Archive for the ‘Monte Carlo Simulation’ Category
(Note: An Excel sheet showing sample calculations and plots discussed in this post can be downloaded here.)
Some months ago, I wrote a post explaining the basics of Monte Carlo simulation using the example of a drunkard throwing darts at a board. In that post I assumed that the darts could land anywhere on the dartboard with equal probability. In other words, the hit locations were assumed to be uniformly distributed. In a comment on the piece, George Gkotsis challenged this assumption, arguing that that regardless of the level of inebriation of the thrower, a dart would be more likely to land near the centre of the board than away from it (providing the player is at least moderately skilled). He also suggested using the Normal Distribution to model the spread of hits, with the variance of the distribution serving as a rough measure of the inaccuracy (or drunkenness!) of the drunkard. In George’s words:
I would propose to introduce a ‘skill’ factor, which represents the circle/square ratio (maybe a normal-Gaussian distribution). Of course, this skill factor would be very low (high variance) for a drunken player, but would still take into account the fact that throwing darts into a square is not purely random.
In this post I revisit the drunkard’s dartboard, taking into account George’s suggestions.
Setting the stage
To keep things simple, I’ll make the following assumptions:
- The dartboard is a circle of radius 0.5 units centred at the origin (see Figure 1)
- The chance of a hit is greatest at the centre of the dartboard and falls off as one moves away from it.
- The distribution of hits is a function of distance from the centre but does not depend on direction. In mathematical terms, for a given distance from the centre of the dartboard, the dart can land at any angle with equal probability, being the angle between the line joining the centre of the board to the dart and the x axis. See Figure 2 for graphical representations of a hit location in terms of and . Note that that the and coordinates can be obtained using the formulas and as s shown in Figure 2.
- Hits are distributed according to the Normal distribution with maximum at the centre of the dartboard.
- The variance of the Normal distribution is a measure of inaccuracy/drunkenness of the drunkard: the more drunk the drunk, the greater the variation in his aim.
These assumptions are consistent with George’s suggestions.
[Note to the reader: you may want to download the demo before continuing.]
The steps of a simulation run are as follows:
- Generate a number that is normally distributed with a zero mean and a specified standard deviation. This gives the distance, , of a randomly thrown dart from the centre of the board for a player with a “inaccuracy factor” represented by the standard deviation. Column A in the demo contains normally distributed random numbers with zero mean and a standard deviation of 0.2 . Note that I selected the latter number for no other reason than the results show up clearly on a fixed-axis plot shown in Figure 2.
- Generate a uniformly distributed random number lying between 0 and . This represents the angle . This is the content of column B of the demo.
- The numbers obtained from steps 1 and 2 for completely specify the location of a hit. The location’s and coordinates can be worked out using the formulas and . These are listed in columns C and D in the Excel demo.
- Re-run steps 1 through 4 as many times as needed. Note that the demo is set up for 5000 runs. You can change this manually or, better yet, automate it. The latter is left as an exercise for you.
It is instructive to visualize the resulting hits using a scatter plot. Among other things this can tell you, at a glance, if the results make sense. For example, we would expect hits to be symmetrically distributed about the origin because the drunkard’s throws are not biased in any particular direction around the centre). A non-symmetrical distribution is thus an indication that there is an error in the calculations.
Now, any finite collection of hits is unlikely to be perfectly symmetrical because of outliers. Nevertheless, the distributions should be symmetrical on average. To test this, run the demo a few times (hit F9 with the demo open). Notice how the position of outliers and the overall shape of the distribution of points changes randomly from simulation to simulation. In all cases, however, there is a clear maximum at the centre of the dartboard with the probability of a hit falling with distance from the centre.
Figure 3 shows the results of simulations for a standard deviation of 0.2. Figures 4 and 5 show the results of simulations for standard deviations of 0.1 and 0.4.
Note that the plot has fixed axes- i.e. the area depicted is the 1×1 square that encloses the dartboard, regardless of the standard deviation. Consequently, for larger standard deviations (such as 0.4) many hits will be out of range and will not show up on the plot.
