PERT/CPM FOR PROJECT SCHEDULING

INTRODUCTION

Basically, CPM (Critical Path Method) and PERT (Programme Evaluation Review Technique)

are project management techniques, which have been created out of the need of Western

industrial and military establishments to plan, schedule and control complex projects.

History of CPM/PERT

CPM/PERT or Network Analysis as the technique is sometimes called, developed along two

parallel streams, one industrial and the other military.

CPM was the discovery of M.R.Walker of E.I.Du Pont de Nemours & Co. and J.E.Kelly of

Remington Rand, circa 1957. The computation was designed for the UNIVAC-I computer. The

first test was made in 1958, when CPM was applied to the construction of a new chemical plant.

In March 1959, the method was applied to a maintenance shut-down at the Du Pont works in

Louisville, Kentucky. Unproductive time was reduced from 125 to 93 hours.

PERT was devised in 1958 for the POLARIS missile program by the Program Evaluation Branch

of the Special Projects office of the U.S.Navy, helped by the Lockheed Missile Systems division

and the Consultant firm of Booz-Allen & Hamilton. The calculations were so arranged so that

they could be carried out on the IBM Naval Ordinance Research Computer (NORC) at Dahlgren,

Virginia.

Planning, Scheduling & Control

Planning, Scheduling (or organising) and Control are considered to be basic Managerial

functions, and CPM/PERT has been rightfully accorded due importance in the literature on

Operations Research and Quantitative Analysis.

Far more than the technical benefits, it was found that PERT/CPM provided a focus around

which managers could brain-storm and put their ideas together. It proved to be a great

communication medium by which thinkers and planners at one level could communicate their

ideas, their doubts and fears to another level. Most important, it became a useful tool for

evaluating the performance of individuals and teams.

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There are many variations of CPM/PERT which have been useful in planning costs, scheduling

manpower and machine time. CPM/PERT can answer the following important questions:

How long will the entire project take to be completed? What are the risks involved?

Which are the critical activities or tasks in the project which could delay the entire project if they

were not completed on time?

Is the project on schedule, behind schedule or ahead of schedule?

If the project has to be finished earlier than planned, what is the best way to do this at the least

cost?

The Framework for PERT and CPM

Essentially, there are six steps which are common to both the techniques. The procedure is listed

below:

I. Define the Project and all of it�s significant activities or tasks. The Project (made up of

several tasks) should have only a single start activity and a single finish activity.

II. Develop the relationships among the activities. Decide which activities must precede and

which must follow others.

III. Draw the "Network" connecting all the activities. Each Activity should have unique event

numbers. Dummy arrows are used where required to avoid giving the same numbering to

two activities.

IV. Assign time and/or cost estimates to each activity

V. Compute the longest time path through the network. This is called the critical path.

VI. Use the Network to help plan, schedule, monitor and control the project.

The Key Concept used by CPM/PERT is that a small set of activities, which make up the longest

path through the activity network control the entire project. If these "critical" activities could be

identified and assigned to responsible persons, management resources could be optimally used

by concentrating on the few activities which determine the fate of the entire project.

Non-critical activities can be replanned, rescheduled and resources for them can be reallocated

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flexibly, without affecting the whole project.

Five useful questions to ask when preparing an activity network are:

Is this a Start Activity?

Is this a Finish Activity?

What Activity Precedes this?

What Activity Follows this?

What Activity is Concurrent with this?

Some activities are serially linked. The second activity can begin only after the first activity is

completed. In certain cases, the activities are concurrent, because they are independent of each

other and can start simultaneously. This is especially the case in organisations which have

supervisory resources so that work can be delegated to various departments which will be

responsible for the activities and their completion as planned.

When work is delegated like this, the need for constant feedback and co-ordination becomes an

important senior management pre-occupation

Drawing the CPM/PERT Network

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Each activity (or sub-project) in a PERT/CPM Network is represented by an arrow symbol. Each

activity is preceded and succeeded by an event, represented as a circle and numbered.

At Event 3, we have to evaluate two predecessor activities - Activity 1-3 and Activity 2-3, both

of which are predecessor activities. Activity 1-3 gives us an Earliest Start of 3 weeks at Event 3.

