(CST 281) with 5 Assignments

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Tutorial on Project Scheduling

with PERT/CPM

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Background

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Introduction

The best known planning tool in the project manager’s tool box is

the PERT/CPM network diagram. This tutorial is designed to provide

scheduling novices with a more detailed treatment of developing

PERT/CPM charts than what is contained in the basic planning

module.

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What Is PERT/CPM?

PERT was developed by the US Navy in 1957. It is an acronym that

stands for Project Evaluation and Review Technique. It is a systems

diagram that shows how project tasks should be sequenced.

CPM was developed by DuPont Corporation in 1958. Conceptually,

it is quite similar to PERT networks. Like PERT, it shows how

project tasks should be sequenced.

What we call PERT/CPM in this course has taken on the name

precedence diagrams. Precedence diagrams are a hybrid of the

PERT and CPM approaches. Use of precedence diagrams to

schedule projects is called the precedence diagram method (PDM).

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Tasks

Tasks represent work effort of some sort. Examples of tasks include:

Testing a piece of software code (working time task)

Laying brick (working time task)

Paint drying (elapsed time task)

Note that the first two examples of tasks entail the application of

people to carry out the work. If the people stop work (e.g., during

lunch break, over weekends), the work does not get done. This type

of task is called a working time task

Note that in the third example, work is carried out passively. There is

no application of physical effort by workers. This type of task is

called an elapsed time task. It operates according to a 24/7

schedule – paint will dry 24 hours a day, seven days a week.

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Milestones

Milestones are markers. Unlike tasks, they don’t reflect active or

passive work effort.

Milestones are usually set up to establish targets toward which we

can direct work efforts. For example, “Design Phase Is Finished, 15

October” may be a milestone indicating that a series of design-

related tasks should be completed by 15 October.

Milestones are good for tracking work. As a project is carried out,

project staff can check off the milestones they have achieved and

report these accomplishments to senior managers.

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Representing Tasks and Milestones

on a PERT/CPM Chart

Start Dig hole Pour concrete

Concrete cure

Excavation phase ended

Milestone Markers

Tasks

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PERT/CPM Charts Are Systems Diagrams

The development of PERT/CPM network charts was a consequence

of advances in systems engineering in the 1950s. The use of flow

charts became big at this time. Flow charts are diagrams that clearly

illustrate processes.

In the 1950s, engineers began asking how flow chart concepts could

be applied to scheduling projects. Ultimately, this quest led to the

development of PERT and CPM

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Traditional Flowchart: Picnic Project

Check weather

Rain?

Prepare indoor activities

Caterer sets up indoors

Prepare outdoor activities

Caterer sets up outdoors

Hold picnic

Yes No

Note that in this flow chart, the rectangular boxes represent processes (e.g., “Prepare indoor activities”) and the diamond represents a decision that needs to be made (e.g., “Rain?”; if “Yes,” proceed down one path; if “No,” proceed down the other path). The terminator (i.e., “Hold picnic”) represents the end of the decision sequence. When properly constructed, flow charts like the one portrayed here give system developers the precise information they need to build solutions that meet the specifications.

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Basic PERT/CPM Network Logic

Start Plan picnic

Get food

Get sports equip

Drive to site

Hold picnic

End

“Get food” and “Get sports equipment” are carried out in parallel

Good practice requires that all PERT/CPM charts start with a “Start” milestone

Good practice requires that all PERT/CPM charts end with an “End” milestone

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Differences between

Flow Charts and PERT/CPM Charts Flow charts have conditional logic, e.g., “If it is raining, then do X. If

it is not raining, then do Y.” PERT/CPM charts do not have

conditional logic. The chart does not incorporate decision making.

Flow charts allow you to go back to an earlier activity, e.g., “If fewer

than 20 orders have been filled, then go back to step 3 and repeat

the process.” PERT/CPM charts do not go back to earlier tasks.

They move inexorably forward in time. If a given set of steps need to

be repeated, these steps are laid out all over again in the chart.

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Critical Path Concept

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Identifying the Critical Path

Start Lay foundation

14 days

Frame house

20 days

Wire house

2 days

Install plumbing

3 days

Put up dry wall

4 days

End

Note that there are two paths in this PERT/CPM chart

Upper path: Foundation  Frame  Wire  Dry wall Duration: 40 days

Lower path: Foundation  Frame  Plumbing  Dry wall Duration: 41 days

The path indicated by thick red arrows is called the critical path. It is the longest path in the network. As such, it defines the length of the project. In this case, the project is scheduled to last 41 days.

