Network Analysis - CPM
Project management is evolved to coordinate and control all project activities in an efficient and cost effective manner. The salient features of a project are:
- A project has identifiable beginning and end points.
- Each project can be broken down into a number of identifiable activities which will consume time and other resources during their completion.
- A project is scheduled to be completed by a target date.
- A project is usually large and complex and has many interrelated activities.
- The execution of the project activities is always subjected to some uncertainties and risks.
The predecessor to network techniques, the Gantt chart was developed, during world war I, by Henry L Gantt, for the purpose of production scheduling. An example of Gantt chart is shown below.
The Gantt chart was later modified to bar chart , which was used as an important tool in both the project and production scheduling.
The bar charts, then developed into milestone charts and next into network techniques (such as CPM and PERT). Bar chart didn't give information about critical activity and interdependence among activities. Bar chart is activity oriented while milestone chart is event oriented.
A network is the graphical representation of the project activities arranged in a logical sequence and depicting all the interrelationships among them. A network consists of activities and events.
An activity is a physically identifiable part of a project, which consumes both time and resources. Activity is represented by an arrow in a network diagram. The head of an arrow represents the start of activity and the tail of arrow represents its end. Activity description and its estimated completion time are written along the arrow. An activity in the network can be represented by a number of ways: (i) by numbers of its head and tail events (i.e. 10-20 etc.), and (ii) by a letter code (i.e. A, B etc.). All those activities, which must be completed before the start of activity under consideration, are called its predecessor activities. All those activities, which have to follow the activity under consideration, are called its successor activities.
An activity, which is used to maintain the pre-defined precedence relationship only during the construction of the project network, is called a dummy activity. Dummy activity is represented by a dotted arrow and does not consume any time and resource.
An unbroken chain of activities between any two events is called a path.
An event represents the accomplishment of some task. In a network diagram, beginning and ending of an activity are represented as events. Each event is represented as a node in a network diagram. An event does not consume any time or resource. Each network diagram starts with an initial event and ends at a terminal event.
Each node is represented by a circle and numbered by using the Fulkerson's Rule. Following steps are involved in the numbering of the nodes:
- The initial event, which has all outgoing arrows and no incoming arrow, is numbered as 1.
- Delete all the arrows coming out from the node just numbered (i.e. 1). This step will create some more nodes (at least one) into initial events. Number these events in ascending order (i.e. 2, 3 etc.).
- Continue the process until the final or terminal node which has all arrows coming in, with no arrow going out, is numbered.
An illustration of Fulkerson's Rule of numbering the events is shown in. As a recommendation it must be noted that most of the projects are liable for modifications, and hence there should be a scope of adding more events and numbering them without causing any inconsistency in the network. This is achieved by skipping the numbers (i.e. 10, 20, 30).
Rules for drawing network diagram
Rule 1: Each activity is represented by one and only one arrow in the network.
Rule 2: No two activities can be identified by the same end events.
Rule 3: Precedence relationships among all activities must always be maintained.
Rule 4: Dummy activities can be used to maintain precedence relationships only when actually required. Their use should be minimized in the network diagram.
Use of Dummy Activites
Rule 5: Looping among the activities must be avoided.
Critical Path Method(CPM)
This is based on deterministic approach in which only one time estimate is made for activity completion. The CPM (critical path method) system of networking is used, when the activity time estimates are deterministic in nature. For each activity, a single value of time, required for its execution, is estimated. Time estimates can easily be converted into cost data in this technique. CPM is an activity oriented technique.
- A network diagram in CPM is activity oriented.
- Cost is the most important criteria. Minimum is found corresponding to optimum time.
- There is only single time estimate for each activity.
- The probability of completion of activity in this estimated duration is 100%.
- It is based on deterministic approach.
- Suitable for repetitive type of work.
- Normal distribution is followed.
(i) Earliest start time
(ii) Earliest start time
EFT = EST + Activity time
(iii) Latest finish time
LFT = TL of head event
(iv) Latest finish time
LST = LFT - tij
Float denotes the range within which activity time or its finish time may fluctuate without effecting the completion of the project.
(i) Total Float (FT):
FT = LST – EST or FT = LFT – EFT
(ii) Free Total (FF):
Where Sj = Head event slack
(iii) Independent Float (FID):
Where Si = Tail event slack
FT = 0 – for critical path FT > 0 –for subcritical path
FT < 0 – for Supercritical path
(iv) Interfering float (FIN)
It is the another name of head event slack.
