Program evaluation and review technique (PERT)

Prepared By

Luay Kadhum Hameed

2009

CHAPTER 1

Introduction

A project defines a combination of interrelated activities that

must be executed in a certain order before the entire task can be

completed. The activities are interrelated in a logic sequences in the sense

that some activities cannot start until others are completed .An activity in

a project is usually viewed as a job requiring time and resources for its

completion .In general ,a project is a one –time effort ;that is, the same

sequence of activities may not be repeated in the future.

In the past ,the scheduling of a project (over time) was done with little

specifies the start and finish for each activity on a horizontal time scale .

Its disadvantage is that the interdependency between the different

activities(which mainly controls the progress of the project)cannot be

determined from the bar chart .The growing complexities of today's

project have demanded more systematic and more effective planning

techniques with the objective of optimizing the efficiency of executing

the project .Efficiency here implies effecting almost reduction in the time

required to complete the project while accounting for the economic

feasibility of using available resources .

Project management has evolved as a new field with the development of

two analytic technique for planning, scheduling, and controlling of

projects.

These are the critical path method (CPM) and the project evaluation and

review technique (PERT) . the two techniques were developed by two

different groups almost simultaneously (1956- 1958)CPM was first

developed by E.Idu Pout de Nemours and company as an application to

construction projects and was later extended to a more advanced status by

Mauchly associates . PERT , on the other hand was developed for the

U.S. Navy by a consulting firm for scheduling the research and

development activities for the Polaris missile program.

PERT and CPM are basically time –oriented methods in the sense that

they both lead to the determination of a time schedule . Although the two

methods were developed independently , they are strikingly similar .

Perhaps the most important difference is that originally the time estimates

for the activities were assumed deterministic in CPM and probabilistic in

PERT. Today , PERT and CPM actually comprise one technique and the

differences , if any,are only historical . Consequently , both techniques

will be referred to as "project scheduling" techniques.

Project scheduling by PERT-CPM consist of three basic phase, planning,

Scheduling, and controlling.

The planning phase is initiated by breaking down the project into distinct

activities . The time estimates for these activities are then determined and

a network (or arrow)diagram is constructed with each of its arcs(arrows)

representing an activity. The entire arrow diagram gives a graphic

representation of the interdependencies between the activities of the

project. The construction of the arrow diagram as a planning phase has

the advantage of studying the different jobs in detail .perhaps suggesting

improvement before the project is actually executed . More important will

be its use to develop a schedule for the project.

The ultimate objective of the Scheduling phase is to construct a time chart

showing the start and finish times for each activity as well as its

relationship to other activities in the project. In addition, the Schedule

must pinpoint the critical (in view of time) activities that require special

attention if the project is to be completed on time. For the non critical

activities the schedule must show the amount of slack or float time that

can be used advantageously when such activities are delayed or when

limited resources are to be used effectively .

The final phase in project management is project control .This include the

use of the arrow diagram and the time chart for making periodic progress

reports. The network may thus be updated and analyzed and, if necessary,

a new schedule is determined for the remaining portion of the project.

CHAPTER 2Project management

Project management is concerned with the overall planning and co-ordination of a project from conception to completion aimed at meeting the stated requirements and ensuring completion on time, within cost and to required quality standards. Project management is normally reserved for focused, non-repetitive, time-limited activities with some degree of risk and that are beyond the usual scope of operational activities for which the organization is responsible.A project is a temporary endeavour involving a connected sequence of activities and a range of resources, which is designed to achieve a specific and unique outcome and which operates within time, cost and quality constraints and which is often used to introduce change.

Characteristic of a project

A unique, one-time operational activity or effort Requires the completion of a large number of interrelated activities Established to achieve specific objective Resources, such as time and/or money, are limited Typically has its own management structure Need leadership

project management

The application of a collection of tools and techniques to direct the use of diverse resources towards the accomplishment of a unique, complex, one time task within time, cost and quality constraints.

Its origins lie in World War II, when the military authorities used the techniques of operational research to plan the optimum use of resources.