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Task Durations May Depend

on Number of Resources Used (1) In the previous example of a PERT/CPM chart, the duration of “Frame house” was listed as 20 days. Let’s say this duration is based on using 4 carpenters to do the job. The overall level of effort to do the job is defined as:

LOE = duration x number of resources

LOE = 20 days x 4 carpenters = 80 carpenter-days

Level of effort is useful in computing task duration when we know how many resources can be applied to the task. For example, if we have only two carpenters available to frame the house

Duration = 80 carpenter-days/2 carpenters = 40 days

If we have eight carpenters,

Duration = 80 carpenter-days/8 carpenters = 10 days

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Task Durations May Depend

on Number of Resources Used (2) In working with level of effort data, the trick, of course, is computing what the value is. The best way to compute level of effort is to follow the approach taken here. Let’s say that we usually work with teams of four carpenters, and that we find on a typical house building project it takes them 20 days to frame the house. Consequently, this one example tells us that the level of effort for framing a house is 80 carpenter-days of effort.

In working with level of effort, you need to employ good sense. While saying that the house can be framed in twenty days with four carpenters or forty days with two carpenters sounds reasonable, it is silly to extend this logic to its extreme and to say that 160 carpenters can do the job in one day!

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Importance of the Critical Path

The critical path is called critical because it defines the length of the project.

Any delays along the critical path can translate into delays in the project

overall. For example, if it takes 5 days to install the plumbing instead of the

scheduled 3 days, this can result in a two day schedule slippage.

Note that the “Wire house” task is non-critical. The network shows that it is

scheduled to take 2 days, while the parallel plumbing task (which is a critical

path task) is scheduled to take 3 days. Consequently, “Wire house” has one

day of float (also called slack by some schedulers). You can have a one day

delay on this non-critical task without causing the project to slip its schedule.

On large projects, you do not need to monitor progress on every path. The

important thing is to monitor progress on the critical path.

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Earliest Start, Latest Start, Float (Slack)

Start

Task A

3 days

Task C

8 days

Task E

5 days Task D

2 days

Task B

5 days

End

The critical path is the longest path. Add up the durations on different paths, and identify the longest path. In this case, the longest path is pictured in red. Its duration is 15 days. This means the project duration is scheduled to be 15 days.

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Calculating Earliest Start

Start

Task A

3 days

Task C

8 days

Task E

5 days Task D

2 days

Task B

5 days

End

To calculate the earliest time a task can begin, start at the left of the chart and work your way to the right. Add the duration of a newly encountered task to the total up to that point.

Begin computing earliest start dates with the critical path. In the network above, the earliest Task C can begin is at time t = 0. If C takes 8 days to complete, the earliest Task D can begin is at time t = 8. If D takes 2 days to complete, then the earliest Task E can begin is at time t = 10.

On the top, non-critical path above, the earliest Task A can begin is at time t=0. If it takes 3 days to complete, then earliest Task B can begin is at t = 3. Note that we have already determined by our critical path computation that Task E begins at time t = 10.

ES = Day 0

ES = Day 0

ES = Day 3

ES = Day 8

ES = Day 10

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Calculating Latest Start

Start

Task A

3 days

Task C

8 days

Task E

5 days Task D

2 days

Task B

5 days

End

To calculate the latest time a task can begin, start at the right side of the chart and work your way to the subtracting durations, task by task.

Note that calculating latest task dates for the critical path is easy, because on the critical path earliest starts and latest starts are the same (ES = LS) – there is no leeway.

On the top, non-critical path above, the latest Task E can begin is 5 days before the project end. Since the project will last 15 days, LS for Task E is 10. The latest start for Task B is 5 days before the latest start of Task E, or 10 – 5, so LS = 5. Finally, the latest start for Task A is 3 days before the latest start of Task B, or 5-3, so LS = 2.

LS = 2 days

LS = Day 0

LS = Day 5

LS = Day 8

LS = Day 10

Note that the latest start is sometimes called the “drop dead” date. If you begin later than the latest start date, then you can cause the project to encounter schedule slippage. For example, in the network diagram provided here, If Task B begins on Day 7 (two days later than Task B’s latest start date), this can cause a two day delay in the project overall. What was initially a non-critical path has now become critical.

