PERT – CPM NETWORK

Reported by:COSCOS, Joseph Noel O.BSCE – 4

CRITICAL PATH SCHEDULING TECHNIQUES

Two graphical techniques which are commonly used in planning and controlling projects:

• Critical Path Method (CPM)• Program Evaluation and Review Technique

(PERT)

Three pertinent factors of concern in a project:• Resource• Time• Cost

Program Evaluation and Review Technique (PERT)

•This is used when there is existence of uncertainties associated with completion times of activity of the project.

•Its application is emphasized in research and development work.

Critical Path Method (CPM)

•This is used where one is certain about the duration times of activities.

•It is commonly used to manage construction project.

Application of PERT - CPM

Both the CPM and the PERT find application in areas such as highway construction, power generation, plant and oil refinery maintenance, shipbuilding and aerospace projects.

Characteristic of a Project

A project has to posses specific characteristics for these techniques to be most applicable.

These characteristics are as follows:i. When a job is started, it has to

continue without any interference until its full completion.

ii. Jobs or tasks are defined such that their completion will result in the end of the project.

Characteristic of a Project

iii.Jobs or tasks are ordered in a specified sequence in which they follow each other.

iv.Jobs or tasks are independent. In other words, within a defined sequence they can be performed, started and stopped independently.

PERT - CPMBoth these methods roughly involve the following steps:i. Decomposing a project into individual job or

tasks.ii. Arranging the individual jobs or task into a

logical network.iii.Estimating the time duration of each individual

job or task.iv.Developing a schedule.v. Finding out those critical jobs or tasks which

control the completion of the project.vi.Redistributing resources or funds so that the

schedule is improved.

Program Evaluation and Review Technique (PERT)

Steps to Develop and Analyze a PERT Network

STEP 1Decompose project into individual job or

tasks and identify each of them.

STEP 2Find out activities or tasks sequence and

develop network.

STEP 3Estimate each activities duration time

STEP 4Obtain expected duration time of each

activity

STEP 5Determine each activity time variance

STEP 6Find out the critical path of the network

under study.

Step 7Calculate project completion probability on

the specified time (date).

STEP 3The PERT Network requires three estimates for the duration time each activity. There estimates may be judgments of three individual persons.

Three estimates are the following:i. The optimistic timeii. The pessimistic timeiii. The most likely time

BACK

STEP 4•An activity or a job expected time duration (i.e. expected time to complete an activity or task)is given by:

Where:Te is the expected duration of an activity.To is the optimistic duration of an activity.Tp is the pessimistic duration of an activity.Tm is the most likely duration of an activity.

STEP 4: EXAMPLE

An activity optimistic and pessimistic duration time estimates are 50 days and 80 days, respectively. In addition, the most likely duration time estimate for the activity is 60 days.

Find the expected duration time.

Activity duration time estimates are as follows:To = 24 daysTp = 30 daysTm = 24 days

Solving:

BACK

STEP 4: EXAMPLE

STEP 5•This step is concerned with calculating the variance of each activity duration time. The following formula is used to determine the variance:

Where:is the variance of the activity duration time

STEP 5: EXAMPLE

An activity optimistic and pessimistic duration time estimates are 50 days and 80 days, respectively. In addition, the most likely duration time estimate for the activity is 60 days.

Find the variance of the activity.

STEP 5: EXAMPLEGiven:

To = 50 daysTp = 80 days

Solving:

BACK

STEP 6

The critical path of the network is given by the longest path of the network. The duration time of the project is given by the total sum of the activity expected duration times of the longest path. The term “critical is used to signify that if any delay in the completion of activities along the longest path occurs then the whole project will be delayed.

BACK

STEP 7This is concerned with calculating the project completion probability on the specified time (date). The following transformation formula is used:

Where:S is the total sum of variances of the activity

time along critical pathtec is the last network activity's earliest expected

completion time.Td is the due date for the completion of the project

x Probability

-4 0.00003

-3.5 0.00023

-3 0.00135

-2.5 0.006

-2 0.02

-1.5 0.07

-1 0.16

-0.5 0.31

0 0.5

0.5 0.69

1 0.84

1.5 0.93

2 0.98

2.5 0.99

3 0.999

3.5 0.9998

4 0.9999

STEP 7: EXAMPLEThe PERT network of a project contains nine activities. After the analysis it is found that the last network activity's earliest expected completion time is 44 days. In addition, the total sum of the variances of the activity duration times along critical path is 16. the due date for the completion of the project is 52 days. Calculate the probability that the project will be accomplished on due date or specified time.

STEP 7: EXAMPLEGiven:

Td = 52 dayss = 16 daystec = 44 days

Solving:

Critical Path Method (CPM)

Steps to Develop and Analyze a CPM NetworkThe following steps are associated with the construction and analysis of a CPM network:i. Decompose the project into individual

jobs or task and identify each of them.ii. Find out activities’ or tasks’ sequence

and develop the network.iii.Estimate each activity’s duration time.iv.Find out the critical path of the

network under study.