Mainly two systems are used in CPM analysis:
- A-O-A System (Activity on arrow system)
An activity is graphically represented by an arrow.
The tail end and head end of arrow represent start and finish of an activity respectively.
- A-O-N System (Activity on node system or precedence diagram). Activity is represented by circle or node. Events have no places. Arrows are used only to show the dependency relationship between activity nodes.
When two or more activities start parallel then an activity called DEBUT (D0) is provided at the beginning. Likewise, a finish activity (F0) is provided at the end when more than one activities finish parallel. Activity D & F have zero duration.
Network Analysis - PERT
Programme Evaluation Review Technique(PERT)
- PERT was developed by the U.S Navy during the late 1950’s to accelerate the development of the Polaris Fleet Ballistic Missile PERT,
- PERT, is abbreviated form which stands for Programme Evaluation and Review Technique used for planning, scheduling and monitoring the project.
- Coordination with the help of PERT was so successful that the entire project was complete years ahead of schedule.
- In PERT all activity time is probabilistic.
- For PERT. Employs Beta-distribution for the time – expectation for activity.
- A project is composed of many diversified activities which contribute to its completion.
- An important characteristics of any project is its duration.
- As its effect and economy is often dependent on the project duration.
- Total project duration depends on the time taken by each activity Estimation of time required for any activity requires wide experience of similar activities.
- The limits with which the duration will lie, is estimated.
- Pert follows the probabilistic approach and absorbs the uncertainties into the time estimates for activity and project durations.
- Therefore PERT is well suited for those projects where there is insufficient or no background information for estimation of time duration
- PERT is used in R&D type projects such as space industry, defence industry etc. As such projects are of non repetitive type or once-through type for which correct time estimates cannot be made.
- Further a PERT analysis is event oriented i e in this analysis interest is more focussed on the PERT (start or completion of activity) rather than the activities
In order to take into account, the uncertainties involved in the activity times three kinds of time estimates are made for each activity in PERT.
(i) Optimistic time (to): If everything in the project goes well.
It is the minimum time required for an activity if everything goes perfectly well without any problems or adverse conditions developed during the execution of the activity
In this time estimate, no provisions are made for delays or setbacks and better than normal conditions are assumed to prevail during the execution of the activity
(ii) Most Likely Time (tm): It is the time for completing an activity that is best.
It is the maximum time required for an activity if everything goes wrong and abnormal situations prevail This time estimate does not include the possible effects of major catastrophes such as flood earthquakes, fire, labour strikes etc.
(iii) Pessimistic Time (tp): If everything in the project goes wrong.
It is the time required to complete the activity if normal conditions prevail
This time estimate lies between pessimistic and optimistic time estimates
- In PERT activity time is probabilistic but in CPM activity time is deterministic.
- The other difference: PERT is Event – Oriented. While the CPM is Activity – Oriented (in CPM we actually know the Activity time)
(i) Expected completion time of an Activity: (tE)
Where, t0 = Optimistic time
tp = Pessimistic time
tm = Most likely time
(ii) Standard deviation of an Activity (𝛔)
(iii) Variance of an activity: (𝛔2)
(iv) Central limit theorem:
(a) The mean time of the project as a whole is along the critical path.
Probability of completion of project in time tE is 50%.
(b) The standard deviation of the project as a whole is along the critical path.
Critical Path: The time wise longest path is called critical path. In this path any type of delay in any event will cause delay to the project. These are shown by double line or dark lines in a network.
An event is critical if its slack is zero.
(i) Earliest expected event occurring time (TE)
When there is only one path.
Where, = Expected completion time of an activity I – j
…. when there are more than one path.
Where = Earliest expected time of event i.
= Earliest expected time of event j.
(ii) Latest allowable occurrence time (TL):
When there is only one path.
When there are more than one path.
(iii) Slack (s): This is the time by which an event may be delayed without affecting the completion time of the project.
- Probability Factor (z)
Where, TS = Given scheduled completion time of the project
TE = Expected completion time of the project.
σ = Standard deviation
Frequency Distribution Curve for PERT
It is assumed to be a β - distribution curve with a unimodal point occurring at tm and its end points occurring at to and tp. The most likely time need not be the midpoint of to and tp and hence the frequency distribution curve may be skewed to the left, skewed to the right or symmetric.
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