One of these techniques was the use of networks to represent a system of related activities

Project Management Process

Project planning Project scheduling Project control Project team

-made up of individuals from various areas and departments within a company

Matrix organization-a team structure with members from functional areas, depending on skills required Project Manager-most important member of project team Scope statement

-a document that provides an understanding, justification, and expected result of a project

Statement of work-written description of objectives of a project Organizational Breakdown Structure

-a chart that shows which organizational units are responsible for work items Responsibility Assignment Matrix-shows who is responsible for work in a project

Work breakdown structure

A method of breaking down a project into individual elements ( components, subcomponents, activities and tasks) in a hierarchical structure which can be scheduled and cost

It defines tasks that can be completed independently of other tasks, facilitating resource allocation, assignment of responsibilities and measurement and control of the project

It is foundation of project planning It is developed before identification of dependencies and

estimation of activity durations It can be used to identity the tasks in the CPM and PERT

Project Planning

• Resource Availability and/or Limits– Due date, late penalties, early completion incentives– Budget

• Activity Information– Identify all required activities– Estimate the resources required (time) to complete each

activity– Immediate predecessor(s) to each activity needed to create

interrelationships

Project Scheduling and Control Techniques 1.Critical Path Method (CPM) 2. Program Evaluation and Review Technique (PERT)

Project Network

• Network analysis is the general name given to certain specific techniques which can be used for the planning, management and control of projects

• Use of nodes and arrows Arrows An arrow leads from tail to head directionally

– Indicate ACTIVITY, a time consuming effort that is required to perform a part of the work.

Nodes n A node is represented by a circle- Indicate EVENT, a point in time where one or more

activities start and/or finish

• Activity– A task or a certain amount of work required in the project– Requires time to complete– Represented by an arrow

• Dummy Activity– Indicates only precedence relationships– Does not require any time of effort

• Event– Signals the beginning or ending of an activity– Designates a point in time– Represented by a circle (node)

• Network– Shows the sequential relationships among activities using

nodes and arrows Activity-on-node (AON)

nodes represent activities, and arrows show precedence relationships

Activity-on-arrow (AOA)arrows represent activities and nodes are events for points in time

32 0

1

31 1

11 2 4 6 7

3

5

Lay foundation

Design house and obtain

financing

Order and receive

materials

Dummy

Finish work

Select carpet

Select paint

Build house

Fig (2-1) AOA Project Network for House

Fig (2-2) AON Project Network for House

Situations in network diagram

Fig (2-3) A must finish before either B or C can start

13

22

43

31 5

1

61

71Start

Design house and obtain financing

Order and receive materials Select paint

Select carpet

Lay foundations Build house

Finish work

AB

C

A

B

C

Fig (2-4) both A and B must finish before C can start

Fig (2-5) both A and C must finish before either of B or D can start

Fig (2-6) A must finish before B can start both A and C must finish before D can start

D

C

B

A

A

C

B

D

Dummy

Fig (2-7) Concurrent Activities

CHAPTER 3

CPM - Critical Path Method

History of CPM

E I Du Pont de Nemours & Co. (1957) for construction of new chemical plant and maintenance shut-down

Deterministic task times Activity-on-node network construction Repetitive nature of jobs

Planning a project usually involves dividing it into a number of small tasks that can be assigned to individuals or teams. The project’s schedule depends on the duration of these tasks and the sequence in which they are arranged. This sequence can be driven by several factors: customer deadlines, availability of personnel or resources, and dependencies among tasks. DuPont developed a Critical Path Method (CPM) designed to address the challenge of shutting down chemical plants for maintenance and then restarting the plants once the maintenance had been completed.

2 3

Lay Lay foundationfoundation

Order Order materialmaterial

((aa))Incorrect Incorrect precedence precedence relationshiprelationship

((bb))Correct Correct precedence precedence relationshiprelationship

3

42

DummDummyy

LayLay foundatiofoundatio

nn

Order Order materialmaterial

11

22 00

Complex project require a series of activities, some of which must be performed sequentially and others that can be performed in parallel with 0ther activities. This collection of series and parallel tasks can be modeled as a network. CPM models the activities and events of a project as a network. Activities are shown as nodes on the network and events that signify the beginning or ending of activities are shown as arcs or lines between the nodes.The Figure below shows an example of a CPM network diagram:

Fig (3-1) CPM network diagramSteps in CPM Project Planning

1. Specify the individual activities. 2. Determine the sequence of those activities. 3. Draw a network diagram. 4. Estimate the completion time for each activity. 5. Identify the critical path (longest path through the network) 6. Update the CPM diagram as the project progresses.

1. Specify the individual activities

All the activities in the project are listed. This list can be used as the basis for adding sequence and duration information in later steps.

