activity, with its label and time shown on the node The branches show the precedence relationships Convention used in Microsoft Project software


Figure 8.8

Label

Duration

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The Project NetworkPaths Through a Network

Table 8.1Paths Through the House-Building

NetworkCopyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Path EventsA 1247

B 12567

C 1347

D 13567

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The critical path is the longest path through the network; the minimum time the network can be completed. From Figure 8.8:

Path A: 1 2 4 7 3 + 2 + 3 + 1 = 9

months

Path B: 1 2 5 6 7 3 + 2 + 1 + 1

+ 1= 8 months

Path C: 1 3 4 7 3 + 1 + 3 + 1 = 8

months

Path D: 1 3 5 6 7 3 + 1 + 1 + 1

+ 1 = 7 months

The Project NetworkThe Critical Path



The Project NetworkActivity Start Times

Figure 8.9 Activity start time

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The Project NetworkActivity-on-Node Configuration

Figure 8.10 Activity-on-Node ConfigurationCopyright © 2010 Pearson Education, Inc. Publishing

as Prentice Hall


■ ES is the earliest time an activity can start: ES = Maximum (EF)

■ EF is the earliest start time plus the activity time: EF = ES + t

The Project NetworkActivity Scheduling : Earliest Times

Figure 8.11 Earliest activity start and finish times


■ LS is the latest time an activity can start without delaying critical path time: LS = LF - t

■ LF is the latest finish time. LF = Minimum (LS)

The Project NetworkActivity Scheduling : Latest Times

Figure 8.12 Latest activity start and finish times


Slack is the amount of time an activity can be delayed without delaying the project: S = LS – ES = LF - EF

Slack Time exists for those activities not on the critical path for which the earliest and latest start times are not equal.

The Project NetworkActivity Slack Time (1 of 2)

Table 8.2

*Critical path

Activity

LS ES LF EF Slack, S

*1 0 0 3 3 0

*2 3 3 5 5 0

3 4 3 5 4 1

*4 5 5 8 8 0

5 6 5 7 6 1

6 7 6 8 7 1

*7 8 8 9 9 0


The Project NetworkActivity Slack Time (2 of 2)

Figure 8.13 Activity slack

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■Activity time estimates usually cannot be made with certainty.

■PERT used for probabilistic activity times.

■In PERT, three time estimates are used: most likely time (m), the optimistic time (a), and the pessimistic time (b).

■These provide an estimate of the mean and variance of a beta distribution:

variance:

mean (expected time):

6b 4m a t

2

6a - b

v

Probabilistic Activity Times



Probabilistic Activity TimesExample (1 of 3)

Figure 8.14 Network for Installation Order Processing System



Table 8.3 Activity Time Estimates for Figure 8.14



Figure 8.15 Earliest and Latest Activity Times


■ Expected project time is the sum of the expected times of the critical path activities.

■ Project variance is the sum of the critical path activities’ variances

■ The expected project time is assumed to be normally distributed.

■ Epected project time (tp) and variance (vp) interpreted as the mean () and variance (2) of a normal distribution: = 25 weeks

2 = 62/9

= 6.9 (weeks)2

Probabilistic Activity TimesExpected Project Time and Variance

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■Using the normal distribution, probabilities are determined by computing the number of standard deviations (Z) a value is from the mean.

■The Z value is used to find corresponding probability in Table A.1, Appendix A.

Probability Analysis of a Project Network (1 of 2)


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Probability Analysis of a Project Network (2 of 2)

Figure 8.16 Normal Distribution of Network Duration


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What is the probability that the new order processing system will be ready by 30 weeks?

µ = 25 weeks

2 = 6.9 = 2.63 weeksZ = (x-)/ = (30 -25)/2.63 = 1.90

Z value of 1.90 corresponds to probability of .4713 in Table A.1, Appendix A. Probability of completing project in 30 weeks or less: (.5000 + .4713) = .9713.

Probability Analysis of a Project NetworkExample 1 (1 of 2)


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Figure 8.17 Probability the Network Will Be Completed in 30 Weeks or LessCopyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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■ A customer will trade elsewhere if the new ordering system is not working within 22 weeks. What is the probability that she will be retained?

Z = (22 - 25)/2.63 = -1.14

■ Z value of 1.14 (ignore negative) corresponds to probability of .3729 in Table A.1, appendix A.

■ Probability that customer will be retained is (0.5 - .3729) = .1271



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Figure 8.18 Probability the Network Will Be Completed in 22 Weeks or LessCopyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall


Given this network and the data on the following slide, determine the expected project completion time and variance, and the probability that the project will be completed in 28 days or less.

Example Problem Problem Statement and Data (1 of 2)


Example ProblemProblem Statement and Data (2 of 2)


6b 4m a t

2

6a - b

v

Example Problem Solution (1 of 4)Step 1: Compute the expected activity times and

variances.


Example Problem Solution (2 of 4)Step 2: Determine the earliest and latest activity times

& slacks

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Example Problem Solution (3 of 4)

Step 3: Identify the critical path and compute expected completion time and variance.