As I have stressed in my previous posts on Monte Carlo simulation, the usefulness of a simulation depends on the choice of an appropriate distribution. If the selected distribution does not reflect reality, neither will the simulation. This is true regardless of whether one is simulating a drunkard’s wayward aim or the duration of project task. You may have noted that the assumption of normally-distributed hits has no justification whatsoever; it is just as arbitrary as my original assumption of uniformity. In fact, the hit locations of drunken dart throws is highly unlikely to be either uniform or Normal. Nevertheless, I hope that some of my readers will find the above example to be of pedagogical value.
Thanks to George Gkotsis for his comment which got me thinking about this post.
(Note to the reader: An Excel sheet showing sample calculations and plots discussed in this post can be downloaded here.)
Monte Carlo simulation techniques have been applied to areas ranging from physics to project management. In earlier posts, I discussed how these methods can be used to simulate project task durations (see this post and this one for example). In those articles, I described simulation procedures in enough detail for readers to be able to reproduce the calculations for themselves. However, as my friend Paul Culmsee mentioned, the mathematics tends to obscure the rationale behind the technique. Indeed, at first sight it seems somewhat paradoxical that one can get accurate answers via random numbers. In this post, I illustrate the basic idea behind Monte Carlo methods through an example that involves nothing more complicated than squares and circles. To begin with, however, I’ll start with something even simpler – a drunken darts player.
Consider a sozzled soul who is throwing darts at a board situated some distance from him. To keep things simple, we’ll assume the following:
- The board is modeled by the circle shown in Figure 1, and our souse scores a point if the dart falls within the circle.
- The dart board is inscribed in a square with sides 1 unit long as shown in the figure, and we’ll assume for simplicity that the dart always falls somewhere within the square (our protagonist is not that smashed).
- Given his state, our hero’s aim is not quite what it should be – his darts fall anywhere within the square with equal probability. (Note added on 01 March 2011: See the comment by George Gkotsis below for a critique of this assumption)
We can simulate the results of our protagonist’s unsteady attempts by generating two sets of uniformly distributed random numbers lying between 0 and 1 (This is easily done in Excel using the rand() function). The pairs of random numbers thus generated – one from each set – can be treated as the (x,y) coordinates of the dart for a particular throw. The result of 1000 pairs of random numbers thus generated (representing the drunkard’s dart throwing attempts) is shown in Figure 2 (For those interested in seeing the details, an Excel sheet showing the calculations for 100o trials can be downloaded here).
A trial results in a “hit” if it lies within the circle. That is, if it satisfies the following equation:
(Note: if we replace “<” by “=” in the above expression, we get the equation for a circle of radius 0.5 units, centered at x=0.5 and y=0.5.)
Now, according to the frequency interpretation of probability, the probability of the plastered player scoring a point is approximated by the ratio of the number of hits in the circle to the total number of attempts. In this case, I get an average of 790/1000 which is 0.79 (generated from 10 sets of 1000 trials each). Your result will be different from because you will generate different sets of random numbers from the ones I did. However, it should be reasonably close to my result.
Further, the frequency interpretation of probability tells us that the approximation becomes more accurate as the number of trials increases. To see why this is so, let’s increase the number of trials and plot the results. I carried out simulations for 2000, 4000, 8000 and 16000 trials. The results of these simulations are shown in Figures 3 through 6.
Since a dart is equally likely to end up anywhere within the square, the exact probability of a hit is simply the area of the dartboard (i.e. the circle) divided by the entire area over which the dart can land. In this case, since the area of the enclosure (where the dart must fall) is 1 square unit, the area of the dartboard is actually equal to the probability. This is easily seen by calculating the area of the circle using the standard formula where is the radius of the circle (0.5 units in this case). This yields 0.785398 sq units, which is reasonably close to the number that we got for the 1000 trial case. In the 16000 trial case, I get a number that’s closer to the exact result: an average of 0.7860 from 10 sets of 16000 trials.
As we see from Figure 6, in the 16000 trial case, the entire square is peppered with closely-spaced “dart marks” – so much so, that it looks as though the square is a uniform blue. Hence, it seems intuitively clear that as we increase, the number of throws, we should get a better approximation of the area and, hence, the probability.