2. Determine the sequence of the activities

Some activities are dependent on the completion of other activities. A list of the immediate predecessors of each activity is useful for constructing the CPM network diagram.

3. Draw the Network Diagram

Once the activities and their sequences have been defined, the CPM diagram can be drawn. CPM originally was developed as an activity on node network.

4. Estimate activity completion time

The time required to complete each activity can be estimated using past experience. CPM does not take into account variation in the completion time.

5. Identify the Critical Path

The critical path is the longest-duration path through the network. The significance of the critical path is that the activities that lie on it cannot be delayed without delaying the project. Because of its impact on the entire project, critical path analysis is an important aspect of project planning.

The critical path can be identified by determining the following four parameters for each activity:

• ES - earliest start time: the earliest time at which the activity can start given that its precedent activities must be completed first. • EF - earliest finish time, equal to the earliest start time for the activity plus the time required to complete the activity. • LF - latest finish time: the latest time at which the activity can be completed without delaying the project. • LS - latest start time, equal to the latest finish time minus the time required to complete the activity.

The slack time for an activity is the time between its earliest and latest start time, or between its earliest and latest finish time. Slack is the amount of time that an activity can be delayed past its earliest start or earliest finish without delaying the project.

The critical path is the path through the project network in which none of the activities have slack, that is, the path for which ES=LS and EF=LF for all activities in the path. A delay in the critical path delays the project. Similarly, to accelerate the project it is necessary to reduce the total time required for the activities in the critical path.

6. Update CPM diagram

As the project progresses, the actual task completion times will be known and the network diagram can be updated to include this information. A new critical path may emerge, and structural changes may be made in the network if project requirements change.

CPM calculation

• Path – A connected sequence of activities leading from the starting

event to the ending event• Critical Path

– The longest path (time); determines the project duration• Critical Activities

– All of the activities that make up the critical path

Forward Pass

The forward pass goes from the initial task (the task with no predecessors) to the final task (the one with no successors), visiting every task in every path and setting the ES and EF dates on the tasks. The algorithm is similar to graph theory’s depth-first search, except that the forward pass follows every path from initial to final task, while depth-first search stops when it arrives at a task that it’s already visited. When the forward pass arrives at a task, it may change that task’s ES and EF dates, and that change must be carried forward to the final task. During the forward pass, a task may be visited several times as different paths through the network are followed. A task’s ES is determined by the predecessor task with the latest EF, since a task can’t start until all of its predecessors have finished.

• Earliest Start Time (ES)– earliest time an activity can start – ES = maximum EF of immediate predecessors

• Earliest finish time (EF)– earliest time an activity can finish– earliest start time plus activity time EF= ES + t

Backward Pass

he backward pass goes from the final task to the initial task, visiting every task in every path and setting the LS and LF dates on the tasks. It’s similar to the forward pass in that it arrives at a task, it may change that task’s LS and LF dates, and that change must be carried back to the initial task. The difference is: • The forward pass sets the task’s latest ES, as determined by the EFs of its predecessors • The backward pass sets the task’s earliest LF, as determined by the LSs of its successors. The reason for the backward pass’s rule for setting LF is not as obvious as the forward pass’s rule. Any start date, ES or LS, must be after the corresponding finish dates of all of the task’s predecessors. To maintain this consistency, the backward pass must set a task’s LF to a value that’s earlier than the LS of any of the task’s successors. Latest Start Time (LS)

Latest time an activity can start without delaying critical path time LS= LF - t

Latest finish time (LF) latest time an activity can be completed without delaying critical path time LS = minimum LS of immediate predecessorsCPM analysis

• Draw the CPM network• Analyze the paths through the network• Determine the float for each activity

– Compute the activity’s float float = LS - ES = LF - EF

– Float is the maximum amount of time that this activity can be delay in its completion before it becomes a critical activity, i.e., delays completion of the project

• Find the critical path is that the sequence of activities and events where there is no “slack” i.e.. Zero slack

– Longest path through a network• Find the project duration is minimum project completion time

CHAPTER 4Program Evaluation and Review Technique (PERT)

U S Navy (1958) for the POLARIS missile program Multiple task time estimates (probabilistic nature) Activity-on-arrow network construction Non-repetitive jobs

The Program Evaluation and Review Technique (PERT) is a network model that allows for randomness in activity completion times. PERT was developed in the late 1950's for the U.S. Navy's Polaris project having thousands of contractors. It has the potential to reduce both the time and cost required to complete a project.