Critical path (activities with no slack): 1 3 5 7

Expected project completion time: tp = 9+5+6+4 = 24 days

Variance: vp = 4 + 4/9 + 4/9 + 1/9 = 5 (days)2


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Example Problem Solution (4 of 4)Step 4: Determine the Probability That the Project Will

be Completed in 28 days or less (µ = 24, = 5)

Z = (x - )/ = (28 -24)/5 = 1.79

Corresponding probability from Table A.1, Appendix A, is .4633 and P(x 28) = .4633 + .5 = .9633.


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■ Project duration can be reduced by assigning more resources to project activities.

■ However, doing this increases project cost.

■ Decision is based on analysis of trade-off between time and cost.

■ Project crashing is a method for shortening project duration by reducing one or more critical activities to a time less than normal activity time.

Project Crashing and Time-Cost Trade-Off Overview



Project Crashing and Time-Cost Trade-Off Example Problem (1 of 5)

Figure 8.19 The Project Network for Building a House

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Figure 8.20Copyright © 2010 Pearson Education, Inc. Publishing as

Prentice Hall

Crash cost & crash time have a linear relationship:

$2000

5 $400 /

Total Crash Cost

Total Crash Time weekswk


Table 8.4



Figure 8.21 Network with Normal Activity Times and Weekly Crashing Costs



Figure 8.22Revised Network with Activity 1 Crashed

Project Crashing and Time-Cost Trade-Off Example Problem (5 of 5)As activities are crashed, the critical path

may change and several paths may become critical.

8-46Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit

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Project Crashing and Time-Cost Trade-Off Project Crashing with QM for Windows

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Project Crashing and Time-Cost Trade-Off General Relationship of Time and Cost (1 of 2)■Project crashing costs and indirect costs

have an inverse relationship.

■Crashing costs are highest when the project is shortened.

■Indirect costs increase as the project duration increases.

■Optimal project time is at minimum point on the total cost curve.


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Project Crashing and Time-Cost Trade-Off General Relationship of Time and Cost (2 of 2)

Figure 8.23The Time-Cost Trade-Off


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General linear programming model with AOA convention:

Minimize Z = xi

subject to: xj - xi tij for all activities i j xi, xj 0

Where: xi = earliest event time of node ixj = earliest event time of node jtij = time of activity i j

The objective is to minimize the project duration (critical path time).

The CPM/PERT Network Formulating as a Linear Programming Model

i


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The CPM/PERT Network Example Problem Formulation and Data (1 of 2)

Figure 8.24


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Minimize Z = x1 + x2 + x3 + x4 + x5 + x6 + x7

subject to:

x2 - x1 12x3 - x2 8x4 - x2 4x4 - x3 0x5 - x4 4x6 - x4 12x6 - x5 4x7 - x6 4xi, xj 0

The CPM/PERT Network Example Problem Formulation and Data (2 of 2)


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Minimize Z = $400y12 + 500y23 + 3000y24 + 200y45 + 7000y46 + 200y56 + 7000y67

subject to:y12 5 y12 + x2 - x1 12 x7 36 y23 3 y23 + x3 - x2 8 xi, yij ≥ 0y24 1 y24 + x4 - x2 4y34 0 y34 + x4 - x3 0y45 3 y45 + x5 - x4 4y46 3 y46 + x6 - x4 12y56 3 y56 + x6 - x5 4y67 1 x67 + x7 - x6 4

xi = earliest event time of node Ixj = earliest event time of node jyij = amount of time by which activity i j is crashed

Project Crashing with Linear ProgrammingExample Problem – Model Formulation


Objective is to minimize the cost of crashing

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Minimize Z = $400y12 + 500y23 + 3000y24 + 200y45 + 7000y46 + 200y56 + 7000y67

subject to:y12 5 x2 - x1 12-y12 x7 36 y23 3 x3 - x2 8-y23 xi, yij ≥ 0y24 1 x4 - x2 4-y24

y34 0 x4 - x3 0-y34 y45 3 x5 - x4 4-y45 y46 3 x6 - x4 12-y46 y56 3 x6 - x5 4-y56

y67 1 x7 - x6 4-x67

xi = earliest event time of node Ixj = earliest event time of node jyij = amount of time by which activity i j is crashed

Project Crashing with Linear ProgrammingExample Problem – Model Formulation


Objective is to minimize the cost of crashing

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Activity on Activity Activity onNode (AON) Meaning Arc (AOA)

A comes before B, which comes before C

(a) A B CBA C

A and B must both be completed before C can start

(b)

A

CC

B

A

B

B and C cannot begin until A is completed

(c)

B

A

CA

B

C

A Comparison of AON and AOA Network Conventions

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C and D cannot begin until both A and B are completed

(d)

A

B

C

D B

A C

D

C cannot begin until both A and B are completed; D cannot begin until B is completed. A dummy activity is introduced in AOA

(e)

CA

B D

Dummy activity

A

B

C

D


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B and C cannot begin until A is completed. D cannot begin until both B and C are completed. A dummy activity is again introduced in AOA.

(f)

A

C

DB A B

C

D

Dummy activity


8-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Project Management Chapter 8.

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