There are a couple of points worth mentioning here. First, in principle this technique can be used to calculate areas of figures of any shape. However, the more irregular the figure, the worse the approximation – simply because it becomes less likely that the entire figure will be sampled correctly by “dart throws.” Second, the reader may have noted that although the 16000 trial case gives a good enough result for the area of the circle, it isn’t particularly accurate considering the large number of trials. Indeed, it is known that the “dart approximation” is not a very good way of calculating areas – see this note for more on this point.
Finally, let’s look at connection between the general approach used in Monte Carlo techniques and the example discussed above (I use the steps described in the Wikipedia article on Monte Carlo methods as representative of the general approach):
- Define a domain of possible inputs – in our case the domain of inputs is defined by the enclosing square of side 1 unit.
- Generate inputs randomly from the domain using a certain specified probability distribution – in our example the probability distribution is a pair of independent, uniformly distributed random numbers lying between 0 and 1.
- Perform a computation using the inputs – this is the calculation that determines whether or not a particular trial is a hit or not (i.e. if the x,y coordinates obey inequality (1) it is a hit, else it’s a miss)
- Aggregate the results of the individual computations into the final result – This corresponds to the calculation of the probability (or equivalently, the area of the circle) by aggregating the number of hits for each set of trials.
To summarise: Monte Carlo algorithms generate random variables (such as probability) according to pre-specified distributions. In most practical applications one will use more efficient techniques to sample the distribution (rather than the naïve method I’ve used here.) However, the basic idea is as simple as playing drunkard’s darts.
Thanks go out to Vlado Bokan for helpful conversations while this post was being written and to Paul Culmsee for getting me thinking about a simple way to explain Monte Carlo methods.
When developing duration estimates for a project task, it is useful to make a distinction between the uncertainty inherent in the task and uncertainty due to known risks. The former is uncertainty due to factors that are not known whereas the latter corresponds uncertainty due to events that are known, but may or may not occur. In this post, I illustrate how the two types of uncertainty can be combined via Monte Carlo simulation. Readers may find it helpful to keep my introduction to Monte Carlo simulations of project tasks handy, as I refer to it extensively in the present piece.
Setting the stage
Let’s assume that there’s a task that needs doing, and the person who is going to do it reckons it will take between 2 and 8 hours to complete it, with a most likely completion time of 4 hours. How the estimator comes up with these numbers isn’t important here – maybe there’s some guesswork, maybe some padding or maybe it is really based on experience (as it should be). For simplicity we’ll assume the probability distribution for the task duration is triangular. It is not hard to show that, given the above mentioned estimates, the probability, , that the task will finish at time is given by the equations below (see my introductory post for a detailed derivation):
for 2 hours 4 hours
for 4 hours 8 hours
These two expressions are sometimes referred to as the probability distribution function (PDF). The PDF described by equations (1) and (2) is illustrated in Figure 1. (Note: Please click on the Figures to view full-size images)
Now, a PDF tells us the probability that the task will finish at a particular time . However, we are more interested in knowing whether or not the task will be completed by time . – i.e. at or before time . This quantity, which we’ll denote by (capital P), is sometimes known as the cumulative distribution function (CDF). The CDF is obtained by summing up the probabilities from hrs to time . It is not hard to show that the CDF for the task at hand is given by the following equations:
for 2 hours 4 hours
for 4 hours 8 hours
For a detailed derivation, please see my introductory post. The CDF for the distribution is shown in Figure 2.
Now for the complicating factor: let us assume there is a risk that has a bearing on this task. The risk could be any known factor that has a negative impact on task duration. For example, it could be that a required resource is delayed or that the deliverable will fails a quality check and needs rework. The consequence of the risk – should it eventuate – is that the task takes longer. How much longer the task takes depends on specifics of the risk. For the purpose of this example we’ll assume that the additional time taken is also described by a triangular distribution with a minimum, most likely and maximum time of 1, 2 and 3 hrs respectively. The PDF for the additional time taken due to the risk is:
for 1 hour 2 hours
for 2 hrs 3 hours
The figure for this distribution is shown in Figure 3.
The CDF for the additional time taken if the risk eventuates (which we’ll denote by ) is given by:
for 1 hour 2 hours
for 2 hours 3 hours
The CDF for the risk consequence is shown in Figure 4.