The Network Diagram

In a project, an activity is a task that must be performed and an event is a milestone marking the completion of one or more activities. Before an activity can begin, all of its predecessor activities must be completed. Project network models represent activities and milestones by arcs and nodes.

PERT is typically represented as an activity on arc network, in which the activities are represented on the lines and milestones on the nodes. The Figure 2-1 shows a simple example of a PERT diagram.

Figure(4-1) PERT diagram.

The milestones generally are numbered so that the ending node of an activity has a higher number than the beginning node. Incrementing the numbers by 10 allows for new ones to be inserted without modifying the numbering of the entire diagram. The activities in the above diagram are labeled with letters along with the expected time required to complete the activity.

Steps in the PERT Planning Process

PERT planning involves the following steps:

1. Identify the specific activities and milestones. 2. Determine the proper sequence of the activities. 3. Construct a network diagram. 4. Estimate the time required for each activity. 5. Determine the critical path. 6. Update the PERT chart as the project progresses.

1. Identify activities and milestones

The activities are the tasks required to complete the project. The milestones are the events marking the beginning and end of one or more activities.

2. Determine activity sequence

This step may be combined with the activity identification step since the activity sequence is known for some tasks. Other tasks may require more analysis to determine the exact order in which they must be performed.

3. Construct the Network Diagram

Using the activity sequence information, a network diagram can be drawn showing the sequence of the serial and parallel activities.

4. Estimate activity times

Weeks are a commonly used unit of time for activity completion, but any consistent unit of time can be used. A distinguishing feature of PERT is its ability to deal with uncertainty in activity completion times. For each activity, the model usually includes three time estimates:

• Optimistic time (OT) - generally the shortest time in which the activity can be completed. (This is what an inexperienced manager believes!) • Most likely time (MT) - the completion time having the highest probability. This is different from expected time. Seasoned managers have an amazing way of estimating very close to actual data from prior estimation errors. • Pessimistic time (PT) - the longest time that an activity might require.

The expected time for each activity can be approximated using the following weighted average:

Expected time = (OT + 4 x MT+ PT) / 6

This expected time might be displayed on the network diagram.

Variance for each activity is given by: [(PT - OT) / 6]2

5. Determine the Critical Path

The critical path is determined by adding the times for the activities in each sequence and determining the longest path in the project. The critical path determines the total time required for the project.

If activities outside the critical path speed up or slow down (within limits), the total project time does not change. The amount of time that a non-critical path activity can be delayed without delaying the project is referred to as slack time.

If the critical path is not immediately obvious, it may be helpful to determine the following four quantities for each activity:

• ES - Earliest Start time • EF - Earliest Finish time • LS - Latest Start time • LF - Latest Finish time

These times are calculated using the expected time for the relevant activities. The ES and EF of each activity are determined by working forward through the network and determining the earliest time at which an activity can start and finish considering its predecessor activities.

The latest start and finish times are the latest times that an activity can start and finish without delaying the project. LS and LF are found by working backward through the network. The difference in the latest and earliest finish of each activity is that activity's slack. The critical path then is the path through the network in which none of the activities have slack.

The variance in the project completion time can be calculated by summing the variances in the completion times of the activities in the critical path. Given this variance, one can calculate the probability that the project will be completed by a certain date.

Since the critical path determines the completion date of the project, the project can be accelerated by adding the resources required to decrease the time for the activities in the critical path. Such a shortening of the project sometimes is referred to as project crashing. 6. Update as project progresses

Make adjustments in the PERT chart as the project progresses. As the project unfolds, the estimated times can be replaced with actual times. In cases where there are delays, additional resources may be needed to stay on schedule and the PERT chart may be modified to reflect the new situation.

PERT analysis

PERT is based on the assumption that an activity’s duration follows a probability distribution instead of being a single value

Three time estimates are required to compute the parameters of an activity’s duration distribution:

Draw the network. Analyze the paths through the network and find the critical path. The length of the critical path is the mean of the project duration

probability distribution which is assumed to be normal The standard deviation of the project duration probability

distribution is computed by adding the variances of the critical activities (all of the activities that make up the critical path) and taking the square root of that sum

Probability computations can now be made using the normal distribution table.

