Pert_cpm Network Analysis Final

8/8/2019 Pert_cpm Network Analysis Final

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PERT/CPM NETWORKS


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PERT (Program Evaluation and ReviewTechnique) /

CPM (Critical Path Method)

Use project networks to help schedule project

activities


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HISTORY: PERT

Used for random completion of activity.

Developed in the late 1950s by U.S. Navy, Booz -

Allen Hamilton, and Lockeheed Aircraft

Probabilistic activity durations. It has a diagram of

interrelationship of a complex series ofactivities;

It has a free-form network showing each activity

as line between events.


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HISTORY: CPM

Uses fixed time estimation for every activity.

Developed in the year 1957 by Dupont De

Nemours Inc. (M.R. Walker and J. E. Kelly of

Remington Rand)

Deterministic activity durations (deterministic

time) and cost estimates. Its advantage includes

the concept of crash efforts and costs.


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Terminology used in PERT & CPM

Activity- task or set of tasks that are required to be accomplished tocomplete the project uses resources & time

o Series- an activity cannot be performed unless another activity is undertaken.

o Parallel- can be performed simultaneously

Predecessors- Activities that must immediately precede a givenactivity.

dummy activity- imaginary activities which require no time and noresources

Event- state resulting from completion activities consume no resources or time

Predecessor activities must be completed for event to be realized


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Cont: Terminology

Milestones- events that mark significant progress

Network- diagram of nodes and arcs

used to illustrate technological relationships

Path- series of connected activities between two events

Critical Path- set of activities on a path that if delayed will delaycompletion of project

Longest Path through the PERT network

Critical Time- time required to complete all activities on the criticalpath


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Conventions used in drawing network diagrams (Arrows &

Circles )

Activityon Arrow (AOA) : The activities are denoted by Arrows and

events are denoted by circles Typically used in PERT

If activity Xmust precede activity Y, there are Xleads into arc Y. Thus, the

nodes represent events or milestones (e.g., finished activityX). Dummy

activities of zero length may be required to properly represent precedence

relationships.

Activityon Node(AON) : Activities are denoted by circles(or nodes) andthe precedence relation ships between activities are indicated by arrows

Typically used in CPM

may be less error prone because it does not need dummy activities or arcs.

PERT/CPM Planning Process

Cont: Construct Network Diagram


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Both the PERT and CPM are time oriented techniques.

Both techniques when used can help answer questions suchas the following about large and complex projects.

1. When will the entire project be completed?

2. What are the critical activities or the activities that willcause delay to the completion of the entire project, if

these activities are delayed?

3. What are the non-critical activities or the activities that

can be delayed without affecting the entire projectscompletion time?


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4. What is the probability that the entire project can be

finished on the target date?5. At any given date, is the project on schedule, behind or

ahead?

6. At any given date, are the expenses equal to, less than,

or greater than the original budget?7. Are there sufficient resources available to finish the

entire project on time?

8. What is the best way to finish the entire project in the

shorter time at least cost?


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The following steps are common to both PERT and

CPM:

1. Define the Project and all of its significant

activities.

2. Develop a precedence relationship among theactivities. (Determine which activity must precede

and which must follow.)

3. Construct the network diagram of the entire

project.


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4. Assign the estimates to each activity. (PERT uses

three time estimates, CPM has only one timefactor.)

5. Compute the critical path. (The paths where

activities have zero slack time.)

6. Use the network as an aid in planning, scheduling,

monitoring, and controlling the project.


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PERT and CPM have the same objectives, and the

analyses used in both techniques are very similar. However,they differ to some extent in their terminologies and in the

construction of the network. But the major difference is

that PERT uses three time estimates for each activity. Each

estimate has an associated probability of occurrence (usingthe beta probability distribution) which in turn is use in

computing the expected activity time and the variance for

each activity. CPM, on the other hand, makes the

assumption that activity times are known with certainty, sothat only one time factor is given for each activity.


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Illustration:

Activity Code Immediate Predecessors Estimated Time (in

weeks

A - 2

B - 3

C A 3

D A 1

E B 2

F C 4

G D, E 1

Critical Path Analysis- The objective of the critical path analysis isto determine the critical activities, that is the activity if delay will also delay

the completion of the en tire project.


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Solution:

a. The network diagram of problem 1

b. Find all the path between event or node 1 and event 6 and the total activity time

for each path.

