Lecture 11 Pert-cpm

EMG181 Quarter 1 SY2012-13 1

NETWORK MODELSPERT-CPM

Meeting 21Lecture 11


Network Models for Project Management

Program Evaluation and Review Technique (PERT) Used if the duration of each project is not known with certainty. Can be used to estimate the probability that the project will be completed

by a given deadline.

Critical Path Method (CPM) Used if the duration of each activity is known with certainty. Can be used to determine the length of time required to complete a

project. Can also be used to determine how long each activity in the project can

be delayed without delaying the completion of the project.


Definition of Terms used in PERT and CPM

• Activity – an effort that requires resources and takes a certain amount of time for completion.

• Event – a specific accomplishment at a recognizable point in time; a milestone, checkpoint. Events do not have a time duration per se. To reach an event, all activities that precede it must be completed.

• Project – a collection of activities and events with a definable beginning and a definable end.

• Network – a logical and chronological set of activities and events, graphically illustrating relationships among the various activities and events of the project.

• Critical Activity – is an activity that, if even slightly delayed, will hold up the scheduled completion date of the entire project.

• Path – sequence of adjacent activities that form a continuous path between two events.

• Critical Path – sequence of critical activities that forms a continuous path between the start of a project and its completion.


Major Difference between PERT and CPM

PERT and CPM are very similar in their approach; however two distinctions are usually made between them:

1. In PERT, three estimates are used to form a weighted average of the expected completion time of each activity, based on a probability distribution of completion times. Therefore, PERT is considered a probabilistic tool. In CPM, there is only one estimate of duration; that is, CPM is a deterministic tool.

2. CPM allows an explicit estimate of costs in addition to time. Thus, while PERT is basically a tool for planning and control of time, CPM can be used to control both the time and the cost of the project.


PERT-CPM provides answers to the following questions:

1. Which activities are critical?2. Which activities are non-critical?3. What is the amount of slack (or float) on each

non-critical activity?


PERT-CPM Process

Step 1. Analysis of the project Formulation Step 2. Sequence the activities Step 3. Estimate activity times and costs Step 4. Construct the network Solution Step 5. Event analysis Step 6. Activity analysis Analysis and Step 7. Monitoring and control application Step 8. Resource utilization


Constructing the PERT Diagram The Pert network is a graphical representation of the project data which shows

the interrelationships among the activities, the events, and the entire project.

How to Construct:1. Start by viewing an activity as an arrow (arc) between two events (circles).

The arrow points in the direction of the time flow, but its length is arbitrarily set at a suitable length for drawing, and is not related to the duration of the activity. The number circled before the arrow is the event that precedes the activity. The number circled after the arrow is the succeeding event.

2. Construction of the network starts with event the beginning of the project or the activity (or activities) that does not require any preceding activity. It is placed at the left side of the diagram, with the flow pointing to the right. The event before this first activity is always number 1, and the event after is numbered 2 (add 3, 4... if there are more than one first activities).

3. Draw the activities following the order specified in the project data. 4. The rest of the network is constructed in the same manner. The diagram

grows to the right until all events and activities are depicted. The last event is the completion of the project.


Three rules in constructing a network diagram:

1. Each activity is represented by one and only one arrow in the network.

2. Each activity must be identified by distinct starting and ending nodes.

3. In building a network, care must be taken to assure that the activities and events are in proper sequence. One device that helps in proper sequencing is dummy activities. These activities are characterized by their use of zero time and zero resources; their only function is to designate a precedence relationship. Graphically, such activities are shown as broken lines.


EVENT ANALYSIS

The following procedure is used in event analysis:a. Enter time estimates on the network.b. Compute the earliest and latest dates for all events.c. Find the slack on the events and identify critical events.d. Find the slack on the activities and identify critical activities.e. Find the critical path.


ExampleTask ID Task Description Prerequisites

Optimistic Duration

Most Likely Duration

Pessimistic Duration

A Build internal components none 1 2 3

B Modify roof and floor none 2 3 4

C Construct collection stack A 1 2 3

D Pour concrete and install frame B 2 4 6

E Build high-temperature burner C 1 4 7

F Install control system C 1 2 9

G Install air pollution device D, E 3 4 11

H Inspection and testing F, G 1 2 3


PERT Time EstimatesOptimistic estimate tO : An estimate of the shortest possible time (duration) in

which the activity can be accomplished. The probability that the activity will take less than this time is 0.01.

Most likely estimate tm : The duration that would occur most often if the activity were repeated under exactly the same conditions many times. Equivalently, it is the time that would be estimated most often by experts.

Pessimistic estimate tP : The longest time that the activity could take “when everything goes wrong.” The probability that the activity will exceed this duration is 0.01.