Before proceeding with the simulation it is worth clarifying what all this means, and what we want to do with it.
Firstly, equations 1-4 describe the inherent uncertainty associated with the task while equations 5 through 8 describe the consequences of the risk, if it eventuates.
Secondly, we have described the task and the risk separately. In reality, we need a unified description of the two – a combined distribution function for the uncertainty associated with the task and the risk taken together. This is what the simulation will give us.
Finally, one thing I have not yet specified is the probabilty that the risk will actually occur. Clearly, the higher the probability, the greater the potential delay. Below I carry out simulations for risk probabilities of varying from 0.1 to 0.5.
That completes the specification of the problem – let’s move on to the simulation.
The simulation procedure I used for the zero-risk case (i.e. the task described by equations 1 and 2 ) is as follows :
- Generate a random number between 0 and 1. Treat this number as the cumulative probability, for the simulation run. [You can generate random numbers in Excel using the rand() function]
- Find the time, , corresponding to by solving equations (3) or (4) for . The resulting value of is the time taken to complete the task.
- Repeat steps (1) and (2) for a sufficiently large number of trials.
The frequency distribution of completion times for the task, based on 30,000 trials is shown in Figure 5.
As we might expect, Figure 5 can be translated to the probability distribution shown in Figure 1 by a straightforward normalization – i.e. by dividing each bar by the total number of trials.
What remains to be done is to incorporate the risk (as modeled in equations 5-6) into the simulation. To simulate the task with the risk, we simply do the following for each trial:
- Simulate the task without the risk as described earlier.
- Generate another random number between 0 and 1.
- If the random number is less than the probability of the risk, then simulate the risk. Note that since the risk is described by a triangular function, the procedure to simulate it is the same as that for the task (albeit with different parameters).
- If the random number is greater than the probability of the risk, do nothing.
- Add the results of 1 and 4. This is the outcome of the trial.
- Repeat steps 1-5 for as many trials as required.
I performed simulations for the task with risk probabilities of 10%, 30% and 50%. The frequency distributions of completion times for these are displayed in Figures 6-8 (in increasing order of probability). As one would expect, the spread of times increases with increasing probability. Further, the distribution takes on a distinct second peak as the probability increases: the first peak is at , corresponding to the most likely completion time of the risk-free task and the second at corresponding to the most likely additional time of 2 hrs if the risk eventuates.
It is also instructive to compare average completion times for the four cases (zero-risk and 10%, 30% and 50%). The average can computed from the simulation by simply adding up the simulated completion times (for all trials) and dividing by the total number of simulation trials (30,000 in our case). On doing this, I get the following:
Average completion time for zero-risk case = 4.66 hr
Average completion time with 10% probability of risk = 4.89 hrs
Average completion time with 30% probability of risk = 5.36 hrs
Average completion time with 50% probability of risk= 5.83 hrs
No surprises here.
One point to note is that the result obtained from the simulation for the zero-risk case compares well with the exact formula for a triangular distribution (see the Wikipedia article for the triangular distribution):
This serves as a sanity check on the simulation procedure.
It is also interesting to compare the cumulative probabilities of completion in the zero-risk and high risk (50% probability) case. The CDFs for the two are shown in Figure 9. The co-plotted CDFs allow for a quick comparison of completion time predictions. For example, in the zero-risk case, there is about a 90% chance that the task will be completed in a little over 6 hrs whereas when the probability of the risk is 50%, the 90% completion time increases to 8 hrs (see Figure 9).
Next steps and wrap up
For those who want to learn more about simulating project uncertainty and risk, I can recommend the UK MOD paper – Three Point Estimates And Quantitative Risk Analysis A Process Guide For Risk Practitioners. The paper provides useful advice on how three point estimates for project variables should be constructed. It also has a good discussion of risk and how it should be combined with the inherent uncertainty associated with a variable. Indeed, the example I have described above was inspired by the paper’s discussion of uncertainty and risk.