Path A - C - F : 2 +3 + 4 = 9 weeks

Path A - D -G : 2 +1 + 1 = 4 weeks

Path B - E - G : 3 +2 + 1 = 6 weeksNine weeks represent the longest path, therefore the critical path is

PathA - C - F

c. From part (b) the earliest expected time that the project can be completed is 9

weeks.

2

1

3

5

4

6

C


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Computing the ES and EF Times

The ES and EF times are computed in a forward pass through the network.Beginning

with the initial node 1 of figure 1 in problem 1, we assign 0 as the ES time for

activity A. Since the duration time of A is 2 weeks, the earliest time A could be

finished EF is week 2, or EF = 2 of A. Now let us look at activity C. It cannot start

until activity A is finished. Since the duration time of C is 3 weeks, the earliest time

C could be finished is week 5, or EF = 2 +3 = 5 for activity C. Continue this manneruntil we obtain all the results as shown in figure 2. What happens when two or

more activities lead into a single node? For example, look at node 5 in figure 2, the

EF for activity D is 3 and for activity E is 5. Activity G leads out of node 5 and

cannot begin until both D and E are complete. Therefore, the earliest start time

(ES) for G is the maximum EF of D and E.

ESG

= max (EF EFe) = max (3, 5) = 5


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Figure 2The network diagram showing the ES and EF of all the activities in the

problem 1

Earliest start/ Earliest Finish Time for an ActivityES = maximum EF of all immediate predecessors

EF = ES + (activity completion time)

2

1

3

5

4

6

EF = 5

StartFinish


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Computing the LS and LF Times

The LS and LF times are computed in a backward pass through the network. Beginning

with the terminal node, we use the OCT of the project as the LF of the terminal

activity and then work backward. The LF of an activity is the smallest LS or

minimum of the LS time of its immediate predecessor. LS is calculated by

subtracting the activity completion time from its LF.

In figure 2, the OCT is 9 weeks. Therefore, the LS of the terminal activity F and G is 9 or

LF = 9. What happens when two or more activities lead out of a node? For

example, node 2 in figure 3, the LS time for C and D are 2 and 7 respectively. Thus,

the latest start time (LS) we can finish activity A is a week 2; otherwise we will

delay the start of C beyond its latest start time.

LFA = min (LSC LSD)= min (2, 7)= 2


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Figure 3

The network diagram showing the LS and LF of all the activities in problem 1.

Latest Start/Latest Finish Times for an Activity

LF= minimum LS of all immediate successor activities

LS = LF - (activity completion time)


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It is more convenient to record both the forward pass and the backward

pass on the same network diagram. For instance

LS A LF

ES t EF

Figure 4

The ES, EF, LS and LF of all the activities of problem 1 are shown on the same network

diagram


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Slack Times

Slack time for an activity in the network is the amount of time we can delay

the start of the activity without delaying the completion time of the whole

project. The critical path consists of all activities having zero slack time.

The slack time for an activity can be computed in 2 ways:

slack time = LS - ESor

slack time = LF EF

For example, for activity E,

slack time for E= LSE ESE

= 6 3

= 3


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Thus, we can start E as early as week 3 or as late as week 6. The slack time of

3 weeks means that we have up to 3 weeks to start activity E beyond the

earliest time of week 3 without delaying the optimum completion time for

the project.The slack times for all the activities of problem 1 are shown in table 2.

Table 2

Activity Schedule Chart

Activity ES EF LS LF SLACK

A 0 2 0 2 0 *

B 0 3 3 6 3

C 2 5 2 5 0 *

D 2 3 7 8 5

E 5 9 5 9 0 *

F 3 5 6 8 3

G 5 6 8 9 3


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Critical activities are activities having no slack time. Activity A, C and F are

critical activities. A delay in a single critical activity will delay the entire

project. For example, if activity C, a critical activity with zero slack, were

delayed one week, the entire project would be delayed one week. A delayin non critical activity will only delay the project by the amount of delay

more than the slack time of the activity.


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THE PERT NETWORK with

ILLUSTRATION


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Any project that can be described by activities

and events may be analyzed by a PERT network.

Hence, in using PERT, the entire project must first

be broken down into events and activities. It is alsoimportant to know the order or sequence of the

activities and the time estimate for ach activity.

The following are the steps to be taken in

developing and solving a PERT network.