However, Activity 2-3 also has to be completed before Event 3 can begin. Along this route, the

Earliest Start would be 4+0=4. The rule is to take the longer (bigger) of the two Earliest Starts.

So the Earliest Start at event 3 is 4.

Similarly, at Event 4, we find we have to evaluate two predecessor activities - Activity 2-4 and

Activity 3-4. Along Activity 2-4, the Earliest Start at Event 4 would be 10 wks, but along

Activity 3-4, the Earliest Start at Event 4 would be 11 wks. Since 11 wks is larger than 10 wks,

we select it as the Earliest Start at Event 4.We have now found the longest path through the

network. It will take 11 weeks along activities 1-2, 2-3 and 3-4. This is the Critical Path.

The Backward Pass - Latest Finish Time Rule

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To make the Backward Pass, we begin at the sink or the final event and work backwards to the

first event.

At Event 3 there is only one activity, Activity 3-4 in the backward pass, and we find that the

value is 11-7 = 4 weeks. However at Event 2 we have to evaluate 2 activities, 2-3 and 2-4. We

find that the backward pass through 2-4 gives us a value of 11-6 = 5 while 2-3 gives us 4-0 = 4.

We take the smaller value of 4 on the backward pass.

Tabulation & Analysis of Activities

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We are now ready to tabulate the various events and calculate the Earliest and Latest Start and

Finish times. We are also now ready to compute the SLACK or TOTAL FLOAT, which is

defined as the difference between the Latest Start and Earliest Start.

EventDuration(Weeks) Earliest

Start

Earliest

Finish

Latest

Start

Latest

Finish

Total

Float

1-2 4 0 4 0 4 0

2-3 0 4 4 4 4 0

3-4 7 4 11 4 11 0

             

1-3 3 0 3 1 4 1

2-4 6 4 10 5 11 1

The Earliest Start is the value in the rectangle near the tail of each activity

The Earliest Finish is = Earliest Start + Duration

The Latest Finish is the value in the diamond at the head of each activity

The Latest Start is = Latest Finish - Duration

There are two important types of Float or Slack. These are Total Float and Free Float.

TOTAL FLOAT is the spare time available when all preceding activities occur at the earliest

possible times and all succeeding activities occur at the latest possible times.

Total Float = Latest Start - Earliest Start

Activities with zero Total float are on the Critical Path

FREE FLOAT is the spare time available when all preceding activities occur at the earliest

possible times and all succeeding activities occur at the earliest possible times.

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When an activity has zero Total float, Free float will also be zero.

There are various other types of float (Independent, Early Free, Early Interfering, Late Free, Late

Interfering), and float can also be negative. We shall not go into these situations at present for the

sake of simplicity and be concerned only with Total Float for the time being.

Having computed the various parameters of each activity, we are now ready to go into the

scheduling phase, using a type of bar chart known as the Gantt Chart.

There are various other types of float (Independent, Early Free, Early Interfering, Late Free, Late

Interfering), and float can also be negative. We shall not go into these situations at present for the

sake of simplicity and be concerned only with Total Float for the time being. Having computed

the various parameters of each activity, we are now ready to go into the scheduling phase, using

a type of bar chart known as the Gantt Chart.

Steps 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

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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.

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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 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.

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P E R T

(Program (or Project) Evaluation and Review Technique)

PERT is a model for project management designed to analyze and represent the tasks involved

in completing a given project.

Overview

PERT is a method to analyze the involved tasks in completing a given project, especially the

time needed to complete each task, and identifying the minimum time needed to complete the

total project.

PERT was developed primarily to simplify the planning and scheduling of large and complex

projects. It was able to incorporate uncertainty by making it possible to schedule a project while

not knowing precisely the details and durations of all the activities. It is more of an event-

oriented technique rather than start- and completion-oriented, and is used more in projects where

time, rather than cost, is the major factor. It is applied to very large-scale, one-time, complex,

non-routine infrastructure and Research and Development projects.

This project model was the first of its kind, a revival for scientific management, founded by

Frederick Taylor "Taylorism" and later refined by Henry Ford "Fordism". DuPont corporation's

critical path method was invented at roughly the same time as PERT.

Conventions

A PERT chart is a tool that facilitates decision making; The first draft of a PERT chart

will number its events sequentially in 10s (10, 20, 30, etc.) to allow the later insertion of

additional events.