Probability computation

Determine probability that project is completed within specified time

Where m = tp = project mean time

s = project standard mean time x = (proposed ) specified time

Mean (expected time): te= tp + 4 tm + to

6

Variance: Vt =s2= tp - to

6

2

Z = x - m

s

Figure(4-2) Normal Distribution of Project Time

PROJECT COST

Cost consideration in project

Project managers may have the option or requirement to crash the project, or accelerate the completion of the project.

This is accomplished by reducing the length of the critical path(s). The length of the critical path is reduced by reducing the duration

of the activities on the critical path. If each activity requires the expenditure of an amount of money to

reduce its duration by one unit of time, then the project manager selects the least cost critical activity, reduces it by one time unit, and traces that change through the remainder of the network.

As a result of a reduction in an activity’s time, a new critical path may be created.

When there is more than one critical path, each of the critical paths must be reduced.

If the length of the project needs to be reduced further, the process is repeated.

Project Crashing

m = tp Timex

Zs

Probability

• Crashing– reducing project time by expending additional resources

• Crash time– an amount of time an activity is reduced

• Crash cost– cost of reducing activity time

• Goal– reduce project duration at minimum cost

Activity crashing

Figure(4-3) Time-Cost Relationship

Crashing costs increase as project duration decreases Indirect costs increase as project duration increases Reduce project length as long as crashing costs are less than

indirect costs

Activit

y cos

t

Activity time

Crashing activity

Crash time

Crash cost

Normal Activity

Normal time

Normal cost

Slope = crash cost per unit time

Figure(4-3) Time-Cost Tradeoff

Case study

CPMPERT

PERT/COSTcost

time

Direct cost

Indirect cost

Total project costMin total cost = optimal

project time

PERT/CPM Chart

A project has been defined to contain the following list of activities along with their required times for completion

Activity No

Activity Expected completion time

Dependency

1. Requirements collection

5 -

2. Screen design 6 13. Report design 7 14. Database design 2 2,35. User documentation 6 46. Programming 5 47. Testing 3 68. Installation 1 5,7

Fig(5-1) CPM Chart

Using information from the table, indicate expected completion time for each activity.

Fig(5-2) CPM Chart with TE

Calculate earliest expected completion time for each activity (TE) and the entire project.The earliest expected completion time for a given activity is determined by summing the expected completion time of this activity and the earliest expected completion time of the immediate predecessor.

1

2

3

4

5

6

8

7

5

6

2

6

5 3

1

TE = 11

TE = 12

TE = 14

TE = 20

TE = 19 TE = 22

TE = 23

1

2

3

4

5

6

8

7

5

6

2

6

5 3

1

TE = 5

TE = 12

TE = 14

TE = 19 TE = 22

Rule: if two or more activities precede an activity, the one with the largest TE is used in calculation (e.g., for activity 4, we will use TE of activity 3 but not 2 since 12 > 11).

the critical path 1,3,4,6,7,8

Fig (5-3) critical path The critical path represents the shortest time, in which a project can be completed. Any activity on the critical path that is delayed in completion delays the entire project. Activities not on the critical path contain slack time and allow the project manager some flexibility in scheduling. PERT

Immed. Optimistic Most Likely PessimisticActivity Predec. Time (Hr.) Time (Hr.) Time (Hr.) A -- 4 6 8B -- 1 4.5 5C A 3 3 3D A 4 5 6 E A 0.5 1 1.5F B,C 3 4 5G B,C 1 1.5 5H E,F 5 6 7I E,F 2 5 8 J D,H 2.5 2.75 4.5 K G,I 3 5 7

1

2

3

4

5

6

8

7

5

6

2

6

5 3

1

TE = 5

TE = 12

TE = 14

TE = 20

TE = 19 TE = 22

TE = 23

Fig (5-4) PERT Network

Activity Expected Time Variance A 6 4/9

B 4 4/9 C 3 0 D 5 1/9 E 1 1/36 F 4 1/9 G 2 4/9 H 6 1/9 I 5 1 J 3 1/9 K 5 4/9

Activity ES EF LS LF Slack

A 0 6 0 6 0 *critical

77 77

33 33

55 55

22 22

AA

DD

CC

BBFF

EE

GG

II

HH

KK

JJ

11 11

66 66

44 44

B 0 4 5 9 5 C 6 9 6 9 0 * D 6 11 15 20 9 E 6 7 12 13 6 F 9 13 9 13 0 * G 9 11 16 18 7 H 13 19 14 20 1 I 13 18 13 18 0 * J 19 22 20 23 1 K 18 23 18 23 0 *

