System Engineering 3(3+0)

LESSON 3. CRITICAL PATH METHOD

a) Let zero be the starting time for the project.  Then for each activity there is an earliest starting time (ES) relative to the project starting time. The earliest finishing time is denoted by   

Thus the formula is

            EF_i  (or) ES_j = max {ES_i + t_ij}

Where  Es_j denotes the earliest start time of all the activities emanating from node i and t_ij is the estimated duration of the activity i-j.

Example:

In the above example the activity is from i-j, the duration of time is 5 hours. Here start time is ES_j= max {ES_i + t_ij}

Initial start time ES_i=0.

ES_j= max {0+ 5} =5.

Initially the starting time will be 0 The finishing time for the i^th event is 5.Staring time for the j^th event is 5.

b) Let us suppose that we have a target time for completing the project.  Then this time is called the latest finish time (LF) for the final activity.  The latest start time (LS) is the latest time at which an activity can start if the target is to be maintained.  It means that for the final activity, its LS is simply LF – activity time.

                    LF_i = min {LF_j – t_ij}, for all defined (i, j) activities.      

Critical Path:

Certain activities in a network diagram of a project are called critical activities because delay in their execution will cause further delay in the project completion time. Thus, all activities having zero total float value are identified as critical activities.

The critical path is the continuous chain of critical activities in a network diagram. It is the longest path starting from first to the last event and is shown by a thick line or double lines in a network.

The length of the critical path is the sum of the individual times of all the critical activities lying on it and defines the minimum time required to complete the project.

The critical path on a network diagram can be identified  as:

(a)  ES _i = LF_i

(b) ES _j = LF_j

(c)  ES _j – ES_i  = LF_j – LF_i = t_ij.

Critical Path Method (CPM)

The iterative procedure of determining the critical path is as follows:

Step 1: List all the jobs and then draw a network diagram.  Each job is indicated by an arrow with the direction of the arrow showing the sequence of jobs.  The length of the arrows has no significance.  Place the jobs on the diagram one by one keeping in mind what precedes and follows each job as well as what job can be done simultaneously.

Step 2: Consider the job’s times to be deterministic.  Indicate them above the arrow representing the task.

Step 3: Calculate the earliest start time (EST) and earliest  finish time (EFT) for each event and write them in the box marked   Calculate the latest start time (LST) and latest finish time (LFT) and write them in the box marked   .

Step 4: Tabulate various times, i.e., activity normal times, earliest times and latest  times, and mark EST and  LFT on the arrow diagram.

Step 5: Determine  the total float for each activity  by taking differences between EST and  LFT.

Step 6: Identify  the critical activities and connect them with the beginning node and the ending node in the network  diagram by double line arrows.  This gives the critical path.

Step 7: Calculate the total project duration.

Slack / Float of an Activity and Event

The float (Slack) or free time is the length of time to which a non-critical activity and the time between its ES & LF is longer than its actual duration or an event can be delayed or extended without delaying the total project completion time.(ie) the  difference between the latest finish and earliest start time.

There are four types of floats namely

 a) Total float

b) Free float

c) Independent float

d) Interference float

a) Total float

     Difference between the latest finish and earliest finish time for the activity

          Total float = TF_ij = LF_j - EF_j

b) Free float

     It is defined by assuming that all the activities start as early as possible.  The free float for the activity (i, j) is the excess available time over its duration.

                   LF_ij = ES_j – ES_i - t_ij

c) Interference float

     The difference between total float and free float.     

d) Independent float

     The time by which an activity can be rescheduled without affecting the preceding or the succeeding activities is known as independent float.

     Independent float = Free float – Tail event Slack                                              

Advantages of Critical Path Method(CPM)

1. CPM was developed for conventional projects like construction project which consists of well know routine tasks whose resource requirement and duration were known with certainty.

2. CPM is suited to establish a trade off for optimum balancing between schedule time and cost of the project.

3. CPM is used for projects involving well know activities of repetitive in nature.

       However the distinction between PERT and CPM is mostly historical.

Problem

CPM

          The following table gives the activities of a construction project and duration.                 

Activity	1-2	1-3	2-3	2-4	3-4	4-5
Duration (days)	20	25	10	12	6	10

 

(i) Draw the network for the project.

(ii) Find the critical path.

(iii) Find the total, free and independent floats each activity.

Solution:

          The  first step is to draw the network and fix early start and  early finish schedule and then  late start-late finish schedule as in figure 1 and figure 2.

Fig.-2

Activity	Total Slack	Free Slack	Independent Slack
1-2	0	0	0
1-3	5	5	5
2-3	0	0	0
2-4	4	4	4
3-4	0	0	0
4-5	0	0	0

 

To find the critical path, connect activities with ) total slack and we get 1-2-3-4-5 as the critical path.

          Check with alternate paths.

          1-2-4-5    = 42

          1-2-3-4-5 = 46*

          1-3-4-5    = 41