Two consecutive events in a PERT chart are linked by activities, which are

conventionally represented as arrows in the diagram above.

The events are presented in a logical sequence and no activity can commence until its

immediately preceding event is completed.

The planner decides which milestones should be PERT events and also decides their

“proper” sequence.

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A PERT chart may have multiple pages with many sub-tasks.

PERT is valuable to manage where multiple task are going simultaneously to reduce the

redundancy

Terminology

A PERT event: is a point that marks the start or completion of one or more tasks. It

consumes no time, and uses no resources. It marks the completion of one or more tasks,

and is not “reached” until all of the activities leading to that event have been completed.

A predecessor event: an event (or events) that immediately precedes some other event

without any other events intervening. It may be the consequence of more than one

activity.

A successor event: an event (or events) that immediately follows some other event

without any other events intervening. It may be the consequence of more than one

activity.

A PERT activity: is the actual performance of a task. It consumes time, it requires

resources (such as labour, materials, space, machinery), and it can be understood as

representing the time, effort, and resources required to move from one event to another. A

PERT activity cannot be completed until the event preceding it has occurred.

Optimistic time (O): the minimum possible time required to accomplish a task, assuming

everything proceeds better than is normally expected

Pessimistic time (P): the maximum possible time required to accomplish a task, assuming

everything goes wrong (but excluding major catastrophes).

Most likely time (M): the best estimate of the time required to accomplish a task,

assuming everything proceeds as normal.

Expected time (TE): the best estimate of the time required to accomplish a task, assuming

everything proceeds as normal (the implication being that the expected time is the

average time the task would require if the task were repeated on a number of occasions

over an extended period of time).

TE = (O + 4M + P) ÷ 6

Float or Slack is the amount of time that a task in a project network can be delayed

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without causing a delay - Subsequent tasks – (free float) or Project Completion – (total

float)

Critical Path: the longest possible continuous pathway taken from the initial event to the

terminal event. It determines the total calendar time required for the project; and,

therefore, any time delays along the critical path will delay the reaching of the terminal

event by at least the same amount.

Critical Activity: An activity that has total float equal to zero. Activity with zero float

does not mean it is on critical path.

Lead time: the time by which a predecessor event must be completed in order to allow

sufficient time for the activities that must elapse before a specific PERT event is reached

to be completed.

Lag time: the earliest time by which a successor event can follow a specific PERT event.

Slack: the slack of an event is a measure of the excess time and resources available in

achieving this event. Positive slack(+) would indicate ahead of schedule; negative slack

would indicate behind schedule; and zero slack would indicate on schedule.

Fast tracking: performing more critical activities in parallel

Crashing critical path: Shortening duration of critical activities

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.

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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:

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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.

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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.

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.

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• The underestimation of the project completion time due to alternate paths becoming critical

is perhaps the most serious.

Performance Monitoring

Once the project is completed, performance review should be done periodically to compare

actual performance with projected performance. Feedback on project is useful in several ways:

a) It helps us to know how realistic were the assumptions underlying the project

b) It provides a documented log of experience that is highly valuable in decision making in

future projects

c) It suggests corrective action to be taken in the light of actual performance

d) It helps in uncovering judgmental biases

e) It includes a desired caution among project sponsors.

Performance Indicators (PIs) are an effective way of communicating a project’s benefits, usually

as part of a performance measuring and reporting process. Performance Indicators are available

for a wide range of industries and allow a measure of energy performance to be assigned to a

process against which others can be judged.

Depending on the nature of the project, savings are determined using engineering calculations, or

through metering and monitoring, utility meter billing analysis, or computer simulations.

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CASE STUDY: THE RELIABLE CONSTRUCTION COMPANY

Project Management: Project management can be defined as the coordination of activities

with the potential use of many organizations, both internal and external to the business, in

order to conduct a large scale project from beginning to end. There are two management

science techniques that are used for project management: CPM/PERT

PERT/CPM

PERT

– PERT was designed to examine projects from the standpoint of uncertainty.

CPM

– CPM was designed to examine projects from the standpoint of costs.

PERT and CPM techniques have been combined over time.

PERT and CPM both rely heavily on the use of networks to help plan and display the

coordination of all the activities for a project.

