Notations: tp = pessimistic time to = optimistic time tm = most likely time
te = expected time of an activity Te = expected length of the project σt = standard deviation of time Vt = variance of time
EST = early start time of an event LFT = late finish time of an event
tij = time duration needed to complete the job ij
CPM stands for Critical Path Method. It has mostly been used in deterministic situations like construction projects. For the most part, houses, bridges, and skyscrapers use standard materials whose properties are well known. They employ more or less standard components and stable technology.
Changes occur mainly in design, size, shapes, and arrangements of different components- rather than in design concepts. CPM takes just one time into account, and it deals with deterministic situation. It is activity oriented and can be used for both large and small projects. It is widely recognized and is the most versatile and potent management planning technique. CPM is used for planning and controlling the most logical and economic sequence of operations for accomplishing a project.
CPM Technique
CPM follows the following steps for accomplishing a project planning:
••••• Break down the project into various activities systematically.
••••• Label all activities.
••••• Arrange all the activities in logical sequence.
••••• Construct the arrow diagram.
••••• Number all the nodes (events) and activities.
••••• Find the time for each activity.
••••• Mark the activity times on the arrow diagram.
••••• Calculate early and late, start and finishing times.
••••• Tabulate various times and mark EST and LFT on the arrow diagram.
••••• Calculate the total project duration.
••••• If it is intended to reduce the total project duration. Crash the critical activities of the net- work.
••••• Optimize the cost.
••••• Update the Network.
••••• Smooth the network resources.
PERT is Program Evaluation and Review Technique. This is mostly used in non-deterministic or probabilistic or stochastic situations such as: space research, R & D projects. These projects (going to Mars, Moon, etc) are relatively new; their technology is rapidly changing, and their products are nonstandard. There is some standard hardware in ICBMs (Inter-continental Ballistic Missiles) and lunar rockets, but much of their design and construction needs new type of materials and technology, and projects are contracted, planned, and scheduled before all technological problems have been solved.
Thus, there is a large amount of uncertainty in design, construction, and configuration of the new weapons systems and spaceships. There is little past history on which to base network construction and time estimates.
PERT was first used in 1957 for the planning and control of the Polaris Missiles program in US navy with a goal to finish the project two years in advance.
••••• PERT is commonly used to conduct the initial review of a project .It is very useful device to plan the time and resources.
••••• PERT is used in activity where timings could not be estimated with enough certainty. It can
be employed at those places where a project cannot be easily defined in terms of time or resources required.
••••• However, events can be readily defined which means it is known that, first, part A will be
manufactured, only then subassembly S can be built, and so on.
••••• PERT offers a lot of advantages for non-repetitive type of projects, R & D, prototype produc- tion, space research, defense projects, etc.
••••• Because of the uncertainty of activity timings, PERT fits into a probabilistic model. Probabil- ity concept helps in estimating activity timings. The statistical probability feature of PERT foretells the probability of reaching the specified target dates.
••••• PERT is mainly concerned with events and is thus seen as an event oriented system.
PERT Techniques
The PERT planning technique consists of the following steps:
••••• The project is broken down into different activities systematically.
••••• Activities are arranged in logical sequence.
••••• The network diagram is drawn. Events and activities are numbered.
••••• Using three time estimate, the expected time for each activity is calculated.
••••• Standard deviation and variance for each activity are computed.
••••• Earliest starting times, and latest finishing times are calculated.
••••• Expected time, earliest starting time, and latest finishing times are marked on the network
diagram.
••••• Slack is calculated.
••••• Critical path(s) are identified and marked on the network diagram.
••••• Length of critical path or total project duration is found out.
••••• Lastly, the probability that the project will finish at due date is calculated.
9.2.1 TIME ESTIMATES IN PERT
To take care of uncertainty, PERT takes three time estimates into account: optimistic, most likely, and pessimistic time. PERT time estimates follow beta distribution.
to tmte tp distribution
Optimistic time(to): This is the shortest time taken by an activity if everything goes exception- ally well.
Most likely time(tm): It is the time in which the activity is normally expected to complete under normal contingencies.
Pessimistic time (tp): It is the maximum time that would be required to complete the activity if bad luck were encountered at every turn. This does not include catastrophes like earthquakes, floods, fires, etc.
These time estimates are not always easy to prepare, but together they give useful information about the expected uncertainties of an activity. For standard activities, the three time estimates should not vary much from each other. But the greater the uncertainty of an activity, the wider will be the range of the estimated completion times.
PERT calculates the expected value of duration as a weighted average of three time estimates. It assumes that to and tp are equally likely to occur, and tm is four times more likely to occur than the other two. Hence, the expected time
te = (to + 4tm + tp)/6
The expected time is the time that we would expect if the activity were repeated a large number of times. But in reality, activities do not get repeated many times; they usually occur just once.
9.2.2 ALGORITHM USED IN CALCULATING CRITICAL PATH There are two methods for computing the critical path. They are:
(a) Forward pass, and (b) Backward pass
Algorithms for two are given below:
(a) Forward pass
S = start time of the project (usually = 0) ETi = S for all beginning events
ETj = max{ ETi + tij}, i
where the maximization is over all the events i that precede event j
T = max {ET(all events)}, which is the earliest finish time for the project.
(b) Backward pass
LTj = T for all ending events LTi = min{ LTj – tij},
j
where the minimization is over all the events j that succeed event i.
The relationship between activity and event times is summarized as follows:
ESij = ETi EFij = ETi + tij LFij = LTj LSij = LTj – tij
TSij = LTj – ETi – tij
FSij = ETj – ETi – ti = TSij – LTj + ETj
where ES = Earliest start, EF = Earliest first, LS = Latest start, LF = Latest first, TS = Total slack, FS = Free slack.