For each activity an estimate must be made of time that will be spent in the actual accomplishment of that activity.

**Critical path analysis**

For each activity an estimate
must be made of time that will be spent in the actual accomplishment of that
activity. Estimates may be expressed in hours, days, weeks or any other
convenient unit of time. The time estimate is usually written in the network
immediately above the arrow. For the purpose of calculating various times of
events and activities, the following terms shall be used in critical path
calculations:

*E _{i}
*=
Earliest start time of event

*L _{j}
*= Latest
start time of event

*t _{ij}
*=
Duration of activity (

The next step after making the
time estimates is the calculations which are done in the following ways:

(i) Forward
Pass Calculations

(ii) Backward Pass Calculations.

We start from the initial node 1
with starting time of the project as zero. Proceed through the network visiting
nodes in an increasing order of node number and end at the final node of the
network. At each node, we calculate earliest start times for each activity by
considering *E _{i}* as the
earliest occurrence of node i.

**Step 1.** Set *E*_{1} = 0 ; *i* = 1 (initial node)

**Step 2.** Set the earliest start
time(EST) for each activity that begins at node *i* as *ES _{ij} *=

**Step 3.** Compute the earliest
finish time(EFT) of each activity that begins at node *i* by adding the earliest start time of the activity to the duration
of the activity. Thus *EF _{ij}*
=

**Step 4.** Move on to next node,
say node *j*(*j > i*) and compute the earliest start time at node *j*, using *E _{j}* = max

**Step 5.** If *j = n* (final node), then the earliest
finish time for the project is given by *E _{n}*
= max {

We start from the final (last)
node n of the network, proceed through the network visiting nodes in the
decreasing order of node numbers and end at the initial node 1. At each node,
we calculate the latest finish time and latest start time for each activity by
considering *L _{j}* as the
latest occurrence of node

**Step 1.** *L _{n} *=

**Step 2.** Set the latest finish
time (LFT)of each activity that ends at node *j* as *LF _{ij} *=

**Step 3.** Compute the latest
start time(LST) of all activities ending at node *j*, subtracting the duration of each activity from the latest finish
time of the activity. Thus, *LS _{ij}*
=

**Step 4.** Proceed backward to
the next node *i*( *i*<*j*) in the sequence
and compute the latest occurrence time at node *i* using

*L _{i}
*=

**Step 5.** If *j* = 1 (initial node), then *L*_{1}
= min {*LS _{ij}* }, = min {

The longest path connected by the
activities in the network is called the critical path. A path along which it
takes the longest duration.

For the activity (i,j), to lie on
the critical path, following conditions must be satisfied:

(i) *E _{i} *=

(ii) *E _{j} *-

**Example 10.13**

Compute the earliest start time,
earliest finish time ,latest start time and latest finish time of each activity
of the project given below:

*Solution:*

Earliest start time (EST)and
latest finish time(LFT) of each activity are given ithe following network.

Here the critical path is
1-2-4-5, which is denoted by double lines.

The longest duration to complete
this project is 21 days.

The path connected by the
critical activities is the critical path(the longest path).

Critical path is 1-2-4-5 and
project completion time is 21 days.

**Example 10.14**

Calculate the earliest start
time, earliest finish time, latest start time and latest finish time of each
activity of the project given below and determine the Critical path of the
project and duration to complete the project.

*Solution:*

Here the critical path is 1–2–4–6

The project completion time is 25
weeks

**Exercise 10.2**

1. Draw
the network for the project whose activities with their relationships are given
below:

Activities A,D,E can start
simultaneously; B,C>A; G,F>D,C; H>E,F.

2. Draw the event oriented
network for the following data:

3. Construct
the network for the projects consisting of various activities and their
precedence relationships are as given below:

A,B,C can start simultaneously
A<F, E; B<D, C; E, D<G

4. Construct the network for each
the projects consisting of various activities and their precedence
relationships are as given below:

5. Construct the network for the
project whose activities are given below.

Calculate the earliest start
time, earliest finish time, latest start time and latest finish time of each
activity. Determine the critical path and the project completion time.

6. A project schedule has the
following characteristics

Construct the network and
calculate the earliest start time, earliest finish time, latest start time and
latest finish time of each activity and determine the Critical path of the
project and duration to complete the project.

7. Draw the network and calculate
the earliest start time, earliest finish time, latest start time and latest
finish time of each activity and determine the Critical path of the project and
duration to complete the project.

8. The following table gives the
activities of a project and their duration in days

Construct the network and calculate
the earliest start time, earliest finish time, latest start time and latest
finish time of each activity and determine the Critical path of the project and
duration to complete the project.

9. A Project has the following
time schedule

Construct the network and
calculate the earliest start time, earliest finish time, latest start time and
latest finish time of each activity and determine the Critical path of the
project and duration to complete the project.

10. The following table use the
activities in a construction projects and relevant information

Draw the network for the project
,calculate the earliest start time, earliest finish time, latest start time and
latest finish time of each activity and find the critical path. Compute the
project duration

Tags : Network Analysis , 11th Business Mathematics and Statistics(EMS) : Chapter 10 : Operations Research

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