The estimated project completion time is the Max EF at node 7 = 23.

s2 = s2A + s2

C + s2F + s2

H + s2K

= 4/9 + 0 + 1/9 + 1 + 4/9 = 2

s path = 1.414 z = (24 - 23)/s(24-23)/1.414 = .71

From the Standard Normal Distribution table:

P(z < .71) = .5 + .2612 = .7612

Fig (5-5) Probability the project will be completed within 24 hrs

xx

ff((xx))

P(TP(T<< 24) = .7612 24) = .7612

..5

00

05

00

0

..2

61

22

61

2

PERT/COST

Activity Normal time

Normal cost RS

Crash time Crash

cost

Allowable crash time

slope

1 12 3000 7 5000 5 4002 8 2000 5 3500 3 5003 4 4000 3 7000 1 30004 12 50000 9 71000 3 70005 4 500 1 1100 3 2006 4 500 1 1100 3 2007 4 1500 3 22000 1 7000

75000 110700

Table (1) Time Cost data

Fig (5-4) PERT Network

111122

2288

441122

3344 55

44

6644

7744

Fig (5-5) Project duration

Fig (5-6) modified Project duration

Summary

Program evaluation and review technique (PERT) charts depict

task, duration, and dependency information. Each chart starts with an

initiation node which is the first task. Each task is represented by a node

112

28

34 5

4

64

74

R400

R500

R3000

R7000

R200R200

R70012

4Project duration = 36

17

28

34 5

4

64

74

R400

R500

R3000

R7000

R200R200

R70012

4Project

duration = 31Additional cost

= R2000

(Activity on Node Network – AON) with lines connecting dependent

tasks.

Each task is connected to its successor tasks in this manner forming a

network of nodes and connecting lines. The chart is complete when all

final tasks come together at the completion node. When slack time exists

between the end of one task and the start of another, the usual method is

to draw a broken or dotted line between the end of the first task and the

start of the next dependent task.

PERT charts are usually drawn on ruled paper with the horizontal axis

indicating time period divisions in days, weeks, months, and so on. Many

PERT charts terminate at the major review points, such as at the end of

the analysis.

Critical Path Method (CPM) charts are similar to PERT charts and are

sometimes known as PERT/CPM. In a CPM chart, the critical path is

indicated. A critical path consists that set of dependent tasks (each

dependent on the preceding one), which together take the longest time to

complete. Tasks which fall on the critical path should be noted in some

way, so that they may be given special attention. One way is to draw

critical path tasks with a double line instead of a single line.

Tasks which fall on the critical path should receive special attention by

both the project manager and the personnel assigned to them.

The critical path for any given method may shift as the project

progresses; this can happen when tasks are completed either behind or

ahead of schedule, causing other tasks which may still be on-schedule to

fall on the new critical path.

Critical path computations are quite simple , yet they provide valuable

information that simplifies the scheduling of complex projects. The result

is that PERT-CPM techniques enjoy tremendous popularity among

practitioners in the field .The usefulness of the techniques is further

enhanced by the availability of specialized computer system for executing

,analyzing, and controlling network projects.

CPM Benefits

• Provides a graphical view of the project. • Predicts the time required to complete the project. • Shows which activities are critical to maintaining the schedule and which are not.

CPM Limitations

While CPM is easy to understand and use, it does not consider the time variations that can have a great impact on the completion time of a complex project. CPM was developed for complex but fairly routine projects with minimum uncertainty in the project completion times. For less routine projects there is more uncertainty in the completion times, and this uncertainty limits its usefulness.

Benefits of PERT

PERT is useful because it provides the following information:

• Expected project completion time. • Probability of completion before a specified date. • The critical path activities that directly impact the completion time. • The activities that have slack time and that can lend resources to critical path activities. • Activities start and end dates.

Limitations of PERT

The following are some of PERT's limitations:

• The activity time estimates are somewhat subjective and depend on judgment. In cases where there is little experience in performing an activity, the numbers may be only a guess. In other cases, if the person or

group performing the activity estimates the time there may be bias in the estimate. • The underestimation of the project completion time due to alternate paths becoming critical is perhaps the most serious.