Symbols and Definitions Used to Construct and Solve a CPM or a PERT Network

CIRCLEIt is used to represent an event. Broadly speaking the circle represents an unambiguous point in time in the project’s life.

Symbols and Definitions Used to Construct and Solve a CPM or a PERT Network

CIRCLE with DIVISIONSIt also represents an event. However, this circle is divided into three portions. The top portion is used to label the event with a number. The bottom portion is divided into two halves. The left is used for the latest event time whereas the remaining portion is used for the earliest event time.

Event Number

Latest event time

Earliest event time

Symbols and Definitions Used to Construct and Solve a CPM or a PERT Network

Earliest Event TimeThis is the earliest time in which an event can be reached; in other word the earliest time in which an activity can be completed.

Latest Event TimeThis is the latest time in which an event can be reached without delaying the completion of the project.

Symbols and Definitions Used to Construct and Solve a CPM or a PERT Network

CONTINUOUS ARROWThis represent an activity. An activity always begins from a circle and ends at a circle. In addition, to accomplish an activity it requires time, man power and money.

Symbols and Definitions Used to Construct and Solve a CPM or a PERT Network

DOTTED ARROWThis represent a dummy activity. It is an imaginary activity and does not consume any money, man power or time. More clearly, it is accomplished in zero time.

Critical Path of the Network

This is the path which is the longest path from one end (i.e. from the first event) of a CPM or a PERT network to another end (i.e. to last event). For the completion of the project on time, the activities along the critical path have to be accomplished on time; otherwise the project will be delayed. The total time of the critical path is given by the largest sum of expected activity time of all the path which originated from the first event and terminate at the last event.

Example 1

Explain the dummy activity with the aid of a CPM network diagram.

E

D

B

C

F

A

Example 2A small construction project was broken down into eight activities as shown in the table below. Construct the CPM network for the project and determine its critical path.

Activity ActivityDesignation

Immediate predecessor

activity

Expected duration time

for each activity

A 1-4 - 10

B 1-3 - 15

C 1-2 - 5

D 3-4 1-3 20

E 4-5 3-4, 1-4 25

F 3-5 1-3 10

G 2-5 1-2 5

H 5-6 4-5, 3-5, 2-5 30

Essential Formulas and Procedure for Determining the Critical Path of a Network

Essential Formulas

Where:t1(j) is the latest event time of the event j.t2(j) is the earliest event time of the event j.t1(k) is the latest event time of the event k.t2(k) is the earliest event time of the event k.Y (j,k) is the expected completion time of the

activity between event j and k.

Event k

t1(k)

t2(k)

Eventj

t1 (j)

t2(j)

Y (j,k)

Essential FormulasFORMULA A:

Latest start time = t1(k) – y (j,k)

FORMULA B:Latest finish time = t1(k)

FORMULA C:Earliest finish time = t2(j) + y (j,k)

FORMULA D:Earliest start time = t2(j)

FORMULA E:Total Float = t1(k) – t2(j) – y(j,k)

Procedure for Determining the Critical Path of a Network

This is basically a four step approach:i. Construct network.ii. Determine the earliest event time,

t2, of each event of the network. This can be determined by making a forward pass of the network.

iii.Determine the latest event time, t1, of each event of the network. This can be determined by making a backward pass of the network.

Procedure for Determining the Critical Path of a Network

iv. Choose those network event whose t1=t2. If the network has only one path with event satisfying the condition t1=t2, then such a path is critical. However, if there is more than one such path satisfying the t1=t2 condition then go to next step.

v. For each path which satisfies the condition of step(iv), determine the total floats for all its activities. Sum the total floats of each path activities. The path which has the least sum of the total float is the critical path of the network.

EXAMPLE:

31 5 6

4

2

2010

1530

25

10

55

Determine the critical path by calculating each event’s earliest and latest times.

EXAMPLE:

A small research and development project is composed of six activities. These activities along with their estimated times are identified in the Table. Construct the PERT network for the project and determine the project probability of completion time. The project due date is 54 days.

EXAMPLE:

Activity

Activity designat

ion

Immediate

predecessor

Time estimates for each activity

(days) Expected times

Variance

To Tm Tp

A 1-3 - 20 25 30 25 2.78

B 1-2 - 10 15 20 15 2.78

C 3-2 1-3 8 10 12 10 0.444

D 3-4 1-3 10 10 10 10 0

E 2-4 1-2, 3-2 10 12 14 12 0.444

F 4-5 3-4, 2-4 5 5 5 5 0

Benefits and Drawbacks of the Critical Path Method

Benefits:i. It helps to improve communication and

understanding.ii. It helps in cost control and cost savings.iii. It depicts interrelationship in work flow.iv. It identifies those work activities which are

vital to completing the project on time.v. Needs for labor and resources can be

determined in advance.vi. It can determine the duration of the project

systematically.

Benefits and Drawbacks of the Critical Path Method

vii. It can be computerizedviii. It helps to avoid duplications and

omissions.ix. It helps to monitor the progress of the

project effectively.x. It allows alternative simulations.

Drawbacks:1. Costly and time consuming.2. Poor time estimates.3. Bias to use pessimistic time estimates.

Thank You!!!