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Start

Task A

3 days

Task C

8 days

Task E

5 days Task D

2 days

Task B

5 days

End

ES = Day 0

LS = Day 2

Float = 2

ES = Day 0

LS = Day 0

Float = 0

ES = Day 3

LS = Day 5

Float = 2

ES = Day 8

LS = Day 8

Float = 0

ES = Day 10

LS = Day 10

Float = 0

Calculating Float (Also Called Slack)

Float = LS - LE

Float measures scheduling leeway for a task. Note that critical path tasks have zero float – there is no leeway!

In a sense, float is a measure of forgiveness. For example, Task A has 2 days of float associated with it. I can begin Task A a day late, and this will not affect the project schedule. However, if I begin it three days late, this can contribute to a slip of the overall schedule. Critical path tasks are unforgiving. This is indicated by the fact that each of them has zero float. If you begin a critical path task even a little late, this translates into overall schedule slippage. The idea that there is no forgiveness on the critical path is a common sense one. Remember, the critical path defines project length. So if there is even a small delay on any of the critical path tasks, this can extend the project schedule.

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Hard Logic and Soft Logic

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Not All Precedence Links Are Equal

Start

Fix Coffee

8 minutes

Pour Coffee

2 minutes

Fix Cereal

6 minutes

Make Toast

6 minutes

End

Critical path

Duration = 12 min

Non-critical path

Duration = 10 min

Float = 2 min

Hard logic link: Pour coffee must follow Fix coffee

Soft logic link: It doesn’t matter which of these two tasks is predecessor and which is successor

With hard logic links, tasks must occur in a prescribed sequence. These links cannot be broken. With soft logic links, the sequence is immaterial. These links can be broken.

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Soft Logic Links Can Be Broken

Start

Fix Coffee

8 minutes

Pour Coffee

2 minutes

Fix Cereal

6 minutes

Make Toast

6 minutes

End

By breaking the soft logic link between “Fix Cereal” and “Make Toast,” I am able to reconfigure the network diagram to have these two tasks carried out in parallel. This has the effect of shortening the project duration, because the new critical path is the top path – it has a 10 minute duration. Two minutes have been shaved off duration! (See previous chart.)

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Estimating Task Duration

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Estimating Task Durations with PERT/CPM

When PERT networks were first developed in the late 1950s, one of

their distinguishing features was the way they estimated task

duration. To compute the expected time [e(t)] of a task, they

employed the following formula:

e(t) = a + 4b + c , [where a = best case, b = typical case, and

6 c = worst case]

Example:

e(t) = 40 + 4x43 + 47 , [where a = 40 hrs, b= 44 hrs, c = 47 hrs]

6

= 44.33 hours

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Estimating Standard Deviation for Task Durations

Standard deviation is an indicator of the “slop” of a measure. A

measure that is right on target has a low standard deviation. The

rougher the measure, the greater its standard deviation.

The inventors of PERT developed a simple formula that provides an

approximation of the standard deviation (SD) associated with the

estimate of the expected duration of a task:

SD(t) = c – a , where c = worst case, a = best case

6

Example: If c = 47 hrs and a = 43 hrs, SD(t) = 0.67 hrs

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Using Expected Duration with Its

Standard Deviation (1) In the previous examples, we have

e(t) = 44.33 and SD(t) = 0.67

Combining the two pieces of information we report that:

e(t) = 44.33 +/- 0.67 hours

That is, we have reason to believe that the true amount of time it will

take to carry out the target task lies somewhere between 43.67

hours and 45.00 hours. We know from our understanding of the

normal distribution that about 68% of observations lie within +/- 1

standard deviation from the mean. Very roughly, we have a sense

that two-thirds of the time, the amount of time it will take to carry out

our task lies within the range of 43.67 and 45.00 hours.

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Using Expected Duration with Its

Standard Deviation (2) Note, that we have to be very careful when talking about assuming that deductions arising from

the normal distribution apply to the Beta distribution (the (a + 4b +c)/6 formula we are using here

provides an estimate of the mean of the Beta distribution). Clearly, the normal distribution is

symmetric about the mean, while the Beta distribution is skewed. Still, it turns out that about two-

thirds of the observations on a typically encountered Beta distribution (such as presented in this

example) lie within a range slightly larger than +/- 1 standard deviation from the mean, so the 1

standard deviation rule for normal distributions roughly applies to the Beta distribution. However,

the 2 standard deviation scenarios differ significantly between the normal and Beta distributions.

For the normal distribution, roughly 95 percent of the observations lie within +/- 2 standard

deviations from the mean. However, with the Beta distribution 95% of the observations lie in the

following range: μ - 1.5 x (c – a)/6 and μ + 1.95 x (c – a)/6, where μ is a measure of the mean.