Of course, as with any quantitative predictions of project variables, the numbers are only as reliable as the assumptions that go into them, the main assumption here being the three point estimates that were used to derive the distributions for the task uncertainty and risk (equations 1-2 and 5-6). Typically these are obtained from historical data. However, there are well known problems associated with history-based estimates. For one, as we can never be sure that the historical tasks are similar to the one at hand in ways that matter (this is the reference class problem). As Shim Marom warns us in this post, all our predictions depend on the credibility of our estimates. Quoting from his post:
Can you get credible three point estimates? Do you have access to credible historical data to support that? Do you have access to Subject Matter Experts (SMEs) who can assist in getting these credible estimates?
If not, don’t bother using Monte Carlo.
In closing, I hope my readers will find this simple example useful in understanding how uncertainty and risk can be accounted for using Monte Carlo simulations. In the end, though, one should always keep in mind that the use of sophisticated techniques does not magically render one immune to the GIGO principle.
Managers make decisions based on incomplete information, so it is no surprise that the tools of probability and statistics have made their way into management practice. This trend has accelerated somewhat over the last few years, particularly with the availability of software tools that simplify much of the grunt-work of using probabilistic techniques such as Monte Carlo methods or Bayesian networks. Purveyors of tools and methodologies and assume probabilities (or more correctly, probability distributions) to be known, or exhort users to determine probabilities using relevant historical data. The word relevant is important: it emphasises that the data used to calculate probabilities (or distributions) should be from situations that are similar to the one at hand. This innocuous statement papers over a fundamental problem in the foundations of probability: the reference class problem. This post is a brief introduction to the reference class problem and its implications for project management.
I’ll begin with some background and then, after defining the problem, I’ll present a couple of illustrations of the problem drawn from project management.
Background and the Problem
The most commonly held interpretation of probability is that it is a measure of the frequency with which an event of interest occurs. In this frequentist view, as it is called, probability is defined as the ratio of the number of times the event of interest occurs to the total number of events. An example might help clarify what this means: the probability that a specific project will finish on time is given by the ratio of the number of similar projects that have finished on time to the total number of similar projects undertaken (including both on-time and not-on-time projects).
At first sight the frequentist approach seems a reasonable one. However, in this straightforward definition of probability lurks a problem: how do we determine which events are similar to the one at hand? In terms of the example: what are the criteria by which we can determine the projects that resemble the one we’re interested in? Do we look at projects with similar scope, or do we use size (in terms of budget, resources or other measure), or technology or….? There could be a range of criteria that one could use, but one never knows with certainty which one(s) is (are) the right one(s). Why is it an issue? It’s an issue because probability changes depending on the classification criteria used. This is the reference class problem.
The reference class problem arises when we want to assign a probability to a proposition (or sentence, or event) X, which may be classified in various ways, yet its probability can change depending on how it is classified.
Incidentally, in another paper entitled Conditional Probability is the Very Guide of Life, Hajek discusses how the reference class problem afflicts all major interpretations of probability, not just the frequentist approach. We’ll stick with the latter interpretation since it is the one used in project management practice and research… and virtually all the social and natural sciences to boot.
The reference class problem in project management
Let’s look at a couple of project management-related illustrations of the reference class problem.
First up, consider the technique of reference class forecasting which I’ve discussed in this post. Note that reference class forecasting technique is distinct from the reference class problem although, as we shall see in less than a minute, the technique is fatally afflicted by the problem.
What’s reference class forecasting? To quote from the post referenced earlier, the technique involves:
…creating a probability distribution of estimates based on data for completed projects that are similar to the one of interest, and then comparing the said project with the distribution in order to get a most likely outcome. Basically, [it] consists of the following steps:
- Collecting data for a number of similar past projects – these projects form the reference class. The reference class must encompass a sufficient number of projects to produce a meaningful statistical distribution, but individual projects must be similar to the project of interest.
- Establishing a probability distribution based on (reliable!) data for the reference class. The challenge here is to get good data for a sufficient number of reference class projects.
- Predicting most likely outcomes for the project of interest based on comparisons with the reference class distribution.
Now, the key assumption in reference class forecasting is that it is possible to identify a number of completed projects that are similar to the one at hand. But what does “similar” mean? Clearly the composition of the reference class depends on the similarity criteria used, and consequently so does the resulting distribution. Reference class forecasting is a victim of the reference class problem!