The Reliable Construction Company

Reliable has just secured a contract to construct a new plant for a major manufacturer.

The contract is for $5.4 million to cover all costs and any profits.

The plant must be finished in a year.

– A penalty of $300,000 will be assessed if Reliable does not complete the

project within 47 weeks.

– A bonus of $150,000 will be paid to Reliable if the plant is completed within

40 weeks.

Needed Terminology

Activity

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– A distinct task that needs to be performed as part of the project.

Start Node

– This is a node that represents the beginning of the project.

Finish Node

– This node represents the end of the project.

Immediate Predecessor

– These are activities that must be completed by no later than the start time of

the given activity.

Immediate Successor

– Given the immediate predecessor of an activity, this activity becomes the

immediate successor of each of these immediate predecessors.

– If an immediate successor has multiple immediate predecessors, then all must

be finished before an activity can begin.

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Activity List for Reliable Construction

Activity Activity Description

Immediate

Predecessors

Estimated

Duration (Weeks)

A Excavate — 2

B Lay the foundation A 4

C Put up the rough wall B 10

D Put up the roof C 6

E Install the exterior plumbing C 4

F Install the interior plumbing E 5

G Put up the exterior siding D 7

H Do the exterior painting E, G 9

I Do the electrical work C 7

J Put up the wallboard F, I 8

K Install the flooring J 4

L Do the interior painting J 5

M Install the exterior fixtures H 2

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N Install the interior fixtures K, L 6

Questions Needed to be Answered

How can the project be displayed graphically?

How much time is required to finish the project if no delays occur?

When is earliest start and finish times for each activity if no delays occur?

What activities are critical bottleneck activities where delays must be avoided to

finish the project on time?

For non bottleneck activities, how much can an activity be delayed and yet still keep

the project on time?

What is the probability of completing the project by the deadline?

What is the least amount of money needed to expedite the project to obtain the bonus?

How should costs be monitored to keep the project within budget?

Project Network

A project network is a network diagram that uses nodes and arcs to represent the

progression of the activities is a project from start to finish.

Three pieces of information needed:

– Activity information

– Precedence relationship

– Time information

Two types of project networks

– Activity-on-Arc (AOA)

• On this diagram, the activity is represented on an arc, while a node is

used to separate an activity from its immediate predecessors.

– Activity-on-Node (AON)

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• On this diagram, the activity is represented by the node, while the arc

is used to showed the precedence relationship between the activities.

A

START

G

H

M

F

J

K L

N

Activity Code

A. Excavate

B. Foundation

C. Rough wall

D. Roof

E. Exterior plumbing

F. Interior plumbing

G. Exterior siding

H. Exterior painting

I. Electrical work

J. Wallboard

K. Flooring

L. Interior painting

M. Exterior fixtures

N. Interior fixtures

2

4

10

746

7

9

5

8

4 5

6

2

0

0FINISH

D IE

C

B

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Microsoft Project Visual Diagram: Gantt Chart

Scheduling Using PERT/CPM

A path through a project network is a route that follows a set of arcs from the start

node to the finish node.

The length of a path is defined as the sum of the durations of the activities of the path.

– What are the paths and their corresponding lengths for Reliable?

Critical Path

This is the path that has the longest length through the project.

The shortest time that a project can conceivably be finished is the critical path.

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More Terminology

Earliest start time of an activity (ES)

– The time at which an activity will begin if there are no delays in a project.

Earliest finish time of an activity (EF)

– The time at which an activity will finish if there are no delays in a project.

Latest start time of an activity (LS)

– The latest possible time that an activity can start without delaying the project.

Latest finish time of an activity (LF)

– The latest possible time that an activity can be completed without delaying the

project.

Forward pass

– The process of moving through a project from start to finish to determine the

earliest start and finish times for the activities in the project.

Backward pass

– The process of moving through a project from finish to start to determine the

latest start and finish times for the activities in the project.

Slack for an activity

– The amount of time that a particular activity can be delayed without delaying

the whole project.

– It is calculated by taking the difference between the latest finish time with the

earliest finish time.

Earliest start time rule

– The earliest start time for an activity is equal to the largest of the earliest finish

times of its immediate predecessors.

Latest finish time rule

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– The latest finish time is equal to the smallest of the latest start times of its

immediate successors.