Once the three time estimates are obtained, their weighted average is computed. This average, called the mean time of an activity (te) is computed as follows:

te = tO + 4tm + tP

After all the te are computed, these are then placed in the network and the network analyzed/computed the same as in CPM.

This equation is based on the assumption that the Beta distribution is the probability distribution of duration times. Other weights may be used based on real-life experience.

6


ExampleTask ID Task Description Prerequisites

Optimistic Duration

Most Likely Duration

Pessimistic Duration

Average Time

A Build internal components none 1 2 3 2

B Modify roof and floor none 2 3 4 3

C Construct collection stack A 1 2 3 2

D Pour concrete and install frame B 2 4 6 4

E Build high-temperature burner C 1 4 7 4

F Install control system C 1 2 9 3

G Install air pollution device D, E 3 4 11 5

H Inspection and testing F, G 1 2 3 2


Determine Earliest and Latest Times

Compute the Earliest and Latest Dates

This approach is based on two important concepts:

The earliest possible event date ES. The latest allowable event date LC .

ES - is the time immediately after all the preceding activities have been completed. LF - is the latest time that can occur without causing a delay in the (already determined) completion date of the project. This completion date can be either the earliest possible completion date or any other agreed-upon date.

20

15 20 (completion

date)

LF = 10 20 10

17 1

a (15) 1

b (17)

ES3 = 17

2

3


Estimating ES: Conduct a Forward PassIn order to compute ES for an event, the duration of each path leading to the event is computed. If several paths lead to an event, then the path with the largest elapsed time is selected. Note that for computation of each ES, it is not necessary to return to the beginning of the network. Just use the following formula:

Length (longest) of a path = Duration of the last activity on the path + ES of preceding event

Estimating LF: Conduct a Backward PassTo compute LF of each event, start from the last event and work backward all the way to event 1, using the largest elapsed time (in the same manner as Step A).


Find the Slack on the Activities and Identify Critical Activities

Similar to a slack on events, there is also slack on activities. This slack can tell us how long the activity can linger without delaying the whole project. This slack is called the total float.

A critical activity is one wherein there is 0 slack.

Activity slack = LF for the event at – ES for the event at the – duration of end of the activity start of the activity the activity


ExampleActivity Duration ES LS LF

Total Float

Critical

A 2 0 0 2 0 Yes

B 3 0 1 4 1 No

C 2 2 2 4 0 Yes

D 4 3 4 8 1 No

E 4 4 4 8 0 Yes

F 3 4 10 13 6 No

G 5 8 8 13 0 Yes

H 2 13 13 15 0 Yes

The critical path for the project is A-C-E-G-H and PCT is 15.


Find the Critical Path

The critical path is the path(s) in the network, leading from the beginning of the project to its end, all of whose activities and events are critical.

Note that there can be more than one critical path in the network; and that the critical path is the longest (timewise) path in the network.


Activity AnalysisThis method is another way to derive the critical activities, critical path and the slack.

ES = Earliest Start time for an activity. This time is equivalent to ET of the event from which the activity starts. We assume that all predecessor activities started at their earliest times, and have been completed.

EF = Earliest Finish time for an activity. Assuming the activity started at its ES and lasted its planned duration (te), then EF = ES + te

LF = Latest Finish time for an activity. This is the latest time by which an activity can be completed without delaying the project. It is equal to the LT of the event at the end of the activity.

LS = Latest Start for an activity. This is the latest an activity can start without jeopardizing the project’s deadline. LS = LF – te

Compute ES and EF on a Forward Pass:ES for the project is usually zero.ES for the other activities = largest value of EF for all activities ending at the event from which the

activity starts. (First ES, then EF, then ES, ... to EF of final activities.)Compute LF and LS on a Backward Pass:Start by setting the LF of all final activities to the largest EF. Alternatively, a desired finish date

(larger than the largest EF) may be used as the starting LF.LF of an activity entering a particular event = smallest value of the LS’s for all activities staring

from that event. (First LF, then LS, then LF, ..., LS of starting activities.)

Regular Slack (Total Float) = LS – ES


Free FloatFree Float (FF) = slack that represents the time any activity can be delayed before it delays the earliest start time of any activity immediately following.

Regular Slack (Total Float)

For example, the slack (12 – 6 = 6 on 1–3) of the figure above is free.

Shared Slack (Slack on a Noncritical Path)

Free float for the second figure is 2 on 2–3. Notice that the FF on 1–2 is 0.

In general, if there are several noncritical activities in a series, only the last one will have a free float.

