Critical Path MethodologyGiven deterministic activity times, it is possible to;
Describe a project as a network
Find a critical path through the network.
Refresher Project Scheduling
Critical Path Methodology
Given deterministic activity times, it is possible to;
Describe a project as a network
Find a critical path through the network.
2 H J Paint & Clean 1 BC I Install Windows 3 EFG H Dry Wall 4 D G Ducting 3 D F Plumbing 4 D E Wiring 3 B D Walls 11 NA C Order Windows 4 A B Frame 4 NA A Foundation Duration Predece Idx Descript
Set up network with dummy nodes so that activities do not finish at same place.
Do a ‘forward’ pass and a ‘backward’ pass.
Based on this, identify the ‘critical path.’
See if you can do this now.
Solution in Excel
It is possible to set up the activities in Excel, and explicitly code dependencies.
If you are careful with the choice of constraints, you can ‘solve’ the network using Solver.
In any case it is possible to readily summarise the features of the project.
‘Realistic’ Scenario
However, it should be clear that activities do not, in practice, take fixed time known a priori.
CPM is appropriate for the deterministic situation.
Another methodology, developed at the same time, is PERT.
PERT
PERT stands for Programme Evaluation and Review Technique.
It has its origins in the Polaris missle project.
A key output is time to completion of project.
Times can be uncertain.
PERT Distn
The PERT distn is used to summarise times for parts of projects.
It is a distn taking min, most likely and max.
It is ‘smoother’ than triangle.
It is in fact a Beta distribution.
PERT(a,b,c) = Be(alpha,beta)*(c-a)+a
Parameterisation
The alpha and beta are given (where the PERT is not symmetric) by;
Let mu = (a+4*b+c)/6
Then alpha = (mu-a)*(2b-a-c)/((b-mu)*(c-a))
And beta = alpha*(c-mu)/(mu-a)
1.5,2,2.5 0.5,1,1.5 2.5,3,3.5 2,4,6 2,2.5,6 3,3.5,7 2,2.5,6 7,10,19 3,4,5 1.5,3.5,8.5 Param 2 H J Paint & Clean 1 BC I Install Win 3 EFG H Dry Wall 4 D G Ducting 3 D F Plumbing 4 D E Wiring 3 B D Walls 11 NA C Order Windows 4 A B Frame 4 NA A Foundation Mu Pred Idx Descript
In Excel
Thus, we have a table with all activities.
Each time is considered an independent random variable.
The total project time is also a random variable.
It can be simulated, or approximated analytically.
Questions of Interest
What is the expected project time?
What is the probability the time of the project is less than 22 days (say)?
What are the critical activities?
Can an activity be ‘probabilistically’ critical?
Simulation Approach
Treat activities as conditionally independent.
Simulate a time for each.
Examine the combination of activity times.
Examine the effect of each on the total time if this is of interest (by, for example, adding ‘delta’ to each activity in turn.)
Summary
PERT is another method of examining project scheduling.
It involves the stochastic specification of activity times.
The PERT distribution is a Beta.
Simulation allows the answering of questions that arise in a realistic fashion.
Comments