The reference class problem will affect any technique that uses arbitrary criteria to determine the set of all possible events. As another example, the probability distributions used in Monte Carlo simulations (of project cost, duration or whatever) are determined using historical data. Again, typically one selects projects (or tasks – if one is doing a task level simulation) that are similar to the one at hand. Defining “similar” is left to common sense or expert judgement or some other subjective approach. Yet, by the most commonly used definition, a project is a “temporary endeavor, having a defined beginning and end, undertaken to meet unique goals and objectives”. By definition, therefore, we never do the same project twice – at best we do the same project differently (and the same applies to tasks). So, despite ones best intentions and efforts, historical data can never be totally congruent to the situation at hand. There will always be differences, and one cannot tell with certainty that those differences do not matter.
Truth be told, most organizations do not retain data on completed projects – except superficial stuff that isn’t much use. The reference class problem seems to justify the position of this slack majority. After all, why bother keeping data when one isn’t able to use it to predict project performance. This argument is wrong-headed: although one cannot use it to calculate probabilities, historical data is useful because it keeps us from repeating our errors. Just don’t expect the data to yield reliable quantitative information on probabilities.
Before I close this piece, I should clarify that there are areas in which the reference class problem is not an issue. In physics, for example, the discipline of statistical mechanics is founded on the principle that the average motion of large collections of molecules can be treated statistically. Clearly, there is no problem here: molecules are indeed indistinguishable from each other, so it is clear that a particular molecule (of a gas in a container of carbon dioxide, say) belongs to the reference class of all carbon dioxide molecules in that container. In general this is true of any situation where one is dealing with a large collection of identical (or very similar) entities.
The reference class problem affects most probabilistic methods in project management and other areas of the social sciences. It is a problem because it is often impossible to know beforehand which attributes of the objects or events of interest are the most significant ones. Consequently it is impossible to determine with certainty whether or not a particular object or event belongs to a defined reference class.
I’ll end with an anecdote to illustrate my point:
Some time ago I was asked to provide estimates for design work that was superficially similar to something I’d done before. “You’ve done this before,” a manager said, “so you should be able to estimate this quite accurately.”
As many best practices and methodologies recommend, I used a mix of historical data and “expert” judgement (and added in a dash of padding) to arrive at (what I thought was) a robust estimate. To all you agilists out there, an incremental approach was not an option in this case.
I got it wrong – badly wrong. It turned out that the unique features of the project, which weren’t apparent at first, made a mockery of my estimates. I didn’t know it then, but I’d fallen victim to the reference class problem.
Finally, it should be clear that although my examples are project management focused, the arguments are quite general. They apply to all areas of management theory and practice, and indeed to most areas of inquiry that use probabilistic techniques. To use the words of Alan Hajek: the reference class problem is your problem too.
Project tasks can have a variety of dependencies. The most commonly encountered ones are task scheduling dependencies such as finish-to-start and start-to-start relationships which are available in many scheduling tools. However, other kinds of dependencies are possible too. For example, it can happen that the durations of two tasks are correlated in such a way that if one task takes longer or shorter than average, then so does the other. [Note: In statistics such a relationship between two quantities is called a positive correlation and an inverse relationship is termed a negative correlation]. In the absence of detailed knowledge of the relationship, one can model such duration dependencies through statistical correlation coefficients. In my previous post, I showed – via Monte Carlo simulations – that the uncertainty in the duration of a project increases if project task durations are positively correlated (the increase in uncertainty being relative to the uncorrelated case). At first sight this is counter-intuitive, even paradoxical. Knowing that tasks are correlated essentially amounts to more knowledge about the tasks as compared to the uncorrelated case. More knowledge should equate to less uncertainty, so one would expect the uncertainty to decrease compared to the uncorrelated case. This post discusses the paradox and its resolution using the example presented in the previous post.
I’ll begin with a brief recapitulation of the main points of the previous post and then discuss the paradox in some detail.
The example and the paradox
The “project” that I simulated consisted of two identical, triangularly distributed tasks performed sequentially. The triangular distribution for each of the tasks had the following parameters: minimum, most likely and maximum durations of 2, 4 and 8 days respectively. Simulations were carried out for two cases:
- No correlation between the two tasks.
