Project Management - jeryfarel · PDF fileThe Elements of Project Management The Project Network Probabilistic Activity Times Chapter Topics Activity-on-Node Networks and Microsoft

Introduction to Management Science

(8th Edition, Bernard W. Taylor III)

Chapter 13

Chapter 13 - Project Management 1

Project Management

The Elements of Project Management

The Project Network

Probabilistic Activity Times

Chapter Topics


Activity-on-Node Networks and Microsoft Project

Project Crashing and Time-Cost Trade-Off

Formulating the CPM/PERT Network as a Linear Programming Model


Uses networks for project analysis.

Networks show how projects are organized and are used to determine time duration for completion.

Overview

determine time duration for completion.

Network techniques used are:

CPM (Critical Path Method)

PERT (Project Evaluation and Review Technique)

Developed during late 1950’s.


Elements of Project Management

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

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

Primary elements of Project Management to be discussed:

Project Team


Project Planning

Project Control

Project team typically consists of a group of individuals from various areas in an organization and often includes outside consultants.

The Elements of Project ManagementThe Project Team

consultants.

Members of engineering staff often assigned to project work.

Most important member of project team is the project manager.

Project manager is often under great pressure because of


Project manager is often under great pressure because of uncertainty inherent in project activities and possibility of failure.

Project manager must be able to coordinate various skills of team members into a single focused effort.

A branch reflects an activity of a project.

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

The Elements of Project ManagementThe Project Network

referred to as events.

Branches in the network indicate precedence relationships.

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


Figure 13.2Network for Building a House

Network aids in planning and scheduling.

Time duration of activities shown on branches:

The Project NetworkPlanning and Scheduling


Figure 13.3Network for Building a House with Activity Times

Activities can occur at the same time (concurrently).

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

The Project NetworkConcurrent Activities

reflects no passage of time.

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


Figure 13.4Expanded Network for Building a House Showing Concurrent Activities

The Project NetworkPaths Through a Network


Table 8.1Paths Through the House-Building Network

The critical path is the longest path through the network; the minimum time the network can be completed. In Figure 13.5:

The Project NetworkThe Critical Path (1 of 2)

13.5:

Path A: 1 2 3 4 6 7, 3 + 2 + 0 + 3 + 1 = 9 months

Path B: 1 2 3 4 5 6 7, 3 + 2 + 0 + 1 + 1 + 1 = 8 months

Path C: 1 2 4 6 7, 3 + 1 + 3 + 1 = 8 months


Path C: 1 2 4 6 7, 3 + 1 + 3 + 1 = 8 months

Path D: 1 2 4 5 6 7, 3 + 1 + 1 + 1 + 1 = 7 months

The Project NetworkThe Critical Path (2 of 2)


Figure 13.6Alternative Paths in the Network

ES is the earliest time an activity can start. ESij = Maximum (EFi)

EF is the earliest start time plus the activity time. EF = ES

The Project NetworkActivity Scheduling – Earliest Times

EF is the earliest start time plus the activity time. EFij = ESij

+ tij


Figure 13.7Earliest Activity Start and Finish Times

LS is the latest time an activity can start without delaying critical path time. LSij = LFij - tijLF is the latest finish time. LF = Minimum (LS )

The Project NetworkActivity Scheduling – Earliest Times

LF is the latest finish time. LFij = Minimum (LSj)


Figure 13.8Latest Activity Start and Finish Times

Slack is the amount of time an activity can be delayed without delaying the project.

Slack Time exists for those activities not on the critical path

The Project NetworkActivity Slack

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

Shared Slack is slack available for a sequence of activities.


Figure 13.9Earliest and Latest Activity Start and Finish Times

Slack, Sij, computed as follows: Sij = LSij - ESij or Sij = LFij -EFij

The Project NetworkCalculating Activity Slack Time (1 of 2)


Figure 13.10Activity Slack

The Project NetworkCalculating Activity Slack Time (2 of 2)


Table 8.2Activity Slack

Activity time estimates usually can not be made with certainty.

PERT used for probabilistic activity times.


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:

mean (expected time):6

b4mat


mean (expected time):

variance:

6t

2

6a-b

v

Probabilistic Activity TimesExample (1 of 3)


Figure 13.11Network for Installation Order Processing System



Table 8.3Activity Time Estimates for Figure 13.11



Figure 13.12Network with Mean Activity Times and Variances

Probabilistic Activity TimesEarliest and Latest Activity Times and Slack


Figure 13.13Earliest and Latest Activity Times

Probabilistic Activity TimesEarliest and Latest Activity Times and Slack


Table 8.4Activity Earliest and Latest Times and Slack

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

The project variance is the sum of the variances of the

Probabilistic Activity TimesExpected Project Time and Variance

The project variance is the sum of the variances of the critical path activities.

The expected project time is assumed to be normally distributed (based on central limit theorem).

In example, expected project time (tp) and variance (vp) interpreted as the mean () and variance (2) of a normal


interpreted as the mean () and variance ( ) of a normal distribution:

= 25 weeks

2 = 6.9 weeks

Critical Path Activity Variance

1 3 3 5 5 7 7 9

1 1/916/9 4

total 62/9

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

Probability Analysis of a Project Network (1 of 2)

the mean.

