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September 5th, 2018

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 started.

A simple example of a PERT diagram for sourcing and installing a new kitchen is shown below:

PERT Example

For each activity we can calculate start and end-time metrics.

We use the following notation and scheme:

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:

PERT Diagram

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 topic.)

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 option chosen.

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:

BetaPERT Distribution

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 customers!

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 this.

Imagine that incorporating the uncertainty distributions into a robust critical path analysis gave the following results:

robust critical path analysis results

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:


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:

Sensitivity: Latest Finish time

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 happen!

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

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Phil RoweDr Phil Rowe is a Senior Consultant and Lean Six Sigma Master Black Belt with over 30 years’ industrial experience.

Phil was trained in Six Sigma by Dr. Mikel Harry, founder of the Six Sigma Academy.  Trained in DFSS at GE, Phil became General Domestic Appliances’ DFSS Master Black Belt and programme manager, working with design engineers in applying DFSS tools to high investment new product programmes.

More recently Phil achieved recognition as a Chartered Statistician, the highest professional award of the Royal Statistical Society.  This award provides formal recognition of his statistical qualifications, professional training and experience and follows on from his recent award of a 1st Class BA Hons in Mathematics and Statistics.

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