- A correlation coefficient of 0.79 between the two tasks.
The simulations yielded probability distributions for overall completion times for the two cases. I then calculated the standard deviation for both distributions. The standard deviation is a measure of the “spread” or uncertainty represented by a distribution. The standard deviation for the correlated case turned out to be more than 30% larger than that for the uncorrelated case (2.33 and 1.77 days respectively), indicating that the probability distribution for the correlated case has a much wider spread than that for the uncorrelated case. The difference in spread can be seen quite clearly in figure 5 of my previous post, which depicts the frequency histograms for the two simulations (the frequency histograms are essentially proportional to the probability distribution). Note that the averages for the two cases are 9.34 and 9.32 days – statistically identical, as we might expect, because the tasks are identically distributed.
Why is the uncertainty (as measured by the standard deviation of the distribution) greater in the correlated case?
Here’s a brief explanation why. In the uncorrelated case, the outcome of the first task has no bearing on the outcome of the second. So if the first task takes longer than the average time (or more precisely, median time), the second one would have an even chance of finishing before the average time of the distribution. There is, therefore, a good chance in the uncorrelated case that overruns (underruns) in the first task will be cancelled out by underruns (overruns) in the second. This is essentially why the combined distribution for the uncorrelated case is more symmetric than that of the correlated case (see figure 5 of the previous post). In the correlated case, however, if the first task takes longer than the median time, chances are that the second task will take longer than the median too (with a similar argument holding for shorter-than-median times). The second task thus has an effect of amplifying the outcome of the first task. This effect becomes more pronounced as we move towards the extremes of the distribution, thus making extreme outcomes more likely than in the uncorrelated case. This has the effect of broadening the combined probability distribution – and hence the larger standard deviation.
Now, although the above explanation is technically correct, the sense that something’s not quite right remains: how can it be that knowing more about the tasks that make up a project results in increased overall uncertainty?
Resolving the paradox
The key to resolving the paradox lies in looking at the situation after task A has completed but B is yet to start. Let’s look at this in some detail.
Consider the uncorrelated case first. The two tasks are independent, so after A completes, we still know nothing more about the possible duration of B other than that it is triangularly distributed with min, max and most likely times of 2, 4 and 8 days. In the correlated case, however, the duration of B tracks the duration of A – that is, if A takes a long (or short) time then so will B. So, after A has completed, we have a pretty good idea of how long B will take. Our knowledge of the correlation works to reduce the uncertainty in B – but only after A is done.
One can also frame the argument in terms of conditional probability.
In the uncorrelated case, the probability distribution of B – let’s call it p(B) – is independent of A. So the conditional probability of B given that A has already finished (often denoted as P(B|A)) is identical to P(B). That is, there is no change in our knowledge of B after A has completed. Remember that we know p(B) – it is a triangular distribution with min, max and most likely completion times of 2, 4 and 8 days respectively. In the correlated case, however, P(B|A) is not the same as P(B) – the knowledge that A has completed has a huge bearing on the distribution of B. Even if one does not know the conditional distribution of B, one can say with some certainty that outcomes close to the duration of A are very likely, and outcomes substantially different from A are highly unlikely. The degree of “unlikeliness” – and the consequent shape of the distribution – depends on the value of the correlation coefficient.
So we see that, on the one hand, positive correlations between tasks increase uncertainty in the overall duration of the two tasks. This happens because a wider range of outcomes are possible when the tasks are correlated. On the other hand knowledge of the correlation can also reduce uncertainty – but only after one of the correlated tasks is done. There is no paradox here, its all a question of where we are on the project timeline.
Of course, one can argue that the paradox is an artefact of the assumption that the two tasks remain triangularly distributed in the correlated case. It is far from obvious that this assumption is correct, and it is hard to validate in the real world. That said, I should add that most commercially available simulation tools treat correlations in much the same way as I have done in my previous post – see this article from the @Risk knowledge base, for example.
In the end, though, even if the paradox is only an artefact of modelling and has no real world application, it is still a good pedagogic example of how probability distributions can combine to give counter-intuitive results.
Thanks to Vlado Bokan for several interesting conversations relating to this paradox.