THE PERT NETWORK


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1. Identify/specify each activity to be done in the project.

2. Determine the sequence of activities (or precedence

relationship) and construct a network reflecting this

relationship.

3. Ascertain time estimates for each activity.

PERT requires three times estimates for each activity.

a = optimistic time or the minimum reasonableperiod of time to complete the activity.

m = most likely (or most probable) time or the best

guess of the time required.

b = pessimistic time or the maximum reasonableperiod of time to complete the activity.


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4. Compute the expected time for each activity. Use the

formula: t = a + 4m + b

6where t = expected time

(This is based on the beta probability distribution and weights

the most likely time (m) 4 times more than either the optimistic

time (a) or the pessimistic time (b).

5. Compute the variance (v) of the activity times.

Specifically, this is the variance associated with each

expected time (t).

v =

where v =

= standard deviation


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6. Determine the critical path.

The critical path is the path where each activity has a

slack time Ts equal zero or (Ts = 0). This path is critical

because a delay in any activity along this path would

delay the entire project.

7. Determine the probability of finishing the project on the target

date (The project completion time Tcp).

Using the formula :

Z = D - Tcp


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where D = the target project completion date.

Tc = earliest expected project completion time.

V = v cp [ or the sum of the variances (v) of the

activities along the critical path

Then using this value of Z, find the probability of meeting theproject due (or target) date, using the normal probability

distribution (Table I, Appendix).


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The use of the dummy (Da) when two activities having identicalstarting and ending events are encountered ensures that the

network properly reflects the entire project.

Example:

Given the following data, develop the network.

Table D1

DUMMY ACTIVITIES AND EVENTS

Activity Immediate

Predecessors

Activity Immediate

Predecessors

A None E C, D

B None F D

C A G E

D B H F


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Solution: A dummy activity has t = 0

1

3

2

4

DE

6

5

7

D

Dummy Event

DaDummy

Event

Figure D1

Fig. D1 contains all the activities and events in their proper

sequence.


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12

3

4

5

Solution: The network on Fig. D2 is not a feasible representation of a

PERT network. In order to correct it introduce a dummy event and a

dummy activity.

Fig. D2

Example 10.3.2

Introduce a dummy activity and dummy event to correct the

following network.


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Fig. D3 is now the correct network representation.

1

2

De

3

4

5Dummy

Activity

Fig. D3


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PROBABILITY OF PROJECT COMPLETION

If the project expected completion time (Tc) and the project

completion variance (V) are given, the probability of project

completion at a specified date can now be determined. Assume

normal probability distribution.

Example

If the project completion variance (V) is 100 and the expectedproject completion time (Tc) is 20 weeks of example 10.3.3, what is the

probability that the project will be finished on or before 25 weeks?

SOLUTION:

Given:

Tc = 20 weeks

D = 25 weeks (the desired completion time)


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V = 100

Z = D Tc = 25 20

= 0.5

The normal curve would appear as follows:

20 25

From the normal probability distribution table., the area under

the curve for Z = 0.5 is 0.6915. Thus, the probability of completing theproject in 25 weeks is approximately 0.69 or 69%


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Example 10.3.5

A metal works plant in Sucat, Paranaque has long been trying

to avoid the expense of installing air pollution control device. The local

environmental protection group has recently given the plant 16 weeks

to install a complex air-filter system on its main smokestack. The plant

was warned that if will be forced to close unless the device is installed

in the allotted period. The manager, David Joseph, wants to make sure

the installation of the filtering device progress smoothly and on time.


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All the activities involved in the project are shown below on

Table 10.3.6

ACTIVITY DESCRIPTION IMMEDIATE

PREDECESSOR

A Build internal components None

B Modify roof and floor None

c Construct collection stack A

D Pour concrete and install frame B

E Build high temperature burner C

F Install control system C

G Install air pollution device D,E

Table 10.3.6

H Inspection and Testing F,G


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ACTIVITYOPTIMISTIC

a

MOST LIKELY

m

MOST LIKELY

m

Expected Time Expected Time

A 1 2 3

B 2 3 4

C 1 2 3

D 2 4 6

E 1 4 7

F 1 2 9

G 3 4 11

H 1 2 13

= 2

= 3

= 2

= 2

= 4

= 3

= 5

=

Table 10.3.7 Time Estimate (in weeks) for the metal works plant.


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Solution : The first step is to construct the network diagram.

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