Thus in the present case, we estimate that 95% of the time, the actual amount of time it will take

to carry out the task being examined lies in the range 43.3 hours and 45.6 hours.

For a typical Beta distribution – as reflected in the data presented in this example – about 70% of

the observations lie at 0.82.

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Estimating the Duration of the Critical Path (1)

To estimate the duration of the critical path, merely add up the

duration of the individual tasks that lie on the critical path.

To estimate the standard deviation of the critical path, carry out the

following computation:

SDPath = SQRT(SD1 2 + SD2

2 + SD3 2 + … + SDi

2) for i tasks on the

path.

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Estimating the Duration of the Critical Path (2)

Task Best case Most typical Worst case

Expected

duration

Standard

deviation SD Squared

A 3 4 7 4.33 0.67 0.44

B 12 15 18 15.00 1.00 1.00

C 6 8 12 8.33 1.00 1.00

D 10 13 18 13.33 1.33 1.78

E 2 4 6 4.00 0.67 0.44

Duration = 45.00 SD Sqrd = 4.67

SD = 2.16

Duration of path = 45 hours +/- 2.16 hours

Note: The units of analysis in this table are hours.

In this example, Tasks A, B, C, D, and E comprise the critical path. For each task, we compute expected duration based on our assessment of the best case, most typical case, and worst case (using the Beta distribution). By adding up these numbers, we get the expected duration of the critical path (45). Given this data, we are also able to estimate the standard deviation associated with the duration of each task. By squaring these values and summing them, we get the sum of 4.67 (which in statistics is called variance). Standard deviation is the square root of variance, so in this example it is 2.16. Thus we estimate that the critical path will consume 45 hours of effort, plus or minus 2.16 hours. As a rough rule of thumb, about two-thirds of our observations lie within one standard deviation from the mean, so we are saying that there is a about a 30-35% chance of the true value lying outside the specified range.

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The Need to Estimate a Range of Duration Times

Up until now, all of our estimates of task duration have been point

estimates. That is, we use a single value of duration. In reality, we

know that these estimates are likely to be wrong. When we say it

takes 1.2 days to test a software algorithm, we don’t seriously

expect it to take exactly 1.2 days. We figure it will take 1.2 days, plus

or minus some segment of time.

In statistics, the “plus or minus” factor is determined by a measure

called standard deviation (SD). By calculating SD for a task, you

have a better idea of how much faster or slower the task will be than

what you speculate with your point estimate.

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Practical Steps in Building a PERT/CPM Chart

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Step 1. Draw a Logic Diagram by Hand

Sometimes you see project workers trying to develop a PERT/CPM

chart by working directly with a scheduling software package.

Generally, this is an ineffective way to begin. The computer screen

is limited in size and only lets you see a handful of tasks at one time.

Seasoned professionals always begin by drawing the logic relations

of tasks by hand on large sheets of paper or on large white boards

mounted onto a wall. This way they can work with big chunks of the

project and perhaps even see the whole project at one time.

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Step 2. Build a PERT/CPM Chart

Based on Your Hand Drawn Chart If you are using computer software, now is the time to build a

PERT/CPM network on the computer, using the large hand drawn

chart as your guide. As you build the chart, enter task duration data.

If you are building a PERT/CPM network by hand, copy your large

rough chart, and enter task duration figures into each box that

represents a task.

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Step 3. Identify the Critical Path

On a computerized scheduling package, once the scheduling data

are entered into the system, the critical path will be identified

automatically. It is usually pictured as a red path.,

If you have created a PERT/CPM network manually, look at all the

paths that run through the network and find the one that is longest.

This is the critical path. Color it red.

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Dealing with Weekends, Holidays, and Such

Computerized scheduling packages can deal with weekends, holidays, and such automatically. Each of these packages contains a central calendar, where you can define what days are weekend days (in Muslim countries, you set weekends for Friday and Saturday), what days are holidays (e.g., you can set New Years Day as a holiday). Beyond this, with computerized scheduling packages you can identify elapsed time activities (e.g., paint drying, concrete curing), so that these activities are carried out according to a 24/7 calendar. That is, if you set “Paint dry” as an elapsed time activity, the software will schedule it to dry on weekends (when appropriate) as well as weekdays.

When you create PERT/CPM networks by hand, you need to track weekends, holidays, and elapsed time tasks manually.