Robust Critical Path
Analysis using Monte Carlo Simulation
Dr P.G. Rowe, Senior Consultant
In project management a PERT (Programme Evaluation and Review Technique) diagram
shows which tasks are required to have been completed before others are
A simple example of a PERT diagram
for sourcing and installing a new kitchen is shown below:
For each activity we can calculate
start and end-time metrics.
We use the following notation and
By identifying process steps with
no slack in them (zero float) the critical
path can be identified.
It is assumed that the reader
knows how to complete the timing boxes – but if not then our consultants at
Bourton Group can enlighten you!
Here is an example of a completed
PERT diagram with timings and critical path identification for the purchasing
and commissioning of a machine in a manufacturing company:
Why might this approach not reflect reality?
One reason of course is that
estimated timings may (actually will) be inaccurate; we don’t expect any
estimate of duration to be perfectly accurate – there is uncertainty.
If uncertainty in estimating
timings can be incorporated into the analysis somehow then a more realistic
prediction of the total process time will be obtained – viz. we will be in a
position to make a statement such as ‘We
are 95% confident that the process will take up to 45 days’ or ‘We are 90% confident that the process will
take between 35 and 45 days’ and the like.
Furthermore we can identify the
most important sources of uncertainty that drive the uncertainty in total
duration and concentrate on refining that prediction on a regular basis.
If this is done, then the range of
lower to upper confident limit (10 days in our example) will reduce accordingly
– our predictions become more precise.
So, how is it done?
Essentially, we define
distributions to reflect uncertainties in timings for each step of the process
and then run all possible scenarios to build a distribution of total duration of the process.
This distribution will show the
most likely duration, as well as the expected range of duration.
The ‘engine’ that performs this
task is called Monte Carlo Simulation (MCS). (Again, Bourton can enlighten you on this
Note that an important corrollary
of this approach is that the critical path will change naturally and dynamically, depending upon the random
combinations of timings chosen by the MCS algorithm – it doesn’t make sense
that the critical path remains the same for any combination of timings.
Also, what if one can take
alternative forms of action on individual or subgroups of process steps in
order to reduce the total duration? (e.g.
subcontracting the work, paying overtime etc.)
The impact of these hypothetical
solutions can be assessed quickly and easily using MCS and hence the best
There are many software programs
that can do MCS – e.g. Minitab’s Quality Companion and an especially good one
is Excel’s Crystal Ball add-in from
Oracle. (The latter is used in this article.)
There are many alternative
distributions that can be used to define uncertainty in the timings, but a
sensible choice is to use the BetaPERT
distribution, shown below for Crystal Ball:
One only needs to specify the
expected minimum process step duration, the expected maximum and the most
likely duration. This latter value for
each process step would probably correspond to those used for the (crude) deterministic approach.
One might ask why we don’t just
use the maximum duration estimated for each task.
Whilst it is likely that the
actual duration would not exceed this, being too conservative can be unhelpful
as by severely overestimating it we may lose valuable opportunities – or
For the sake of illustration,
let’s suppose that a deterministic
PERT analysis gave an estimated duration of 76.5 days.
Further, assume that we required
high confidence that the project would be completed within +/- 2.5 days of
Imagine that incorporating the
uncertainty distributions into a robust critical path analysis gave the
This tells us that we are only
about 60% confident in completing the project within the specification of 76.5
+/- 2.5 days!
Furthermore by moving the sliders
on the horizontal axis of the graph we can ascertain the 95% upper confidence
limit, say, for total duration:
In this example, we are 95%
confident that the project will be completed in less than 82.4 days.
In other words, although we
require a maximum duration of 79 days, there is a 1 in 20 chance that it will exceed
Similarly we can deduce that we
are 90% confident that the project will be finished between approximately 73
and 82 days.
The analysis can also show the
sources of uncertainty that are mainly responsible for this:
The activity ‘Awaiting
delivery’ contributes over 90% to the ‘variance’ of the
project duration estimate, so this is clearly the major timing element for
attention in terms of reducing the imprecision in the estimate of its duration. (Variance is standard deviation squared - again
we can help you to understand statistical terminology and methods.)
Robust Critical Path Analysis can
help to make estimates of process or project duration more realistic and so
reduce the pressure on people to meet impossible timescales – all too often
senior management exhort their staff to ‘just do it’. Well, sometimes it just can’t be done and we
can now demonstrate why – and understand what needs to be done to make it
If you are interested in taking
this approach and learning how to complete timing diagrams and perform Monte
Carlo Simulation to do Robust Critical Path Analysis, or indeed any other
statistical training, please contact us at email@example.com.