Value is used to find corresponding probability in Table A.1, Appendix A.


Probability Analysis of a Project Network (2 of 2)


Figure 13.14Normal Distribution of Network Duration

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)

weeks or less: (.5000 + .4713) = .9713.

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




Figure 13.15Probability the Network Will Be Completed in 30 Weeks or Less

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.


of .3729 in Table A.1, appendix A.

• Probability that customer will be retained is .1271




Figure 13.16Probability the Network Will Be Completed in 22 Weeks or Less

Probability Analysis of a Project NetworkCPM/PERT Analysis with QM for Windows


Exhibit 13.1

The project networks developed so far have used the “activity-on-arrow” (AOA) convention.

“Activity-on-node” (AON) is another method of creating a

Activity-on-Node Networks and Microsoft Project

“Activity-on-node” (AON) is another method of creating a network diagram.

The two different conventions accomplish the same thing, but there are a few differences.

An AON diagram will often require more nodes than an AOA diagram.


An AON diagram does not require dummy activities because two “activities” will never have the same start and end nodes.

Microsoft Project handles only AON networks.

This node includes the

Activity-on-Node Networks and Microsoft ProjectNode Structure

This node includes the activity number in the upper left-hand corner, the activity duration in the lower left-hand corner, and the earliest start and finish times, and latest start and finish times in the four boxes on the right side of


four boxes on the right side of the node.

Figure 13.17Activity-on-Node Configuration

Activity-on-Node Networks and Microsoft ProjectAON Network Diagram


Figure 13.18House-Building Network with AON

Activity-on-Node Networks and Microsoft ProjectMicrosoft Project (1 of 4)


Exhibit 13.2



Exhibit 13.3



Exhibit 13.4



Exhibit 13.5

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

Doing this however increases project cost.

Project Crashing and Time-Cost Trade-Off Definition

Doing this however 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.


normal activity time.

Crashing achieved by devoting more resources to crashed activities.

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


Figure 13.19Network for Constructing a House


Crash cost and crash time have linear relationship: total crash cost/total crash time = $2000/5 = $400/wk


Figure 13.20Time-Cost Relationship for Crashing Activity 12



Table 8.5Normal Activity and Crash Data for the Network in Figure 13.19



Figure 13.21Network with Normal Activity Times and Weekly Activity Crashing Costs


As activities are crashed, the critical path may change and several paths may become critical.


Figure 13.22Revised Network with Activity 12 Crashed

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


Exhibit 13.6

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. 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.


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


Figure 13.23A Time-Cost Trade-Off

General linear programming model:

The objective is to determine the earliest time the project can be completed (i.e., the critical path time).

The CPM/PERT Network Formulating as a Linear Programming Model

General linear programming model:

Minimize Z = xi

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

Where:


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

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

subject to:

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

x2 - x1 12x3 - x2 8x4 - x2 4x4 - x3 0x5 - x4 4x6 - x4 12


6 4

x6 - x5 4x7 - x6 4xi, xj 0

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


Figure 13.24CPM/PERT Network for the House-Building Project with Earliest Event Times

The CPM/PERT Network Example Problem Solution with Excel (1 of 4)


Exhibit 13.7



Exhibit 13.8



Exhibit 13.9



Exhibit 13.10

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

Probability Analysis of a Project NetworkExample Problem – Model Formulation

yij = amount of time by which activity i j is crashed

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

subject to:y12 5 y12 + x2 - x1 12 x7 30 y23 3 y23 + x3 - x2 8 y67 1y 1 y + x - x 4 x + x - x


y24 1 y24 + x4 - x2 4 x67 + x7 - x6 4

y34 0 y34 + x4 - x3 0 xj, yij 0y45 3 y45 + x5 - x4 4y46 3 y46 + x6 - x4 12y56 3 y56 + x6 - x5 4

Probability Analysis of a Project NetworkExample Problem – Excel Solution (1 of 3)


Exhibit 13.11



Exhibit 13.12



Exhibit 13.13

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

PERT Project Management Example ProblemProblem Statement and Data (1 of 2)

project will be completed in 28 days or less.


Time Estimates (weeks)

PERT Project Management Example ProblemProblem Statement and Data (2 of 2)

Time Estimates (weeks)

Activity a m b1 21 32 32 43 4

57314

810536

1713758


3 43 54 5

433

634

835

b4ma 2

a-b

PERT Project Management Example ProblemSolution (1 of 4)

Step 1: Compute the expected activity times and variances.

6b4mat

2

6a-b

v

Activity t v

1 21 32 3

9105

41

4/9


2 32 43 43 54 4

53634

4/94/94/90

1/9


Step 2: Determine the earliest and latest times at each node.



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

Critical path (activities with no slack): 1 2 3 4 5Critical path (activities with no slack): 1 2 3 4 5

Expected project completion time (tp): 24 days

Variance: v = 4 + 4/9 + 4/9 + 1/9 = 5 days



Step 4: Determine the Probability That the Project Will be Completed in 28 days or less.

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

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



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Project Management - jeryfarel · PDF fileThe Elements of Project Management The Project Network Probabilistic Activity Times Chapter Topics Activity-on-Node Networks and Microsoft