Free float is important in the case of shared slack. It means that if a slack has not been shared, it can all be used in the last sharing activity. Computer printouts typically give both the TF and FF for each activity. Basically, the free float on an activity can be calculated as the early start of the successor event, minus the early finish of the activity.

2 5

3 1

7

6

2 5

3 1

7

14


The Case when LF ES for the Whole Project

If LF > ES for the last event, then the slack for the last event will be positive.

If LF < ES for the last event, a negative slack will result, indicating that the desired date cannot be achieved and a delay of the magnitude of the negative value is expected.


Finding the Probabilities of Completion in PERT (Risk Analysis)

The 3 estimates of activity duration in PERT, to, tm and tp , are assumed to follow a probability distribution called the Beta distribution.

= tP – tO standard deviation of an activity

6

An example of the Beta distribution for the activity b with to = 1, tm = 3, tp = 11. => te = 4

1 3 114

1

20

Frequency (%)

Activity Duration (wks)


For activities where tO = tm = tP , there is no uncertainty involved in their estimates.

Assuming that the duration of the activities are independent of each other, the variance of a group of activities can be computed by adding the variances of the activities in that group.

The variance of the critical path corresponds to the variance for the whole project. The variance for any event can be computed in a similar manner by considering the group of activities along the critical path leading to that event.

The method described above is valid only if the following 3 assumptions hold:1. there is a large number of activities (at least 25) on the critical path2. the activities’ completion times are independent of each other 3. the noncritical paths are not relevant

If the above assumptions are not valid, simulation must be used for the risk analysis.

2204/13/23


Quantifying the Chance of Finishing a Project

Computing the chance of completing the project by a certain desired time, or the duration necessary for obtaining any desired probability of completion.

Let TS = earliest project completion time = ET for the last event

D = desired completion time

Z = number of std deviations of a normal distribution corresponding to the probability of completing the project by the desired completion time

Z = X – mean = D – TS V V = variance of critical path

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Example 1. Finding the Probability of Completion within a Desired Time, D

Based on data, a project can be completed in 27 weeks on the average with a variance of 4.67 weeks. You wish to know the probability of completing the project on or before the 30th week, as specified in their contract.

Thus: D = 30, TS = 27 , V = 4.6667

Therefore Z = (30 – 27) / 6667.4 = 3 / 2.1602 = 1.3888 1.39

From the normal distribution table Z = 1.39 has probability equivalent = 0.9177. Therefore, there is a 91.77 % probability of completing the project within 30 weeks.

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The Variance of Noncritical Paths The probability of completing a project is related to

the variance of the critical path.

If more than one path exists from the start of the project to the finish, then V should be computed for all such paths. To be conservative, use the path with the largest V in the computation of the probability of completion for dates after the expected completion time; and use the smallest V for probabilities of completion for dates before the expected completion time.

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Example 1: Project CrashingMIT Engineering has accepted a project to design a new tool for its client. The company has

limited time and resources to complete the project. It has enumerated the following activities to complete the project with the corresponding activity information.

Immediate Normal Normal Crash Cost/ CrashActivity Predecessor Time (days) Cost (PhP) Day (PhP) Time (days) A - 4 400 125 3 B A 5 800 200 4 C A 4 500 150 2 D B 3 600 225 2 E C 3 250 100 2 F B, E 4 600 175 2

a. Draw the project network and identify the critical path.b. Determine the project completion time and cost.c. Compute for the project cost if MIT Engineering was given a deadline to complete the

project in 13 days.d. If the client wants to accelerate the project to 10 days completion, should MIT

Engineering comply? The client will charge MIT Engineering PhP 200 (PhP 175) per day beyond the 10-day deadline. Determine the optimal crashed time for the project.



CPM: Cost-Time Relationships

Activity

Normal Time

Normal Cost

Crash Time

Crash Cost

Slope

A 5 100 4 140 B 9 200 7 300 C 7 250 4 340 D 9 280 7 340 E 5 250 2 460 F 11 400 7 720 G 6 300 4 420 I 8 80 6 140

Total

Cost in $, and time in days.

Find the cheapest 19-day solution.

1

2 4

3

5

6

I E

G

D

C

B

A

F

H


Project Crashing1. Crash C for 3 days

a. PCT 22 daysb. Project Cost $1,950c. CPs: ACEI, ADHI, BEI

2. Crash I for 2 daysa. PCT 20 daysb. Project Cost $2,010c. CPs: ACEI, ADG, ADHI, ACF, BEI, BF

3. Crash A and B for 1 daya. PCT 19 daysb. Project Cost $2,100c. CPs: ACEI, ADG, ADHI, ACF, BEI, BF

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Lecture 11 Pert-cpm

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