Procedure for Obtaining Earliest Times

Step 1: For the activity that starts the project, assign an earliest start time of zero, i.e.,

ES=0.

Step 2: For each activity whose ES has just been obtained, calculate its earliest finish

time as ES plus duration of the activity.

Step 3: For each new activity whose immediate predecessors have EF values, obtain

its ES by using the earliest start time rule.

Step 4: Apply step 2 to calculate EF.

Step 5: Repeat step 3 until ES and EF have been obtained for all activities including

the finish node.

Procedure for Obtaining Latest Times

Step 3: For each new activity whose immediate successors now have LS values,

obtain its LF by applying the latest finish time rule.

Step 4: Apply step 2 to calculate its LS.

Step 5: Repeat step 3 until LF and LS have been obtained for all activities.

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A

START

G

H

M

F

J

FINISH

K L

N

D IE

C

B

2

4

10

746

7

9

5

8

4 5

6

2

S = (0, 0) F = (2, 2)

S = (2, 2) F = (6, 6)

S = (16, 20) F = (22, 26)

S = (16, 16) F = (20, 20)

S = (16, 18) F = (23, 25)

S = (20, 20) F = (25, 25)

S = (22, 26) F = (29, 33)

S = (6, 6) F = (16, 16)

S = (0, 0) F = (0, 0)

S = (25, 25) F = (33, 33)

S = (33, 33) F = (38, 38)

S = (38, 38) F = (44, 44)

S = (33, 34) F = (37, 38)

S = (29, 33) F = (38, 42)

S = (38, 42) F = (40, 44)

S = (44, 44) F = (44, 44)

0

0

Ways of Finding the Critical Path

Examine all the paths and find the path with the maximum length.

Calculate the slack for an activity.

– If the slack is zero, it is on the critical path.

– If the slack is positive, it is not on the critical path.

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Time-Cost Trade-Offs

Reliable had an incentive bonus of $150,000 to finish the project in 40 weeks.

– Is it worth while for Reliable to speed-up the project?

Crashing

Crashing an activity refers to taking on extra expenditures in order to reduce the

duration of an activity below its expected value.

Crashing a project refers to crashing a number of activities to reduce the duration of

the project.

CPM Method of Time-Cost Trade-Offs

This is a method concerned with whether it is worthwhile to crash activities to reduce

the anticipated duration of the project to a desired value.

This assumes that there is a trade-off between time and cost that has an inverse

relationship.\

More Terminology

Normal Point is the time and cost of an activity when it is performed in a normal way.

Crash point show the time and cost when the activity is fully crashed.

Graph of Normal and Crash Points

Activity duration

Activity cost

Crash cost

Normal cost Normal

Crash

Crash time Normal time

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Marginal Cost Analysis

It is a method of using the marginal cost of crashing individual activities on the

current critical path to determine the least expensive way of reducing the project

duration to an acceptable level.

This method requires us to calculate the cost per desired time unit and compare each

cost with the other costs.

    Time (weeks)                               Cost                           Maximum

Reduction

in Time

(weeks)

Crash

Cost

per Week

SavedActivity Normal Crash Normal Crash

A 2 1 $180,000 $280,000 1 $100,000

B 4 2 320,000 420,000 2 50,000

C 10 7 620,000 860,000 3 80,000

D 6 4 260,000 340,000 2 40,000

E 4 3 410,000 570,000 1 160,000

F 5 3 180,000 260,000 2 40,000

G 7 4 900,000 1,020,000 3 40,000

H 9 6 200,000 380,000 3 60,000

I 7 5 210,000 270,000 2 30,000

J 8 6 430,000 490,000 2 30,000

K 4 3 160,000 200,000 1 40,000

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L 5 3 250,000 350,000 2 50,000

M 2 1 100,000 200,000 1 100,000

N 6 3 330,000 510,000 3 60,000

Marginal Cost Analysis Cont…

Once the marginal cost for crashing each activity has been conducted, you next want

to choose the crashing that has the smallest marginal cost.

Next, calculate the effect that the crash has on each path.

– Crashing can potentially cause another path to become a critical path.

Solving Crashing Problems Using LP

There are three decisions to be made:

– The start time of each activity

– The reduction in each activity due to crashing

– The finish time of the project

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