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onte Carlo simulation uses repeated randomsampling to calculate results about physical

and mathematical systems. It uses uncertainty in itsinputs to generate a range of possible outcomes,which are then reported as results with a degree ofmathematical confidence. The method tends to beused when it is infeasible or impossible to computean exact result with a deterministic algorithm.

Describing Uncertainty and risk events

The uncertainty used as inputs to Monte Carlo simulationis described using probability distributions, which definethe range of values a variable might take, and thelikelihood of those values occurring.

The most commonly used probability distribution used formodelling project and business uncertainty is theTriangular distribution, so called because it is defined bythree points – a minimum, likely and maximum value. Forexample, the time required to dig the foundations for anew house may be a minimum of 10 days (best case),likely of 12 days and maximum of 15 days (worst case).This is called a three point estimate and is often based onhistoric information and experience (a building firm thathas built hundreds of houses over several years will havea pretty good idea).

Monte Carlo Simulation – Improving Project And Business Confidence

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Monte Carlo simulation is also used to model project andbusiness risk events. In this case, two probabilitydistributions are required. First a Bernoulli distribution is usedto model whether the risk event occurs – resulting in either aTrue or False result (e.g. a coin can be used to model a riskthat has a 50% chance of occurring – heads it happens, tailsit doesn’t). Second, the uncertain impact of the risk isdescribed, typically using a Triangular distribution.

For example, while digging the foundations for your house,you may suffer from extremely bad weather, causing a delaybefore you can continue with your task. The probability ofextremely bad weather (the event) might be 5% and yourestimate of the impact (the uncertainty) might be between2 and 8 days, but most likely 4.

Using Monte Carlo Simulation

Even in a small project or business model, you are likely tohave hundreds of uncertain tasks and dozens of possible riskevents. You need a mechanism for understanding what thismeans in terms of the possible outcome (success or failure).Taking a simplistic approach, you could calculate the worstcase scenario, by adding the maximum values for allvariables and assuming all of the risks happen. But this willproduce a very pessimistic outcome, with the result that youwill probably never embark on building your house. Similarly,taking the most optimistic view will give you a false pictureof your ability to deliver on time and budget, resulting insubstantial overrun and losses.

Uncertainty

Risks

3 point estimates(Max, Min, Most likely)

Identified risk(Probability/Impact)

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Further results can be gleaned from the Monte Carlo outputs.One very useful measure is sensitivity, which is oftenportrayed in a Tornado diagram. Sensitivity measures howmuch influence an input variable has on your outcome. Forexample, any risk event or uncertainty experienced whiledigging your foundations may have a very large influence onyour project end date, whereas problems when you arepainting the walls may not factor into the end date at all.The Tornado chart is particularly useful in highlighting whichtasks in your project are the most likely cause of delay oroverrun. This helps focus on which areas of the project totarget most risk management effort.

Using Monte Carlo for cost and schedule analysis

The examples given in this article are schedule related – howmany days it will take to do something? What date you canbe confident of meeting? This is because of the complexnature of schedules, that include logic links (for example,task A in a MS Project plan needs to finish before task B canstart) as well as parallel critical paths. Other methods ofschedule analysis e.g. PERT cannot easily handle suchcomplex logic. Monte Carlo is invaluable for scheduleanalysis.

You can also use Monte Carlo simulation to model costs. Fora simple set of uncertain cost variables, algorithms areavailable to calculate results. However, these methods willonly work assuming the variables are independent – whencorrelation is included, Monte Carlo is a much more genericanalysis solution.

In cases of both schedule and cost risk analysis, adding riskevents to your base estimating model is a further reason forusing Monte Carlo simulation. The inclusion of risk events,with the binary True/False probabilistic branching is verydifficult to achieve without the use of Monte Carlosimulation.

Monte Carlo simulation is used to create a picture of theoutcomes between the optimistic and pessimisticscenarios, with the results tending to a bell shaped curve(a Normal distribution). The mathematical basis for thisshape is the Central Limit Theorem, where thecombination of a large number of independent randomvariables will converge to a Normal distribution. The onlyrequirement is that you repeat the Monte Carlo‘experiments’ enough times, to ensure a representativeset of random samples have been used.

Understanding the results

The Monte Carlo set of outcomes (results distribution) canbe interrogated to understand the confidence of meetingspecified targets. For example, if you run the Monte Carlosimulation 1000 times, you will produce 1000 potentialcompletion dates for your house. To work out theconfidence of meeting a 30th July deadline, count up thenumber of results that lie to the right of that date. If thereare 200 such results, then you know that there is a 20%chance (200/1000) that you will overrun. Conversely, youcan be 80% confident that you will finish on time orearly. The benefit of Monte Carlo, is that you can generateintermediary results in addition to the overall outcome,for example, you can calculate the confidence of meetingmilestones as well as the project end date.

Simulation for1000 slices

Tornado chartshowingsensitivies

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